Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.39.0-wmf.22 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk Gadget Gadget talk Gadget definition Gadget definition talk 4 28 2410810 2409861 2022-08-01T19:23:42Z Sophivorus 85690 /* TemplateScripts = Templates + JavaScript */ Reply wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Warning templates == We do not have any warning templates here at Wikiversity yet. Why not import them? [[User:Lightbluerain|Lightbluerain]] ([[User talk:Lightbluerain|discuss]] • [[Special:Contributions/Lightbluerain|contribs]]) 17:40, 13 May 2022 (UTC) :@[[User:Lightbluerain|Lightbluerain]]: I'm not sure which warnings you are looking for, but see [[Wikiversity:Import]] to add you request(s). Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:38, 13 May 2022 (UTC) :We do have a couple of templates in [[:Category:User warning templates]]. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 21:40, 17 June 2022 (UTC) == Next steps on the Universal Code of Conduct (UCoC) Enforcement guidelines == Hey all - I have an update on the [[m:Special:MyLanguage/Universal Code of Conduct/Project|Universal Code of Conduct (UCoC) project]]. [[m:Special:MyLanguage/Universal Code of Conduct/Enforcement guidelines/Voting/Report|'''A report is available on Meta-Wiki''']]. about the [[m:Special:MyLanguage/Universal Code of Conduct/Enforcement guidelines/Vote|2022 March ratification vote]] on the [[m:Special:MyLanguage/Universal Code of Conduct/Enforcement guidelines|UCoC Enforcement guidelines]]. Voters cast votes from at least 137 communities. At least 650 participants added comments with their vote. ''([[m:Special:MyLanguage/Universal Code of Conduct/Enforcement guidelines/Voting/Report/Announcement|See full announcement]])'' Following the vote, the [[m:Special:MyLanguage/Wikimedia Foundation Community Affairs Committee|Community Affairs committee (CAC)]] of the Wikimedia Foundation Board of Trustees [https://lists.wikimedia.org/hyperkitty/list/wikimedia-l@lists.wikimedia.org/thread/JAYQN3NYKCHQHONMUONYTI6WRKZFQNSC/ asked that several areas be reviewed for improvements]. A [[m:Special:MyLanguage/Universal Code of Conduct/Drafting committee#Revisions Committee|Revision Drafting Committee]] will refine the enforcement guidelines based on community feedback. To help the Revisions committee, input from the community is requested. Visit the Meta-wiki pages ([[m:Special:MyLanguage/Universal_Code_of_Conduct/Enforcement_guidelines/Revision_discussions|Enforcement Guidelines revision discussions]], [[m:Special:MyLanguage/Universal_Code_of_Conduct/Policy text/Revision_discussions|Policy text revision discussions]]) to provide thoughts for the new drafting committee. ''([[m:Universal Code of Conduct/Enforcement guidelines/Revision discussions/Announcement|See full announcement]])'' Let me know if you have any questions about these next steps. [[User:Xeno (WMF)|Xeno (WMF)]] ([[User talk:Xeno (WMF)|discuss]] • [[Special:Contributions/Xeno (WMF)|contribs]]) 02:23, 2 June 2022 (UTC) == Chris Tolworthy == A blogger by the name of [https://answersanswers.com/index.html Chris Tolworthy] has written a lot of essays about the Bible and other subjects. Personally, I think that while some of these essays are pretty much just drivel, others might be worth including on Wikiversity. However, I don't know what kind of license Tolworthy uses on his work- the main page just says, "If you find these pages useful please share and copy them." How that would be expressed in the form of a Creative Commons license, I'm not entirely sure. Also, if we did include Tolworthy's works, he has [https://www.facebook.com/HeyLookThatsMe/ a Facebook account], which would satisfy criteria no. 3 on the [[Help:Essay]] page. What do you think? --[[User:Lizardcreator|Lizardcreator]] ([[User talk:Lizardcreator|discuss]] • [[Special:Contributions/Lizardcreator|contribs]]) 23:34, 15 June 2022 (UTC) :@[[User:Lizardcreator|Lizardcreator]] You are welcome to link to these resources and create learning projects around them. You cannot copy them and host them at Wikiversity. Only Chris Tolworthy can do that. Any resource that doesn't explicitly include a Creative Commons or other open license isn't licensed for sharing on Wikiversity. [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:57, 16 June 2022 (UTC) == Desktop Improvements update == [[File:Table of contents shown on English Wikipedia 02.webm|thumb]] ; Making this the new default Hello. I wanted to give you an update about the [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements|Desktop Improvements]] project, which the Wikimedia Foundation Web team has been working on for the past few years. Our work is almost finished! 🎉 We would love to see these improvements become the default for readers and editors across all wikis. <span style="background-color:#fc3;">In the coming weeks, we will begin conversations on more wikis, including yours. 🗓️</span> We will gladly read your suggestions! The goals of the project are to make the interface more welcoming and comfortable for readers and useful for advanced users. The project consists of a series of feature improvements which make it easier to read and learn, navigate within the page, search, switch between languages, use article tabs and the user menu, and more. The improvements are already visible by default for readers and editors on more than 30 wikis, including Wikipedias in [[:fr:|French]], [[:pt:|Portuguese]], and [[:fa:|Persian]]. The changes apply to the [{{fullurl:{{FULLPAGENAMEE}}|useskin=vector}} Vector] skin only, although it will always be possible to revert to the previous version on an individual basis. [{{fullurl:{{FULLPAGENAMEE}}|useskin=monobook}} Monobook] or [{{fullurl:{{FULLPAGENAMEE}}|useskin=timeless}} Timeless] users will not notice any changes. ; The newest features * [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements/Features/Table of contents|Table of contents]] - our version is easier to reach, gain context of the page, and navigate throughout the page without needing to scroll. It is currently tested across our pilot wikis. It is also available for editors who have opted into the Vector 2022 skin. * [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements/Features/Page tools|Page tools]] - now, there are two types of links in the sidebar. There are actions and tools for individual pages (like [[Special:RecentChangesLinked|Related changes]]) and links of the wiki-wide nature (like [[Special:RecentChanges|Recent changes]]). We are going to separate these into two intuitive menus. ; How to enable/disable the improvements [[File:Desktop Improvements - how to enable globally.png|thumb|[[Special:GlobalPreferences#mw-prefsection-rendering|{{int:globalpreferences}}]]]] * It is possible to opt-in individually [[Special:Preferences#mw-prefsection-rendering|in the appearance tab within the preferences]] by selecting "{{int:skinname-vector-2022}}". Also, it is possible to opt-in on all wikis using the [[Special:GlobalPreferences#mw-prefsection-rendering|global preferences]]. * On wikis where the changes are visible by default for all, logged-in users can always opt-out to the Legacy Vector. There is an easily accessible link in the sidebar of the new Vector. ; Learn more and join our events If you would like to follow the progress of our project, you can [[mw:Special:Newsletter/28/subscribe|subscribe to our newsletter]]. You can read the [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements|pages of the project]], check [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements/Frequently_asked_questions|our FAQ]], write on the [[mw:Talk:Reading/Web/Desktop_Improvements|project talk page]], and [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|join an online meeting with us]]. Thank you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|talk]]) 16:59, 21 June 2022 (UTC) <!-- Message sent by User:SGrabarczuk (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:SGrabarczuk_(WMF)/sandbox/MM/En_fallback&oldid=23430301 --> == TemplateScripts = Templates + JavaScript == Hi! I'd like to propose enabling [[c:Help:TemplateScripts|TemplateScripts]] on the English Wikiversity. It's not a MediaWiki extension, but a few lines of JavaScript added to [[MediaWiki:Common.js]] that basically allow to run JavaScript from templates, '''as long as the code is on the MediaWiki namespace and with the "TemplateScript-" prefix''', which requires an authorized user and community consensus to get there. The system is enabled on the Spanish Wikipedia where it's used for easy signing of polls and projects (see blue button [[:es:Wikiproyecto:Veganismo/participantes|here]]), for navigating [[Template:Excerpt#Excerpt trees|excerpt trees]] (see box with tree icon [[:es:Discusión:Ciencia|here]]), for injecting interactive widgets on some articles ([[:es:Hormiga de Langton|here]] and [[:es:Juego de la vida|here]]) and more recently for creating interactive forms that inject content into other pages (see template [[:es:Plantilla:Formulario|here]], soon to be used on admin boards). My immediate goal on Wikiversity is to use it to develop a tool to make [[Wikidebate|wikidebates]] more friendly. However I believe some of the existing scripts, particularly the ones for creating forms and signing pages, can be very useful on Wikiversity overall, as well as in some specific projects like [[Automata theory]] and [[Conway's Game of Life]]. So what do you think? [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|talk]]) 21:12, 29 June 2022 (UTC) :Hmm well, there's no support, but no objections either. If no one objects in a week or two, I may implement this since I got the necessary permission (and years of involvement in Wikiversity and other Wikimedia projects, I dare say). Cheers! [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 19:23, 1 August 2022 (UTC) == Results of Wiki Loves Folklore 2022 is out! == <div lang="en" dir="ltr" class="mw-content-ltr"> {{int:please-translate}} [[File:Wiki Loves Folklore Logo.svg|right|150px|frameless]] Hi, Greetings The winners for '''[[c:Commons:Wiki Loves Folklore 2022|Wiki Loves Folklore 2022]]''' is announced! We are happy to share with you winning images for this year's edition. This year saw over 8,584 images represented on commons in over 92 countries. Kindly see images '''[[:c:Commons:Wiki Loves Folklore 2022/Winners|here]]''' Our profound gratitude to all the people who participated and organized local contests and photo walks for this project. We hope to have you contribute to the campaign next year. '''Thank you,''' '''Wiki Loves Folklore International Team''' --[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 16:12, 4 July 2022 (UTC) </div> <!-- Message sent by User:Tiven2240@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=23454230 --> == Wikiversity == What can i do with wikiversity --[[User:Goku Sakaki|Goku Sakaki]] ([[User talk:Goku Sakaki|discuss]] • [[Special:Contributions/Goku Sakaki|contribs]]) 16:58, 10 July 2022 (UTC) :@[[User:Goku Sakaki|Goku Sakaki]] Welcome! Start with [[What is Wikiversity?]]. Then look around and see what interests you. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 01:06, 11 July 2022 (UTC) == How to graphically design a park == How do I design a park {{unsigned|Darelle Meyer}} :That would be a pretty involved process and I don't think that Wikiversity or our sister project [[:b:en:|Wikibooks]] has a resource on that yet. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:36, 11 July 2022 (UTC) == Wikimedia Foundation Board of Trustees Election: Propose statements for the 2022 Election Compass == <section begin="announcement-content" /> :''[[m:Special:MyLanguage/Wikimedia Foundation elections/2022/Announcement/Propose statements for the 2022 Election Compass| You can find this message translated into additional languages on Meta-wiki.]]'' :''<div class="plainlinks">[[m:Special:MyLanguage/Wikimedia Foundation elections/2022/Announcement/Propose statements for the 2022 Election Compass|{{int:interlanguage-link-mul}}]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-{{urlencode:Wikimedia Foundation elections/2022/Announcement/Propose statements for the 2022 Election Compass}}&language=&action=page&filter= {{int:please-translate}}]</div>'' Hi all, Community members in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2022|2022 Board of Trustees election]] are invited to '''[[m:Special:MyLanguage/Wikimedia_Foundation_elections/2022/Community_Voting/Election_Compass|propose statements to use in the Election Compass.]]''' An Election Compass is a tool to help voters select the candidates that best align with their beliefs and views. The community members will propose statements for the candidates to answer using a Lickert scale (agree/neutral/disagree). The candidates’ answers to the statements will be loaded into the Election Compass tool. Voters will use the tool by entering in their answer to the statements (agree/disagree/neutral). The results will show the candidates that best align with the voter’s beliefs and views. {{collapse|heading=Timeline for the Election Compass|content= July 8 - 20: Community members propose statements for the Election Compass July 21 - 22: Elections Committee reviews statements for clarity and removes off-topic statements July 23 - August 1: Volunteers vote on the statements August 2 - 4: Elections Committee selects the top 15 statements August 5 - 12: candidates align themselves with the statements August 15: The Election Compass opens for voters to use to help guide their voting decision }} The Elections Committee will select the top 15 statements at the beginning of August. The Elections Committee will oversee the process, supported by the Movement Strategy and Governance team. MSG will check that the questions are clear, there are no duplicates, no typos, and so on. Best, Movement Strategy and Governance ''This message was sent on behalf of the Board Selection Task Force and the Elections Committee''<br /><section end="announcement-content" /> [[User:Xeno (WMF)|Xeno (WMF)]] ([[User talk:Xeno (WMF)|discuss]] • [[Special:Contributions/Xeno (WMF)|contribs]]) 17:00, 11 July 2022 (UTC) == Movement Strategy and Governance News – Issue 7 == <section begin="msg-newsletter"/> <div style = "line-height: 1.2"> <span style="font-size:200%;">'''Movement Strategy and Governance News'''</span><br> <span style="font-size:120%; color:#404040;">'''Issue 7, July–⁠September 2022'''</span><span style="font-size:120%; float:right;">[[m:Special:MyLanguage/Movement Strategy and Governance/Newsletter/7|'''Read the full newsletter''']]</span> ---- Welcome to the 7th issue of Movement Strategy and Governance News! The newsletter distributes relevant news and events about the implementation of Wikimedia's [[:m:Special:MyLanguage/Movement Strategy/Initiatives|Movement Strategy recommendations]], other relevant topics regarding Movement governance, as well as different projects and activities supported by the Movement Strategy and Governance (MSG) team of the Wikimedia Foundation. The MSG Newsletter is delivered quarterly, while the more frequent [[:m:Special:MyLanguage/Movement Strategy/Updates|Movement Strategy Weekly]] will be delivered weekly. Please remember to subscribe [[m:Special:MyLanguage/Global message delivery/Targets/MSG Newsletter Subscription|here]] if you would like to receive future issues of this newsletter. </div><div style="margin-top:3px; padding:10px 10px 10px 20px; background:#fffff; border:2px solid #808080; border-radius:4px; font-size:100%;"> * '''Movement sustainability''': Wikimedia Foundation's annual sustainability report has been published. ([[:m:Special:MyLanguage/Movement Strategy and Governance/Newsletter/7#A1|continue reading]]) * '''Improving user experience''': recent improvements on the desktop interface for Wikimedia projects. ([[:m:Special:MyLanguage/Movement Strategy and Governance/Newsletter/7#A2|continue reading]]) * '''Safety and inclusion''': updates on the revision process of the Universal Code of Conduct Enforcement Guidelines. ([[:m:Special:MyLanguage/Movement Strategy and Governance/Newsletter/7#A3|continue reading]]) * '''Equity in decisionmaking''': reports from Hubs pilots conversations, recent progress from the Movement Charter Drafting Committee, and a new white paper for futures of participation in the Wikimedia movement. ([[:m:Special:MyLanguage/Movement Strategy and Governance/Newsletter/7#A4|continue reading]]) * '''Stakeholders coordination''': launch of a helpdesk for Affiliates and volunteer communities working on content partnership. ([[:m:Special:MyLanguage/Movement Strategy and Governance/Newsletter/7#A5|continue reading]]) * '''Leadership development''': updates on leadership projects by Wikimedia movement organizers in Brazil and Cape Verde. ([[:m:Special:MyLanguage/Movement Strategy and Governance/Newsletter/7#A6|continue reading]]) * '''Internal knowledge management''': launch of a new portal for technical documentation and community resources. ([[:m:Special:MyLanguage/Movement Strategy and Governance/Newsletter/7#A7|continue reading]]) * '''Innovate in free knowledge''': high-quality audiovisual resources for scientific experiments and a new toolkit to record oral transcripts. ([[:m:Special:MyLanguage/Movement Strategy and Governance/Newsletter/7#A8|continue reading]]) * '''Evaluate, iterate, and adapt''': results from the Equity Landscape project pilot ([[:m:Special:MyLanguage/Movement Strategy and Governance/Newsletter/7#A9|continue reading]]) * '''Other news and updates''': a new forum to discuss Movement Strategy implementation, upcoming Wikimedia Foundation Board of Trustees election, a new podcast to discuss Movement Strategy, and change of personnel for the Foundation's Movement Strategy and Governance team. ([[:m:Special:MyLanguage/Movement Strategy and Governance/Newsletter/7#A10|continue reading]]) </div><section end="msg-newsletter"/> [[User:Xeno (WMF)|Xeno (WMF)]] ([[User talk:Xeno (WMF)|discuss]] • [[Special:Contributions/Xeno (WMF)|contribs]]) 00:29, 17 July 2022 (UTC) == Master/doctoral theses publication here == Hello, I am a somewhat long-time Wikipedian but a novice in Wikiversity. I have a few of questions: #Does Wikiversity allow for the publication here of masters and doctoral theses? ##If the answer is yes, can an editor do so on behalf of the author, if the author has granted the editor permission? #If an academic paper has been published in a peer-reviewed journal that is open access, can the paper be re-published in Wikiversity? #Does Wikiversity allow for the publication of research papers written as part of a university course? #Can syllabi and reading lists from graduate courses be published here? (with the permission of the Professors who created them) Thank you. [[User:Al83tito|Al83tito]] ([[User talk:Al83tito|discuss]] • [[Special:Contributions/Al83tito|contribs]]) 04:39, 22 July 2022 (UTC) :@[[User:Al83tito|Al83tito]] Welcome to Wikiversity! The answer to all of these questions is yes. You can find some or many examples of each of these here. The only hesitation would be republishing "open access" papers. Be careful to verify that the open access license it was published under is consistent with CC-BY-SA, and always reference sources. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 12:55, 23 July 2022 (UTC) == Let's talk about the Desktop Improvements == [[File:Vector 2022 showing language menu with a blue menu trigger and blue menu items 01.jpg|thumb]] Join an online meeting with the team working on the [[mw:Reading/Web/Desktop Improvements|Desktop Improvements]]! It will take place on '''26 July 2022 at [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20220726T1200 12:00 UTC] and [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20220726T1900 19:00 UTC]''' on Zoom. '''[https://wikimedia.zoom.us/j/5304280674 Click here to join]'''. Meeting ID: 5304280674. [https://wikimedia.zoom.us/u/kc2hamfYz9 Dial by your location]. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|Read more]]. See you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|talk]]) 16:19, 25 July 2022 (UTC) <!-- Message sent by User:SGrabarczuk (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:SGrabarczuk_(WMF)/sandbox/MM/En_fallback&oldid=23430301 --> == Announcing the six candidates for the 2022 Board of Trustees election == <section begin="announcement-content" /> : ''[[metawiki:Special:MyLanguage/Wikimedia Foundation elections/2022/Announcement/Announcing the six candidates for the 2022 Board of Trustees election| You can find this message translated into additional languages on Meta-wiki.]]<div class="plainlinks">[[metawiki:Special:MyLanguage/Wikimedia Foundation elections/2022/Announcement/Announcing the six candidates for the 2022 Board of Trustees election|{{int:interlanguage-link-mul}}]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-{{urlencode:Wikimedia Foundation elections/2022/Announcement/Announcing the six candidates for the 2022 Board of Trustees election}}&language=&action=page&filter= {{int:please-translate}}]</div>'' Hi everyone, '''The Affiliate voting process has concluded.''' Representatives from each Affiliate organization learned about the candidates by reading candidates’ statements, reviewing candidates’ answers to questions, and considering the candidates’ ratings provided by the Analysis Committee. The selected 2022 Board of Trustees candidates are: * Tobechukwu Precious Friday ([[metawiki:User:Tochiprecious|Tochiprecious]]) * Farah Jack Mustaklem ([[metawiki:User:Fjmustak|Fjmustak]]) * Shani Evenstein Sigalov ([[metawiki:User:Esh77|Esh77]]) * Kunal Mehta ([[metawiki:User:Legoktm|Legoktm]]) * Michał Buczyński ([[metawiki:User:Aegis Maelstrom|Aegis Maelstrom]]) * Mike Peel ([[metawiki:User:Mike Peel|Mike Peel]]) You may see more information about the [[metawiki:Special:MyLanguage/Wikimedia Foundation elections/2022/Results|Results]] and [[metawiki:Special:MyLanguage/Wikimedia Foundation elections/2022/Stats|Statistics]] of this Board election. Please take a moment to appreciate the Affiliate Representatives and Analysis Committee members for taking part in this process and helping to grow the Board of Trustees in capacity and diversity. These hours of volunteer work connect us across understanding and perspective. Thank you for your participation. Thank you to the community members who put themselves forward as candidates for the Board of Trustees. Considering joining the Board of Trustees is no small decision. The time and dedication candidates have shown to this point speaks to their commitment to this movement. Congratulations to those candidates who have been selected. A great amount of appreciation and gratitude for those candidates not selected. Please continue to share your leadership with Wikimedia. Thank you to those who followed the Affiliate process for this Board election. You may review the results of the Affiliate selection process. '''The next part of the Board election process is the community voting period.''' [[metawiki:Special:MyLanguage/Wikimedia Foundation elections/2022#Timeline|You may view the Board election timeline here]]. To prepare for the community voting period, there are several things community members can engage with in the following ways: * [[metawiki:Special:MyLanguage/Wikimedia Foundation elections/2022/Candidates|Read candidates’ statements]] and read the candidates’ answers to the questions posed by the Affiliate Representatives. * [[metawiki:Special:MyLanguage/Wikimedia_Foundation_elections/2022/Community_Voting/Questions_for_Candidates|Propose and select the 6 questions for candidates to answer during their video Q&A]]. * See the [[metawiki:Special:MyLanguage/Wikimedia Foundation elections/2022/Candidates|Analysis Committee’s ratings of candidates on each candidate’s statement]]. * [[metawiki:Special:MyLanguage/Wikimedia Foundation elections/2022/Community Voting/Election Compass|Propose statements for the Election Compass]] voters can use to find which candidates best fit their principles. * Encourage others in your community to take part in the election. Best, Movement Strategy and Governance ''This message was sent on behalf of the Board Selection Task Force and the Elections Committee'' <section end="announcement-content" /> [[User:MNadzikiewicz (WMF)|MNadzikiewicz (WMF)]] ([[User talk:MNadzikiewicz (WMF)|discuss]] • [[Special:Contributions/MNadzikiewicz (WMF)|contribs]]) 20:36, 25 July 2022 (UTC) == Vote for Election Compass Statements == <section begin="announcement-content" /> :''[[m:Special:MyLanguage/Wikimedia Foundation elections/2022/Announcement/Vote for Election Compass Statements| You can find this message translated into additional languages on Meta-wiki.]]'' :''<div class="plainlinks">[[m:Special:MyLanguage/Wikimedia Foundation elections/2022/Announcement/Vote for Election Compass Statements|{{int:interlanguage-link-mul}}]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-{{urlencode:Wikimedia Foundation elections/2022/Announcement/Vote for Election Compass Statements}}&language=&action=page&filter= {{int:please-translate}}]</div>'' Hi all, Volunteers in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2022|2022 Board of Trustees election]] are invited to [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2022/Community_Voting/Election_Compass/Statements|vote for statements to use in the Election Compass]]. You can vote for the statements you would like to see included in the Election Compass on Meta-wiki. An Election Compass is a tool to help voters select the candidates that best align with their beliefs and views. The community members will propose statements for the candidates to answer using a Lickert scale (agree/neutral/disagree). The candidates’ answers to the statements will be loaded into the Election Compass tool. Voters will use the tool by entering in their answer to the statements (agree/disagree/neutral). The results will show the candidates that best align with the voter’s beliefs and views. Here is the timeline for the Election Compass: *<s>July 8 - 20: Volunteers propose statements for the Election Compass</s> *<s>July 21 - 22: Elections Committee reviews statements for clarity and removes off-topic statements</s> *July 23 - August 1: Volunteers vote on the statements *August 2 - 4: Elections Committee selects the top 15 statements *August 5 - 12: candidates align themselves with the statements *August 15: The Election Compass opens for voters to use to help guide their voting decision The Elections Committee will select the top 15 statements at the beginning of August Best, Movement Strategy and Governance ''This message was sent on behalf of the Board Selection Task Force and the Elections Committee'' </div><section end="announcement-content" /> [[User:MNadzikiewicz (WMF)|MNadzikiewicz (WMF)]] ([[User talk:MNadzikiewicz (WMF)|discuss]] • [[Special:Contributions/MNadzikiewicz (WMF)|contribs]]) 21:21, 27 July 2022 (UTC) jz0f6cxjv4gb4lccn3trga1nrwxsp5x 4 1558 2410951 2409002 2022-08-02T11:32:01Z 45.116.22.73 raman koirala is from nepal wikitext text/x-wiki {{Please leave this line alone (sandbox heading)}} <!-- raman koirala --><ref>{{Cite web|url=https://www.facebook.com/raman.player|title=Facebook|website=www.facebook.com|access-date=2022-08-02}}</ref> d4yoc55zbetfn5mzry9btqrhg144juw 2410952 2410951 2022-08-02T11:46:59Z Dave Braunschweig 426084 Reverted edits by [[Special:Contributions/45.116.22.73|45.116.22.73]] ([[User_talk:45.116.22.73|talk]]) to last version by [[User:Dave Braunschweig|Dave Braunschweig]] using [[Wikiversity:Rollback|rollback]] wikitext text/x-wiki {{Please leave this line alone (sandbox heading)}} <!-- Hello! Feel free to try your formatting and editing skills below this line. As this page is for editing experiments, this page will automatically be cleaned periodically. --> copaadgn6cz69177ksi0b8aalofil0h 0 15133 2410956 1712759 2022-08-02T11:51:13Z Dave Braunschweig 426084 Cleanup wikitext text/x-wiki Security engineering is the field of engineering dealing with the security and integrity of real-world systems. It is similar to systems engineering in that its motivation is to make a system meet requirements, but with the added dimension of enforcing a security policy. It has existed as an informal field for centuries, in the fields of locksmithing and security printing. Technological advances, principally in the field of computers, have now allowed the creation of far more complex systems, with new and complex security problems. Because modern systems cut across many areas of human endeavor, security engineers not only need consider the mathematical and physical properties of systems; they also need to consider attacks on the people who use and form parts of those systems using social engineering attacks. Secure systems have to resist not only technical attacks, but also coercion, fraud, and deception by confidence tricksters. For this reason it involves aspects of social science, psychology and economics, as well as physics, chemistry and mathematics. Some of the techniques used, such as fault tree analysis, are derived from safety engineering. Other techniques such as cryptography were previously restricted to military applications. One of the pioneers of security engineering as a formal field of study is Ross Anderson. ==Sub-fields of security engineering== * [[Computer security]] * [[Physical security]] * [[Information security]] * [[Economics of security]] * [[Locksmithing]] == Patterns & Practices == The [http://msdn.com/SecurityEngineering patterns & practices Security Engineering] consists of the following activities: * Security Objectives * Security Design Guidelines * Security Modeling * Security Architecture and Design Review * Security Code Review * Security Testing * Security Tuning * Security Deployment Review These activities help meet security objectives in the software life cycle. == Coursework == [[Category:Computer programming]] [[Category:Engineering]] ps19tixx0wyapohygzi1bckbbe2zo0f 0 36510 2410792 1518100 2022-08-01T15:22:11Z Isivile Nkungu Khenkebe 2947451 /* =Readings */Easy thing wikitext text/x-wiki ==Summary== This object of this class is to teach students about Non-combat formations and their usage ==Introduction== In this class we will learn about the various kinds of Non-combat formations, their benefits and drawbacks, and when they should be used. ==Readings= This will delineate the readings that should be done prior to undertaking the exercises in this class. ==Lecture== ===In route=== ====Column==== ====Staggered Column==== ====Single File==== ===In Camp=== 3-6-9 Triangle ==Project== Apply the lessons learned above to aerial formations, seperate out the non-combat formations, and list them above in a section for aerial. ==Denouement== You should now be familiar with the types of non-combat formations that are relevant to all kinds of units. Please help improve this class. {{Strategic-studies-stub}} [[Category:Strategic Studies]] 6bqw0fmuqbbt4x7pwx5n8rj1q2vnite 2410793 2410792 2022-08-01T15:22:38Z Isivile Nkungu Khenkebe 2947451 /* =In route */Easy thing wikitext text/x-wiki ==Summary== This object of this class is to teach students about Non-combat formations and their usage ==Introduction== In this class we will learn about the various kinds of Non-combat formations, their benefits and drawbacks, and when they should be used. ==Readings= This will delineate the readings that should be done prior to undertaking the exercises in this class. ==Lecture== ===In route== ====Column==== ====Staggered Column==== ====Single File==== ===In Camp=== 3-6-9 Triangle ==Project== Apply the lessons learned above to aerial formations, seperate out the non-combat formations, and list them above in a section for aerial. ==Denouement== You should now be familiar with the types of non-combat formations that are relevant to all kinds of units. Please help improve this class. {{Strategic-studies-stub}} [[Category:Strategic Studies]] nd7wgmvpye7h1m2l2eqse7dh4v14jjz 2410794 2410793 2022-08-01T15:23:04Z Isivile Nkungu Khenkebe 2947451 /* =Column */Easy thing wikitext text/x-wiki ==Summary== This object of this class is to teach students about Non-combat formations and their usage ==Introduction== In this class we will learn about the various kinds of Non-combat formations, their benefits and drawbacks, and when they should be used. ==Readings= This will delineate the readings that should be done prior to undertaking the exercises in this class. ==Lecture== ===In route== ====Column=== ====Staggered Column==== ====Single File==== ===In Camp=== 3-6-9 Triangle ==Project== Apply the lessons learned above to aerial formations, seperate out the non-combat formations, and list them above in a section for aerial. ==Denouement== You should now be familiar with the types of non-combat formations that are relevant to all kinds of units. Please help improve this class. {{Strategic-studies-stub}} [[Category:Strategic Studies]] 55z5aq4edwoz9y6s6qtykfb0arc2tdd 2410961 2410794 2022-08-02T11:55:30Z Dave Braunschweig 426084 Reverted edits by [[Special:Contributions/Isivile Nkungu Khenkebe|Isivile Nkungu Khenkebe]] ([[User_talk:Isivile Nkungu Khenkebe|talk]]) to last version by [[User:MaintenanceBot|MaintenanceBot]] using [[Wikiversity:Rollback|rollback]] wikitext text/x-wiki ==Summary== This object of this class is to teach students about Non-combat formations and their usage ==Introduction== In this class we will learn about the various kinds of Non-combat formations, their benefits and drawbacks, and when they should be used. ==Readings== This will delineate the readings that should be done prior to undertaking the exercises in this class. ==Lecture== ===In route=== ====Column==== ====Staggered Column==== ====Single File==== ===In Camp=== 3-6-9 Triangle ==Project== Apply the lessons learned above to aerial formations, seperate out the non-combat formations, and list them above in a section for aerial. ==Denouement== You should now be familiar with the types of non-combat formations that are relevant to all kinds of units. Please help improve this class. {{Strategic-studies-stub}} [[Category:Strategic Studies]] d1lin734vyj05qy6cxpg6da90n8igrq 0 77105 2410805 2230202 2022-08-01T17:10:22Z 2.45.69.216 wikitext text/x-wiki '''LaTeX''' is a [[markup language]] (as is [[MediaWiki]]!) for producing mathematical texts of the highest quality. Its use is widespread in the mathematics world. It is built on plain [[w:TeX|TeX]] developed by [[W:Donald Knuth|Donald Knuth]]. You can embed LaTeX markups in MediaWiki by the <nowiki><math></math></nowiki> tags, e.g. <kbd><nowiki><math> e = m c^2</math></nowiki></kbd> is rendered as <math>e = m c^2</math>. == Readings == * [[Wikipedia: LaTeX]] * [[Wikibooks: LaTeX]] ==Installation== *If you are using Linux, you probably have it already! Just type "latex". *If you are using Windows, the simplest thing to do is to open [[w:cygwin|Cygwin]] (the linux emulator) and type "latex". If you haven't installed Cygwin, do it! You will often find it useful. The installation may take less than an hour but it is usually straightforward. ** Note that it is now fairly simple to install a [[linux dual boot]], for example, with [[wubi]]. All you need is a number of Gigabytes in a refactored hard-drive. You can delete it like a regular windows directory whenever you don't want it any more. ==Step by step guide for beginners== #Create a file with .tex suffix, e.g. "helloworld.tex" (see [[/helloworld.tex]]). #Go to your linux/cygwin/<ref>If you are new to cygwin, you can find your c-drive by something like <kbd>cd /cygdrive/c</kbd> </ref> whatever shell, go to the directory of your file "helloworld.tex" and type "latex helloworld". #A [[w:dvi|.dvi]] file will be created - you may view it directly if you have the right tools. #If not, type "dvips helloworld" or type "dvipdf helloworld" to convert it to a [[w:postscript|postscript]] or [[w:pdf|PDF]] file. #Use [[w:Ghostview|Ghostview]] to view the .ps file, or the [[w:Adobe Acrobat Reader|Adobe Acrobat Reader]] to view the .pdf file. ==Choosing an editor== A good editing environment helps! A useful and free option is the [[w:LaTeX-Editor_(LEd)|LaTeX editor (LEd)]] ([http://www.latexeditor.org/ see their home page]). A [http://ooolatex.sourceforge.net LaTeX extension] is also available for [[Open Office]]. Or for a quick and easy solution, try [http://rogercortesi.com/eqn/index.php Roger's Online Equation Editor]. == Lessons == * [[/helloworld.tex/]] * [[/Including graphics/]] * [[/Inserting diagrams/]] * [[/Two-lined-subscripts/]] ==Obtain PDF from Latex== pdflatex hello.tex == Use Latex Source in Wikiversity == You can use latex sources also in Wikiversity. In general you will create a PDF file from your latex source. <kbd>pdflatex</kbd> converts the a tex-file <kbd>hello.tex</kbd> in <kbd>hello.pdf</kbd>. You may need sometimes a different output format. If you want to convert the tex-file into a Wiki markdown output for Wikiversity you can use [https://pandoc.org/try PanDoc]. === Pandoc === This is useful if you want to use parts from source text in LaTeX also in a Wikiversity learning resource (e.g. for a mathematical text for a lecture). Copy the source in [https://pandoc.org/try PanDoc] and * select [https://pandoc.org/try "Latex" as input format] and * select [https://pandoc.org/try "MediaWiki" as an output format]. When press [https://pandoc.org/try "convert"] you can use the Wiki source code in your learning resource. === Expand Newcommands === Sometimes it is necessary to [https://niebert.github.io/closingbracket expand latex macros] because the macros in your source are not available in Wikiversity. * With the BracketHandler <kbd>closingbracket.js</kbd> a newcommand replacement into the source text is possible. * Use [https://niebert.github.io/closingbracket Newcommand Expander] as an [[AppLSAC]] for replacing the newcommand ([https://www.github.com/niebert/closingbracket GitHub Repository closingbracket.js]). ==External Links== * [http://tex.loria.fr/general/texbytopic.pdf tex by topic] - an indepth study of TeX * [https://niebert.github.io/closingbracket closingbracket.js BracketHandler] - GitHub demo for newcommand replacement with [https://www.github.com/niebert/closingbracket GitHub-repository 'closingbracket'] == References == <references/> [[category:LaTeX| ]] [[category:mathematics]] [[category:computer science]] [[category:academic publishing]] da08o78x5bbwmf20sbrbut4xus3iqls 2410806 2410805 2022-08-01T17:10:48Z 2.45.69.216 wikitext text/x-wiki '''LaTeX''' is a [[markup language]] (as is [[MediaWiki]]!) for producing mathematical texts of the highest quality. Its use is widespread in the mathematics world. It is built on plain [[w:TeX|TeX]] developed by [[W:Donald Knuth|Donald Knuth]]. You can embed LaTeX markups in MediaWiki by the <nowiki><math></math></nowiki> tags, e.g. <kbd><nowiki><math>e = m c^2</math></nowiki></kbd> is rendered as <math> e = m c^2</math>. == Readings == * [[Wikipedia: LaTeX]] * [[Wikibooks: LaTeX]] ==Installation== *If you are using Linux, you probably have it already! Just type "latex". *If you are using Windows, the simplest thing to do is to open [[w:cygwin|Cygwin]] (the linux emulator) and type "latex". If you haven't installed Cygwin, do it! You will often find it useful. The installation may take less than an hour but it is usually straightforward. ** Note that it is now fairly simple to install a [[linux dual boot]], for example, with [[wubi]]. All you need is a number of Gigabytes in a refactored hard-drive. You can delete it like a regular windows directory whenever you don't want it any more. ==Step by step guide for beginners== #Create a file with .tex suffix, e.g. "helloworld.tex" (see [[/helloworld.tex]]). #Go to your linux/cygwin/<ref>If you are new to cygwin, you can find your c-drive by something like <kbd>cd /cygdrive/c</kbd> </ref> whatever shell, go to the directory of your file "helloworld.tex" and type "latex helloworld". #A [[w:dvi|.dvi]] file will be created - you may view it directly if you have the right tools. #If not, type "dvips helloworld" or type "dvipdf helloworld" to convert it to a [[w:postscript|postscript]] or [[w:pdf|PDF]] file. #Use [[w:Ghostview|Ghostview]] to view the .ps file, or the [[w:Adobe Acrobat Reader|Adobe Acrobat Reader]] to view the .pdf file. ==Choosing an editor== A good editing environment helps! A useful and free option is the [[w:LaTeX-Editor_(LEd)|LaTeX editor (LEd)]] ([http://www.latexeditor.org/ see their home page]). A [http://ooolatex.sourceforge.net LaTeX extension] is also available for [[Open Office]]. Or for a quick and easy solution, try [http://rogercortesi.com/eqn/index.php Roger's Online Equation Editor]. == Lessons == * [[/helloworld.tex/]] * [[/Including graphics/]] * [[/Inserting diagrams/]] * [[/Two-lined-subscripts/]] ==Obtain PDF from Latex== pdflatex hello.tex == Use Latex Source in Wikiversity == You can use latex sources also in Wikiversity. In general you will create a PDF file from your latex source. <kbd>pdflatex</kbd> converts the a tex-file <kbd>hello.tex</kbd> in <kbd>hello.pdf</kbd>. You may need sometimes a different output format. If you want to convert the tex-file into a Wiki markdown output for Wikiversity you can use [https://pandoc.org/try PanDoc]. === Pandoc === This is useful if you want to use parts from source text in LaTeX also in a Wikiversity learning resource (e.g. for a mathematical text for a lecture). Copy the source in [https://pandoc.org/try PanDoc] and * select [https://pandoc.org/try "Latex" as input format] and * select [https://pandoc.org/try "MediaWiki" as an output format]. When press [https://pandoc.org/try "convert"] you can use the Wiki source code in your learning resource. === Expand Newcommands === Sometimes it is necessary to [https://niebert.github.io/closingbracket expand latex macros] because the macros in your source are not available in Wikiversity. * With the BracketHandler <kbd>closingbracket.js</kbd> a newcommand replacement into the source text is possible. * Use [https://niebert.github.io/closingbracket Newcommand Expander] as an [[AppLSAC]] for replacing the newcommand ([https://www.github.com/niebert/closingbracket GitHub Repository closingbracket.js]). ==External Links== * [http://tex.loria.fr/general/texbytopic.pdf tex by topic] - an indepth study of TeX * [https://niebert.github.io/closingbracket closingbracket.js BracketHandler] - GitHub demo for newcommand replacement with [https://www.github.com/niebert/closingbracket GitHub-repository 'closingbracket'] == References == <references/> [[category:LaTeX| ]] [[category:mathematics]] [[category:computer science]] [[category:academic publishing]] rb5y0c2333xddogn5haduzri1wk8agx 2 95744 2410816 2392598 2022-08-01T20:01:12Z CommonsDelinker 9184 Replacing James_R_Ballantyne.jpg with [[File:James_Ballantyne_(1808-1877).jpg]] (by [[:c:User:CommonsDelinker|CommonsDelinker]] because: [[:c:COM:FR|File renamed]]:). wikitext text/x-wiki ==Meta-information== If you happen upon this weblog sandbox and are interested in corresponding with or befriending me, please refer [https://sites.google.com/site/b9jokerswild108/ Internet Cyber Register] for appropriate channels and forums, particularly my preferred email address in small print. It may take me in excess of 20 years to finish this work but by the grace of that which is divine and divinely pragmatic, nothing in this world will obstruct me come what may: friendlessness, absence of family, lovelessness, homelessness, loss of health and wellbeing, poverty. The code will be published in an appropriate open, free, accessible and reputable forum for peer review in time. I have not sought to plan an overt structure as this engenders insidious bias but a natural structure has and will continue to grow organically as the project projects. I am hopeful that a clear, intelligible and intuitive structure(s) and/or themes will gracefully present themselves. :May these loci as gossamer threads of germane discourse ensue as seed-beads caressed with heartful love as benevolent and sensate action on a threaded rosary of sublime and auspicious living! Until that hallowed point, this endgame will continue as an embedded discourse, ie. a mess and a muddle but no less informative, instructive and indeed edifying for that. Blessings B9hummingbirdhovering (aka Beauford Anton Stenberg). ==Maṇībhadantānkuśa: Bejewelled Ankush of Ivory; or alternatively: Nonduality, Transpersonal Psychology, Subjectivity and the Human Condition: an embedded narrative== '''''Maṇībhadantānkuśa: Bejewelled Ankush of Ivory; or alternatively: Nonduality, Transpersonal Psychology, Subjectivity and the Human Condition: an embedded narrative''''' is the working title of this engaged, reflective investigation. I define terms as I value precision and I also understand that ambiguity has its usage and place. My endeavour at precision is not to close the openness of what I write, but to assist the comprehension and thereby the reception of this overview. This work is twofold, it is a cursory exploration of "nonduality" in the wisdom traditions of the World's peoples and I have chosen selectively and with purpose. Incidentally, it will identify a global development that is not real in any true sense as is just the construction of the works of a number of academic specialists such as historians, anthropologists, linguists and archeologists. That said, the development is real and there is no contradiction. This work is also the fruit of my reflection of human spiritual thought and praxis and my experience of that value. ==Ankusha== A note on the Sanskrit title, '''''Bejewelled Ankush of Ivory: Maṇībhadantānkuśa'''''. Full Sanskrit title: Maṇībhadantānkuśa: Śrīmaharatnaikacittatvasaṃtānasvasaṃvedanaṃ; English: ''The Bejewelled Ankusha of Ivory: The Great Precious Glory; The Continuum of the Apperceptive Reflexivity of the Inclusive Nature of the Heartmind''. Sanskrit written works, like Tibetan works following them, often have many names: one often contains a metaphor or analogy and another that conveys its meaning. Maṇībhadantānkuśa: Śrīmaharatnaikacittatvasaṃtānasvasaṃvedanaṃ. In the artifice of giving this English work a Sanskrit title I am following an ancient lineage and positing this work within that lineage. The Bejewelled Ankusha of Ivory: The Great Precious Glory; The Continuum of the Appercetive Reflexivity of the Inclusive Nature of the Heartmind. Citta essentially means both "heart" and "mind" or the most essential, heartful, cordial and pure aspect of our consciousness and being. [http://books.google.com.au/books?id=KtbIXgOGr5AC&dq=heartmind+dzogchen&source=gbs_navlinks_s Heartmind] is a gloss of Bodhicitta principally from English renderings of Zen works, but the rendering has been adopted by other Buddhadharma traditions, particularly some teachers of Dzogchen and Mahamudra. I like its simplicity and its accessibility and subsequently, it is my favourite rendering for the "pure and perfect mind" (citta) of "awakening" (bodhi). Citta in the title is affixed with "eka" as "total", "one", "unitary and contracted with "tattva" or "principle", "nature", "essence" or "truth" so the "Inclusive Nature of the Heartmind". "Śrī" may be understood as the iconographic "[[w:glory|glory]]" that is the halo and [[w:Aureola|aura]] often depicted in art in many traditions as well as what is felt in truly good people in whom this heartmind or however else it is understood or codified in the various precious traditions of the World is manifest when we are open to the experience. Sometimes this holiness transforms people who are not even consciously ready for the experience. "Samtana" or continuum is both the Mindstream of which I have written about on Wikipedia as well as the the "flux" of all materiality, not even diamonds are forever as they are not timeless. I claim many of these traditions as mine simply as I am a Human in the current fulcrum of global culture, but I hold that in the system and development of human spirituality in a fashion many of these traditions have impacted on each other specifically in identifiable ways as well as indirectly in ways both known and unknown. This is what I term Systems Theology: a theology informed by [[w:Systems Theory|Systems Theory]] and [[w:Cybernetics|Cybernetics]]. Which is curious considering that Systems Theory and Cybernetics were informed by the Buddhadharma teaching of ''[[w:Pratītyasamutpāda|Pratītyasamutpāda]]'' of [[w:Gautama Buddha|Śākyamuni]] (fl c.400 BCE).(Beauford the citation for this may be in the Permaculture Bible.)<ref>Macy, Joanna (1991). ''Mutual causality in Buddhism and general systems theory: the dharma of natural systems''. SUNY series in Buddhist studies. SUNY Press. {{ISBN|0791406377}}, 9780791406373</ref> The essence of the teaching understood as Pratītyasamutpāda isn't peculiar just to the Buddhadharma. The Doctrine of Flux has been associated with [[w:Heroclitus|Heroclitus]] (c. 535–c. 475 BCE)) by [[w:Plato|Plato]] (428/427 – 348/347 BCE) and [[w:Aristotle|Aristotle]] (384 – 322 BCE) as affirmed by Craig (1998: p.694).<ref>Craig, Edward (1998). ''Routledge Encyclopedia of Philosophy: Index, Volume 10''. Taylor & Francis. {{ISBN|0415073103}}, 9780415073103, p.694</ref> It is written with no other purpose than this is of interest and of value to me and though I entertain that very few would find it of value, it is written as an offering to my Human family. It makes me feel that Humanity and I as a part of that class of sentient beings have a purpose even if my experience of my present generation and times is one of inanity, selfishness, cruelty and isolation from true intimacy and love with ourselves, each other and our environment and manifold living beings. This is understood as terror. This terror is not new and I have no delusions of a once-upon-a-time Golden Age yet I have a real appreciation of how significantly the Human cultures are progressing in terms of their spirituality, inclusion and heartfulness which is being brought into line with material prosperity which is yet to be extended to all. I have no particular interest in esotericism and secret knowledge in truth. I have no particular interest in the exclusivity of such traditions even though many such traditions have been areas of my protracted study, contemplation and meditation. "Inclusive" is a practical, pragmatic and ideological inclusion. The living world is both beautiful and terrible, often simultaneously. Humans partaking of this beauty and terror are in my experience more terrible than beautiful. Beauty is understood as heartfulness. "Nature" may be understood as at once the fullness of our potential a well as actuality. "Reflexive apperception" will be dealt with throughout the work if the reader is aware and attentive of their presence in the experience of reading this work as well as specifically in due course and it is implied in the title by the material of the ankusha being "ivory" and in "svasamvedanam". The "ankusha", the "elephant goad" or "elephant hook" is chosen for personal, historical, meditative, inclusive and iconographic purpose and edification. As a practical tool of human endeavour it has roots in the "goad" of Egyptian iconography. Though it may be pereived as a tool of subjugation and domination this has both appropriate and inappropriate applications and employment. It is a metaphor for directing focus, attention and endeavour. ===Nonduality=== "Nondualism", "nonduality" and "nondual" are revisionist terms that have entered the English language from literal English renderings of "advaita" (Sanskrit: nondual) subsequent to the first wave of English translations of the [[w:Upanishads|Upanishads]] commencing with the work of [[w:Max Müller|Müller]] (1823 – 1900), in the monumental ''[[w:Sacred Books of the East|Sacred Books of the East]]'' (1879), who rendered "advaita" as "[[w:Monism|Monism]]" under influence of the then prevailing discourse of English translations of the Classical Tradition of the [[w:Ancient Greek|Ancient Greek]]s such as [[w:Thales|Thales]] (624 – c. 546 BCE) and [[w:Heraclitus|Heraclitus]] (c. 535–c. 475 BCE). The first usage of the terms are yet to be attested. The English term "nondual" was also informed by early translations of the Upanishads in Western languages other than English from 1775. The term "nondualism" and the term "advaita" from which it originates are [[w:polyvalent|polyvalent]] terms. The English word's origin is the Latin ''duo'' meaning "two" prefixed with "non-" meaning "not". Wiktionary (May 2010) ventures the etymology and offers a definition of "[[:wikt:Nondualism|Nondualism]]" thus: *Etymology: derived from the translation of the [[w:Sanskrit|Sanskrit]] term [[:wikt:अद्बैत|अद्बैत]] '[[:wikt:advaita|advaita]]', "not two". *Noun: The belief that [[:wikt:dualism|dualism]] or [[:wikt:dichotomy|dichotomy]] are [[:wikt:illusory|illusory]] [[:wikt:phenomena|phenomena]]; that things such as mind and body may remain distinct while not actually being separate. ===Transpersonal Psychology=== Caplan (2009: p.231) conveys the genesis of the discipline of [[w:Transpersonal psychology|Transpersonal Psychology]], states its mandate and ventures a definition: <blockquote>"Although transpersonal psychology is relatively new as a formal discipline, beginning with the publication of ''The Journal of Transpersonal Psychology'' in 1969 and the founding of the Association for Transpersonal Psychology in 1971, it draws upon ancient mystical knowledge that comes from multiple traditions. Transpersonal psychologists attempt to integrate timeless wisdom with modern Western psychology and translate spiritual principles into scientifically grounded, contemporary language. Transpersonal psychology addresses the full spectrum of human psychospiritual development -- from our deepest wounds and needs, to the existential crisis of the human being, to the most transcendent capacities of our consciousness."<ref>Caplan, Mariana (2009). ''Eyes Wide Open: Cultivating Discernment on the Spiritual Path''. Sounds True. {{ISBN|1591797322}}, 9781591797326. Source: [http://books.google.com.au/books?id=cDX8zNEbUXoC&pg=PA232&dq=transpersonal+psychology+nonduality&hl=en&ei=eDHgS7nnA4ugkQWj6snCCA&sa=X&oi=book_result&ct=result&resnum=10&ved=0CGcQ6AEwCQ#v=onepage&q=transpersonal%20psychology%20nonduality&f=false] (accessed: Wednesday May 5, 2010), p.231</ref></blockquote> ===Subjectivity=== Subjectivity as collapse of subject-object into a continuum of perceiver-perceived is a key theme in many nondual traditions as it is in contemporary perceptual theory. Subjectivity qua [[w:Consciousness Studies|Consciousness]] is the conundrum and delimitation of the Human Condition. <blockquote> There is no God, as I was taught in youth, <br /> Though each, according to his stature, builds <br /> Some covered shrine for what he thinks the truth...<br /> There is no God, but we, who breathe the air, <br /> Are God ourselves and touch God everywhere.<br /> <br /> An extract from ''Sonnets'' (1915) of [[w:John Masefield|Masefield]]<ref>Source: [http://www.sonnets.org/masefield.htm] (accessed: Monday May 17, 2010)</ref> (1878-1967) and I entreat my reader to please be gender inclusive rather than exclusive whilst reading it. There be Theism in Pantheism be under no illusions! </blockquote> *[https://dspace.hampshire.edu:8443/dspace/bitstream/10009/182/1/Division+III+Thesis---We+Are+The+Imagination+of+Ourselves.pdf Foster, Shane T. (2008). We Are The Imagination of Ourselves ~ The Principle of Lila in Advaita Vedanta Metaphysics: A Playful, Creative, and Storytelling Reality]<br /> ===Human Condition=== :"Sweet are the uses of adversity, which like the toad, ugly and venomous, wears yet a precious jewel in his head; and this our life, exempt from public haunt, finds tongues in trees, books in the running brooks, sermons in stones, and good in everything." :Shakespeare -- [[s:As_You_Like_It/Act_II|As You Like It (2.1.13)]] The term '[[w:Human condition|Human condition]]' has been employed at least since 1933 and I have as yet been unable to establish the first attested usage. Like "nondual" and its inflections, it may not as yet be documented. We each know what constitutes the human condition but often culture conceals it from us. I will state it simply here. We are an individual, no-one is identical not even an identical twin. We are in general socialized within a society of individuals, some with similarities some with dissimilarities. Many groups or communities are identified and recognized due to shared similarities eg. family, location, ethnicity, trade, hoby and culture. There are good people, bad people and people of complex alignments and agenda. There are different cultures and different polities, different political systems, different judicatures and legislatures, different laws, different countries, different cultural practices, different religions and faith systems, different values and mores. As a result there are sometimes war, conflict and strife between individuals, between communities, between individuals and communities, between other species, etc. We need water, food and air to sustain life. We tend to live in a dynamic continuum of competition-cooperation. We are embedded in complex systems organic and inorganic. Many of us establish relationships with different species. We live on Earth in the Milky Way. Depending on our locality and culture we need certain kinds of clothes and shelter. For the most part we establish interpersonal relationships. We learn unique knowledge and experiences that further individuate our individuality so that no two people have the same knowledge, skill and experience set. There is the additional texture and natural drive of sexuality and the creative impulse. We use, create and modify and innovate tools and technology. We each undergo the trial of disease and the surety of death. Generally, we have the complexity of emotions, the ability of rationality, the propensity to learn. Curiously, human memory and experience is and are designed to fade. We as a class of organism have a fractal ordering: five sections (four limbs and head) that for the most part branch into five sections (fingers and toes). We as a species tend to perceive our experience in similar ways but there are marked divergences and varietal differences. There are various rites of passage, many with celebrations and festivities. In sum, the human condition is a complex. ===Embedded narrative=== The classic model of the embedded narrative is the [[w:One Thousand and One Nights|Arabian Nights]] wherein Scheherazade narrates stories embedding narratives to enthrall a King and thereby forestall her death. Scheherazade as the principal narrative voice, conveys narratives in complexities sometimes to the order of eight embedded narrative levels. Though the different narrative orders are discreet they bleed into one another and reflect and refract one another through various [[w:Literary technique|literary device]]s. The usage of "embedded narrative" in this investigation of nonduality partakes of this as each thematic section is an embedded narrative but the denotation is more purposeful in this context. I employ the term with its broad usage within critical theory: every [[w:Text (literary theory)|text]] is embedded in a narrative as it too holds embedded narratives. It is the discourse of [[w:deixis|deixes]], that a text as a technical term in [[w:Critical_theory|critical theory]] (eg. picture, song, book, house, etc., indeed anything which conveys meaning) always points within itself, out of itself and other texts point to it by their very nature as texts: intertextuality as an interpermeablility. Any spiritual tradition worth its salt must grapple with these themes of beauty and terror. It is my understanding that the most profound, practical and beautiful wisdom traditions of the World are increasingly understood as "nondual" traditions. To live in this world we must deliver terror as a daily matter of course as plants and animals are living. If we truly need to deliver terror be heartful in so doing, this is honourable pragmatism. ===Ideology and discourse=== :There are more things in Heauen and Earth, Horatio, :Then are dream't of in our Philosophy. :~ [http://www.gutenberg.org/ebooks/2265 Hamlet Act 1. Scene V, Shakespeare] '[[w:Open discourse|Open Discourse]]' is a [[w:Technical terminology|technical term]] employed in [[w:discourse analysis|discourse analysis]] and [[w:Sociolinguistics|Sociolinguistics]] which is contrasted with 'Closed Discourse'. The concept of open and closed discourse is associated with the overlay of open and closed [[w:discourse community|discourse communities]] and open and closed communication events. Key to open and closed discourse is [[w:Information access|access to information]], [[w:Equity theory|equity]] of access, [[w:Open access (publishing)|open access]], quality of discourse and mechanisms and modalities of discourse control: overt, covert, implicit and incidental. As a conceptual filter and cultural construct, [[w:ideology|ideology]] is a function and mechanism of discourse control. [[w:Channel (communications)|Channel]] and [[w:Signal (electronics)|signal]] of communication event and [[w:Register (sociolinguistics)|register]] of communication control discourse and therefore, determine degree of [[w:social inclusion|social inclusion]] and [[w:social exclusion|social exclusion]] and therefore, efficiency of communication event. Open and closed discourse operate on a continuum where absolute closure and complete openness are theoretically untenable due to [[w:Communication noise|noise]] in the channel. Nature of channel, signal, code, replicability, recording, transmissibility, cataloguing, recall or other variable of a communication event and its information control and context of transmission-as-event, impacts on its entrance into open discourse; where open discourse is sustained discourse. [[w:Teun A. van Dijk|Van Dijk]] (c.2003: p.&nbsp;357) holds that: <blockquote>"Although most discourse control is contextual or global, even local details of meaning, form, or style may be controlled, e.g. the details of an answer in class or court, or choice of lexical items or jargen in courtrooms, classrooms or newsrooms (Martin Rojo 1994).<ref>Van Dijk, Teun A. (c2003). "Critical Discourse Analysis." Cited in: Schiffrin, Deborah; Tannen, Deborah; Hamilton, Heidi Ehernberger (2003). ''The handbook of discourse analysis''. Volume 18 of Blackwell handbooks in linguistics. Wiley-Blackwell. {{ISBN|0631205969}}, 9780631205968. Source: [http://books.google.com.au/books?id=1vtHhlqzg5kC&pg=RA1-PA357&lpg=RA1-PA357&dq=pubic+discourse&source=bl&ots=hiZSJ_Dzg6&sig=3KibKn-moR0fd_9n8o3kHFO6-pw&hl=en&ei=wIfCS9TvI8uGkAXN4vC5BQ&sa=X&oi=book_result&ct=result&resnum=11&ved=0CDEQ6AEwCg#v=onepage&q=pubic%20discourse&f=false] (accessed: Monday April 12, 2010), p.357</ref></blockquote> Ideology as discourse control is very important for awareness in the human as it constitutes a conceptual and perceptual filter of experience. ==Acknowledgements== [[File:DetalleNivelDeBurbuja.jpg|thumb|left|A "fact" is a bubble in a spirit level.]] This exploration of nonduality as [[w:critical discourse analysis|critical discourse analysis]] is founded upon the [[w:Nondualism|Nondualism]] article at Wikipedia, an article significantly improved by me (circa May 2010). But there are differences of opinion about the quality of my edits and contributions not to that article in particular but whether I have actually improved Wikipedia at all. After being linked with cults-in-the-pejorative, a baseless assertion mind you, I named the demon of my somesay-peers "bland stupidity" as was my experience of the them "the Mob" in question. By grace, I am not constrained in this Wikimedia Project so I will continue here presently. This work as declared above, was founded upon the work of others and their contributions may be ascertained by mining the [http://en.wikipedia.org/w/index.php?title=Nondualism&action=history History Tab] at the appropriate page cited. I thank them for their contributions. I have left Wikipedia for the time being due to a Mob hiding behind Wikipedia's noble ideal of consensus. Unfortunately, consensus is flawed for very similar reasons as the charge made by classical and contemporary critiques of Democracy. In general, an individual cannot avail against an unfavourable Mob. Might and numbers are not necessarily right. Hence, I will continue my discourse here. This is excellent, as here I may critique my sources with my own voice and be more creative than I may be by the conventions of an Encyclopedia. It is not as immediately discoverable as Wikipedia, but Wikiversity is thoroughly indexed in time. Now discourse necessitates a [[w:dialectic|dialectic]] or a debate, whether this discourse happens as a verbal or textual dialogue or otherwise is irrespective. I also wish to associate the dialectic of [[w:scientific method|scientific method]]: [[w:hypothesis|hypothesis]], [[w:antithesis|antithesis]], [[w:synthesis|synthesis]]. Where the synthesis as product in turn becomes a hypothesis and the refinement is ever-set in motion. A girlfriend of mine advised her Mother said a fact is a bubble in a [[w:spirit level|spirit-level]] and this metaphor conveys a continuity of refining. These are all analogues of nonduality. Synthesis as monism is just nonduality from a different perspective. <blockquote>"This historical pattern - famously redeployed by Mark - involved the idea of historical progression by contraries. A first condition - hypothesis - begets its opposite, or antithesis: a reaction against the first condition. From these emerge a third: a synthesis of the first two. As this synthesis fulfils itself it begins to show the germ of its own opposite, and the pattern is repeated, slightly differently. Hegel himself envisaged this chain-sequence as being ultimately circular."<ref>Source: [http://books.google.com.au/books?id=7Ee8AAAAIAAJ&pg=PA8&lpg=PA8&dq=hypothesis+antithesis+synthesis&source=bl&ots=TysLJxV7uO&sig=fNv8dOsnSQ6GQhZKwXyvAhRvUOQ&hl=en&ei=iLP_S42TL82LkAXi073GDQ&sa=X&oi=book_result&ct=result&resnum=10&ved=0CEQQ6AEwCQ#v=onepage&q=hypothesis%20antithesis%20synthesis&f=false] (accessed: Friday May 28, 2010), p.8</ref></blockquote> [[File:NautilusCutawayLogarithmicSpiral.jpg|thumb|Nautilus shell]] It most definitely is circular when viewed in two dimensions but is more appropriate when the model is transported into three dimensions and become spirallic such as the [[w:nautilus shell|nautilus shell]] and an unfurling fern frond, which both follow the [[w:Fibonacci sequence|Fibonacci sequence]] famous in sacred numerical lore. And this both introduces and necessitates a discussion on [[w:Rose Windows|Rose Windows]] and [[w:Sacred Geometry|Sacred Geometry]]. ==Introduction== ===Invocation=== {{center top}}''"Truth like water takes the form of the vessel within which it is housed..."''{{center bottom}} [[Image:Saraswati with Vitarka Mudra.JPG|180px|thumb|centre|{{center top}}[[w:Sarasvati|Srimati Sarasvati]], personification of the [[w:Sarasvati River|Sarasvati River]] and goddess of the Arts and Learning{{center bottom}}]] Fauteux (1993: p.1) discussed the iterating dialogue of [[w:Freud|Freud]] and [[w:Romain Rolland|Rolland]] that yielded what became known as the 'oceanic experience'<ref>Beauford, secure article: [http://www.ncbi.nlm.nih.gov/pubmed/2423575 Oceanic Experience]</ref>: <blockquote> Some people will take offense at the suggestion that religious experience is a return to primitive psychological processes. Others will say it is obvious. Their differing views can be traced back in this century to the debate between Sigmund Freud and the French philosopher Romain Rolland.<ref>Fauteux, Kevin (1993). ''The Recovery of Self: Regression and Redemption in Religious Experience''. Paulist Press. {{ISBN|0809134233}}. Source: [http://books.google.com.au/books?id=CFgADAsU9hQC&pg=PA1&lpg=PA1&dq=oceanic+experience&source=bl&ots=9KXD4V6ndB&sig=cTQT40_x9252StO3KiVBcF41WrU&hl=en&ei=EY7kSbzGIJeIkQWejrTSCw&sa=X&oi=book_result&ct=result&resnum=8] (accessed: Tuesday April 14, 2009), p.1</ref> </blockquote> Albrecht (2007: p.51) holds: <blockquote>"At the ''basal region'' of your brain, your spinal cord enlarges to form the ''medulla oblongata'', and above it a bulbous structure called the ''pons'', two structures that regulate and control the most primitive aspects of life: breathing, heartbeat, arousal, and primary motor control. This portion of the system is sometimes called the ''brainstem'', considered by scientists to be the most ancient part of the brain, evolutionarily speaking. We share this primary type structure with reptiles, birds, and probably with the dinosaurs."<ref>Albrecht, Karl (2007). ''Practical Intelligence: The Art and Science of Common Sense''. Edition: illustrated. John Wiley and Sons. ((ISBN|0787995657}} Source: [http://books.google.com.au/books?id=-LdYnBY_zwAC&pg=PA51&lpg=PA51&dq=Medulla+Oblongata+reptillian+brain&source=bl&ots=wNDBhWNGhS&sig=UNqxNr8eQ6IJSBRTt2wnwH_2XLk&hl=en&ei=ekLlSevNG8GOkAWf3NHZCw&sa=X&oi=book_result&ct=result&resnum=8#PPA51,M1] (accessed: Wednesday April 15, 2009), p.51 </ref></blockquote> ===Dharmic Traditions: [[w:Ekayāna|Ekayāna]] of Dharma=== Seeing the truth beyond the words of the following quotation through the employ of [[w:essence-function|essence-function]] and the pragmatism of the Buddhadharma, the concepts of 'God' and 'soul', etcetera, may be provisionally engaged as ''[[w:upaya|upaya]]''. [[w:Evelyn Underhill|Underhill]] (1911: unpaginated) in her Preface to the Twelfth Edition of her seminal ''Mysticism''<ref>Underhill, Evelyn (1911). ''Mysticism: A Study in Nature and Development of Spiritual Consciousness''. Twelth Edition. </ref>, drawing on the work of [[w:Friedrich von Hügel|Von Hügel]], conveys the "freedom and originality", (indulging in a tangential aside, both technical terms in Dzogchen that will be explored later in this text) that arises from the [[w:interpenetration|interpenetration]] of 'institutional authority' and 'mystical authority' of direct experience, states: <blockquote> First, that while mysticism is an essential element in full human religion, it can never be the whole content of such religion. It requires to be embodied in some degree in history, dogma and institutions if it is to reach the sense-conditioned human mind. Secondly, that the antithesis between the religions of “authority” and of “spirit,” the “Church” and the “mystic,” is false. Each requires the other. The “exclusive” mystic, who condemns all outward forms and rejects the support of the religious complex, is an abnormality. He inevitably tends towards pantheism, and seldom exhibits in its richness the Unitive Life. It is the “inclusive” mystic, whose freedom and originality are fed but not hampered by the spiritual tradition within which he appears, who accepts the incarnational status of the human spirit, and can “find the inward in the outward as well as the inward in the inward,” who shows us in their fullness and beauty the life-giving possibilities of the soul transfigured in God.<ref>Source: [http://www.ccel.org/ccel/underhill/mysticism.txt] (accessed: Tuesday April 14, 2009)</ref> </blockquote> In declaration at the outset I have a strong sense of God qua the Ground-of-Being as an interpenetration of Personalism ([[w:Saguna brahman|Saguna Brahman]]; [[w:Samboghakaya|Samboghakaya]]) and Impersonalism ([[w:Nirguna Brahman|Nirguna Brahman]]; [[w:Dharmakaya|Dharmakaya]]). I do not consider one of these to be ascendant or primary. This does not confound my understanding of [[w:theurgy|theurgy]] and my working with archetypes and tutelary deities as the imaginary friends of childhood writ large for spiritually developmental adulthood. To the untrained and uninformed sensibility, the wrathful yidam of Buddhism and Bon sadhana appear as demons and monsters, this quotation of [[w:Nietzsche|Nietzsche]]'s informs the interpenetrating reciprocity of [[w:Iṣṭha-deva(tā) (Buddhism)|iṣṭhadevatā]] and [[w:sadhaka|sadhaka]], and the coalescence of the two kinds of [[w:Śūnyatā|Śūnyatā]]: <blockquote> He who fights with monsters should be careful lest he thereby become a monster. And if thou gaze long enough into an abyss, the abyss will also gaze into thee. </blockquote> ==Kipling's 'The King's Ankus'== [[Image:SecondJungleBookCover1895.jpg|thumb|180px|Embossed cover from the original 1895 MacMillan edition of ''The Second Jungle Book''. Illustrations based on sketches by J. Lockwood Kipling (father of Rudyard Kipling).]] As a child my exploration of the Dharma may have commenced with the [[w:Aṅkuśa|Ankus]]. This may be a romantic notion, and the romance or [[w:Rasa (aesthetics)|rasa]] of '[[w:Triune brain|reptilian-brain]]' (technically the [[w:Basal ganglia|Basal ganglia]]) [[w:endocrine|endocrine]] feelings and sentiment or electrico-chemical sediment or instinctive hormonal-pulse in the [[w:bodymind|bodymind]] system of the human [[w:mindstream|mindstream]] are important in the [[W:Indian religions|Dharmic Traditions]] but more on that later; but the Ankus is the first iconic cultural artifact from the Dharmic Traditions that I remember, remembering. The Dharmic Traditions comprise the Sanatana Dharma, Sikha Dharma, Buddha Dharma, Jaina Dharma, etcetera. Historically, this is what these traditions were known as in their indigenous tongue where 'tradition' is an English gloss of words in the Sanskrit lexicon such as '[[w:Parampara|parampara]]'. Now this is somewhat lauding the Sanskritic great and learned tradition but many of the other indigenous tongues have worthy 'dharmic' literatures in their own right and rite. The semantic field for 'tradition' in the indigenous Dharmic tongues is and are vast. I prophesy a great frank female French scholar from without the bastion, auspice or ken of the academic ivory-tower in the fabled not too distant future will unpack these for the discerning English reader. In Western academic scholarship the cult of Grecian or Hellenic [[:wikt:-ism#English|-Isms]] has superseded "Dharma" and rewritten them namely as Hinduism, Sikhism, Jainism and Buddhism etcetera which is flawed and unfortunate and betrays that the traditions are fundamentally systemic and related. This is the principal reason why I prefer the rubric "Dharmic Traditions". This will become the standard. Now back to the discourse of the Ankus. I first remember sighting and herewith cited in the movie, ''[[w:Jungle Book (1942 film)|Jungle Book]] (1942)'' originally black and white trailblazing 'merchant ivory' [[w:Film noir|film noir]], later doctored, polychromatically stained, technically technicolored. It is downloadable from The Internet Archive, I need to watch it again and take detailed notes. But then in truth I need do nothing and am not required to do anything as 'knowing' arises naturally and unbidden in my mindstream. At the Positive Living Centre I happened upon the ''[[s:The Second Jungle Book|The Second Jungle Book]]'' of Kipling's in print and by the grace of goddess Serendipity it had this preeminent story enfolded within it. I mention the Positive Living Centre sited in Melbourne Australia as they feed me, an [[w:Avadhuta|Avadhut]]. Feeding an Avadhaut is THE most holy activity that can traditionally be done in the Dharmic Traditions. Importantly, ANY Avadhuta is sacred in ALL the Traditions. An Avadhuta transcends the tradition even whilst they are positioned within it. Now the Positive Living Centre are not formally a part of the Dharmic Traditions but wise Indian women get their children to brush my corporeal form. Ancient traditions are living traditions and are invested in me. I mention them as I honour them and they do good work. Some people consider me egoic and prefer a 'teacher' less learned but more lauded which just offsets their institutionalized stupidity and penchant for the cult of prestige which I identify and defame later. The Avadhaut are fundamentally dejected and neglected and it is from this as much as anything else that is the origin and wellspring of their power. That and the sanctity, trial and burden of being set apart and singled out. The archetype of the [[w:Scapegoat|scapegoat]] of which the Avadhaut partakes has an ancient heritage more significant than the quasi-historical sacrifice of Christ's the 'christos' or 'annointed-one' somesay circa 2000 year odd blood-rite. I could frame this now but it is not required and is easily discoverable. Let us keep to the order of ceremonies and the matters at hand. Many consider Second Jungle Book superior to the first and that the ''[[s:The Second Jungle Book/The King's Ankus|King's Ankus]]'' the jewel of the suite of Jungle Book stories and amongst Kipling's greatest works. I knew it would naturally appear in my lifepath when timely. The amazing thing is, in the movie the Ankus is not ivory, but in the story a part of it is and it is bejewelled with a magnificent ruby and attendant turquoise. I named this exploration the ''Bejewelled Ankusha of Ivory'' before I knew that the original Kipling ankush from the short-story was ivory. That has profound significance for me as my intuition in my journey has been absolute. The timeless quality of the ivory Ankus of this work will be made apparent in time. The Kipling [[s:The_Second_Jungle_Book/The_King%27s_Ankus|story]] gives the appellation "blood drinker" to the Ankus which brought blood blessing of [[w:Heruka|Heruka]] clearly to mind. The mind is well-blooded, let us not forget. The English term 'to bless' is etymologically rooted in both 'sacrifice' and 'blood'. Ancient wisdom is encoded in etymology and semantic fields. The ivory nature of the Ankus in this work is key and that it is in Kipling's and other evocations of this tale betray that he either had a keen appreciation of the traditions of the Dharma or that he was accessing the Akasha or that I am attributing qualities to his work that are not there. The sum may be true, none or some. The ''Introduction'' to the World's Classics edition Oxford University Press (1987) by W. W. Robson states that this story is the best, the most accomplished out of both Jungle Books. For a time Kipling was considered one of the greatest writers of the language by both scholars and the general reading public, well so Robson relates. That the Ankus is ivory powerfully reinforces my appreciation of my intuitive guidance, the 'blessing' (Sanskrit: [[w:adhishthana|adhishthana]]) I had prior, during and subsequent to my entwinement, the pointing out instruction to the primordial nature of mind by the naga and that "we be of one blood ye and I" of all sentient beings and phenomena, lifebood, whether breathing, or living, sentient or falsely considered inanimate. Yes, I am a pantheist, panentheist and animist but not just delimited and compromised by these, but for me stated simply, there is nothing that is not animated by the grace of 'creative intelligence' (Sanskrit: [[w:prakriti|prakriti]]). If this confounds you, Sanskrit and the Dharma just embrace the intention apart from the tradition, imbibe the salience beyond the measured letter of the words. ==An ancient paradigm as the present darling of the Academy'== '''Nondual''' can refer to a belief, condition, theory, practice, or quality. The [[w:academic discipline|academic discipline]]s that study Nondualism in its spiritual permutations and cultural evocations are [[w:transpersonal psychology|Transpersonal Psychology]] and the [[w:anthropology of religion|Anthropology of Religion]] and [[w:Theology|Theology]] amongst others. Though nondualism proper has historically been glossed as "monism" and its variants as "qualified [[w:monism|monism]]" with which it may appropriately or inappropriately be conflated, the nomenclature "nonduality" is now a pervasive paradigm in Western scholarship throughout diverse academic disciplines. Importantly, such paradigms that transcend either/or constructions are pertinent given [[w:Quantum superposition|Quantum superposition]] theory and the models of light as simultaneous wave-particle constructions that we determine through the '''intent''' of our subjectivity. Traditions of monism according to premier contemporary discourse invested in such works as [[w:Thomas McEvilley|McEvilley]] (2002) for example where it should be stated the absence of the term "nondual" and its inflections are conspicuous. Well it has been employed in relation to the Upanishads and Advaita Vedanta but not elsewhere. Maybe, this is to be lauded. That is yet to be determined though discourse analysis. I searched in the parts of the texts that are discoverable in Google Books and you may do so as well. Monism is to be found or even more adroitly, we now perceive its traces in [[w:ancient Egypt|ancient Egypt]], [[w:ancient Persia|ancient Persia]] and [[w:ancient India|ancient India]]; in the Classical traditions of [[w:ancient Greece|ancient Greece]] and [[w:ancient Rome|ancient Rome]]; in several major [[w:world religion|world religious traditions]]; and indigenous traditions such as the [[w:Navajo|Navajo]]; in a number of philosophers such as [[w:Martin Buber|Martin Buber]], [[w:Hans-Georg Gadamer|Hans-Georg Gadamer]]<ref>Rennie, Bryan S. (2001). ''Changing religious worlds: the meaning and end of Mircea Eliade''. SUNY series. SUNY Press. {{ISBN|0791447294}}, 9780791447291. Source: [http://books.google.com.au/books?id=stVC1DH-SAIC&pg=PA103&dq=derrida+nonduality&hl=en&ei=cK_iS62fJ82LkAW1waAU&sa=X&oi=book_result&ct=result&resnum=7&ved=0CFIQ6AEwBg#v=onepage&q=derrida%20nonduality&f=false] (accessed: Thursday May 6, 2010), p.103, note 19</ref> and [[w:Jacques Derrida|Jacques Derrida]]<ref>Rennie, Bryan S. (2001). ''Changing religious worlds: the meaning and end of Mircea Eliade''. SUNY series. SUNY Press. {{ISBN|0791447294}}, 9780791447291. Source: [http://books.google.com.au/books?id=stVC1DH-SAIC&pg=PA103&dq=derrida+nonduality&hl=en&ei=cK_iS62fJ82LkAW1waAU&sa=X&oi=book_result&ct=result&resnum=7&ved=0CFIQ6AEwBg#v=onepage&q=derrida%20nonduality&f=false] (accessed: Thursday May 6, 2010), p.103, note 19</ref>, and various mystics within orthodox and heterodox traditions or arising outside of any tradition, amongst others. Michaelson (2009: p.130) writes: <blockquote>"Conceptions of nonduality evolve historically."<ref>Michaelson, Jay (2009). ''Everything Is God: The Radical Path of Nondual Judaism''. Shambhala Publications. {{ISBN|1590306716}}, 9781590306710. Source: [http://books.google.com.au/books?id=jmrgoeaQvlwC&pg=PA130&dq=i+and+thou+martin+buber+nondual&hl=en&ei=0I3iS_jIFdGgkQX50dHtDw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDYQ6AEwAA#v=onepage&q=i%20and%20thou%20martin%20buber%20nondual&f=false] (accessed: Thursday May 6, 2010), p.130</ref></blockquote> Present "darling" of the "Academy"? So many "spiritual" books, teachers, traditions and texts tout "nondual" of late. I venture it is often ungrounded, unspecified and rarely made clear exactly what is nondual. Academy? Those who can spell and in whom are invested prestige, a prestige attributed. Is such prestige well founded and sound? We each have to determine that for ourselves. Why is the paradigm of nonduality so valuable? Well said simply, it is very difficult with certainty to determine where any entity ends and where any entity is or where any entity begins or even to define what an entity is. Defining anything is problematic in truth. Do you end at your skin? At your range of sight? At your range of hearing? At your range of experience? At your range of knowledge? At your social-interactions? At your last breath? At the last beating of your heart? Where do we begin in the sense of the personal 'entity'? At the union of zygotes? At our first breath? At our first heartbeat? At birth? When your eyes first focus? Contemplate it. Do we begin with the origins of DNA? The origins of the Universe? When contemplating the nature of your perception as you are aware of it be aware of notions such as [[w:qualia|qualia]]? What is the universal if any in an entity such as a dragonfly perceived by a human as different to a dragonfly perceived by a canine as different to a dragonfly perceived by a bird? Unpacking qualia, colour, in the last or final analysis is determined by subject yes? Not by object or atmosphere nor by refraction of light and this is just one example. The subjectivity interpenetrates and informs the perception event, the cognition event of that perception event, and then the attribution of meaning to the said wave of embedded processes. Maybe there are no events but a seamlessness? It is counter-intuitive, but it may be affirmed with certainty *chuckle* that the subject and the object are not distinct. ==A note on sanctity and profanity, a fun profundity== Since early youth I have been a person (and we have already established the problematic notions of persons and entities) who goes into altered states of consciousness with minimal cue. As a child I called it phasing. I went into such a state when I wrote the first version of the ''[[w:Trance|Trance]]'' article on Wikipedia which I created. The content emerged from me spontaneously as the fruit of a lifetime of research and endeavour. I pray that it will be appropriately cited in the years that come. This phasing is not necessarily day-dreaming, there is no fancy. There is just union. One of my favorites as a child was gnat-vision where I would be entranced by the [[w:Darśana|Darśana]] of the pulsing chaotic-order of a sphere-swarm of gnats. I also have been unable to be hypnotized in the "formal" sense though my Mother by birth but not by quality tried and tried. What has that got to do with anything? I am not going to explain everything. '''I affirm with no prejudice and with no fear that I am as holy and as profane as any other human being that has ever graced our world in the full history of this gracious Blue Planet Earth.''' That said, my style of profanity though wanton and sensual has never been evil. Sensuality is a source and wellspring of spiritual power. Indeed, a fun profundity. This comely import will be cumulatively nailed and nailed with an even and decided precision as a matter of course in the relentless ebb and flow of interpenetrative, embedded discourse. There is a folk saying within the International Dzogchen Community that a Dzogchenpa's realization is determined by the number of sexual contacts/partners they have accrued in their life-path. This is amplified with the fecund Nath (where the end phoneme is an aspirated T as there is no "th" sound as in English "path" within the Sanskrit garland of phonemes) and the ancient traditions of the Avadhaut. Let us name the demon of puritanical nay puratyRRRanical tyranny exactly that. Forget what you may have read or have heard, holiness in all its evocations in ancient Mother India has never been the sole domain of the bodies-sexed-male, that is if we qualify the Vedic Agnihotra and Purohit Brahmanical rites, but throughout the manifold diversity for the most part our blessed women and shining third-gender were evident in the field of sacred play. Sanyassin were of all genders and their attributed continence qua celibacy is of recent construction and not universal in the manifold Sanyassin traditions. That said, even in the ancient Indian tradition sanctity was not the sole domain of any particular aspect of the Varnashram Dharma or without. "Nath", as different to the [[w:Sampradaya|Sampradaya]], is simply a Sanskrit term for a "master" of multiple spiritual traditions. Something curious happens when you are a master of multiple traditions. What do you intuit that may be? Many of the ancient Avadhaut texts have not even entered English discourse. Why? Prudish scholars etic and emic and they have been lost to the archives of history. This will change. Avadhaut is the plural for Avadhut. The wonders of nondual praxis! {{center top}} <poem> '''A Swarm Of Gnats''' Many thousand glittering motes Crowd forward greedily together In trembling circles. Extravagantly carousing away For a whole hour rapidly vanishing, They rave, delirious, a shrill whir, Shivering with joy against death. While kingdoms, sunk into ruin, Whose thrones, heavy with gold, instantly scattered Into night and legend, without leaving a trace, Have never known so fierce a dancing. Hermann Hesse as rendered in English from the German by James Wright<ref>Source: [http://www.poemhunter.com/poem/a-swarm-of-gnats/] (accessed: Sunday May 9, 2010)</ref> </poem> {{center bottom}} ==Etymology and an introduction on the fly== I have been unable to source the first attested usage of "nondualism", "nonduality" and "nondual". They may as yet be undocumented. But I hold that they are revisionist terms that have entered the English language from literal English renderings of "advaita" (Sanskrit: nondual) subsequent to the first wave of English translations of the [[w:Upanishads|Upanishads]] commencing with the work of [[w:Max Müller|Müller]] (1823 – 1900), in the monumental ''[[w:Sacred Books of the East|Sacred Books of the East]]'' (1879) which he edited, wherein "advaita" was rendered as "[[w:Monism|Monism]]" under influence of the then prevailing discourse of the Classical Tradition of the [[w:Ancient Greek|Ancient Greek]]s commencing with the "monism" of such as [[w:Thales|Thales]] (624 – c. 546 BCE) and [[w:Heraclitus|Heraclitus]] (c. 535–c. 475 BCE). Muller and many of the other learned and lettered of those times were versed in many languages and Greek and Latin were mandatory to be considered "cultured" and "finished". I have never read any of my forebears state that Indian literature, wisdom and philosophy, its very "memes", entered English discourse by way of the lens of the Classical Tradition and the cultural mores of the Victorian society then prevalent. But this is being stated by me clearly and stated here. Hence, the necessity of revisionism. Whether stated clearly or documented, this has been happening constantly. Knowledges and disciplines constantly interpenetrate and redefine and mutually inform and qualify. This complexity is now happening at a rate neverbeforeseen by our Humankind through facility of the Internet and the dawn of open discourse as accessible discourse. Finally, the manifold discourses are being made available to the general populace most of which are not interested or even aware that they are becoming available. Traditionally, by whom was information mediated? Contemplate that. Contemplate agendas and embedded values. The "Orient" and its perceived cultural tokens were considered wealthy and therefore prestigious. Note, the unholy constructions of the Other which the discourse of nondualism nails as unfounded. I will talk on the discourse of prestige later. I am writing conversationally. I venture it is more intelligible to most of my potential audience to employ such a voice and register. I mention discourse often don't I? I will explain the value of discourse analysis very soon. Knowledge as part of the human condition is always grafted upon knowledge already evident. Study the history and mechanism of human metaphorical construction and my point will be somewhat clear. The human brain is made in such a way, that is in the way that we construct metaphors ~ an ancient foundation upon which have been established annexations. We construct as we are constructed. Curious, yes? My conjecture upon "nondual" and its inflections may or may not be correct, but it is better than any source I have yet encountered. That is saying something. We have to build knowledge somewhere and I have declared the foundations unsure. This entails a very important point, knowledge is never factual. What we call facts are arbitary units of meaning founded upon other units of meaning. In truth facts are very unstable and this is clearly evident when reading a textbook in any discipline of over 100 years ago. The first usage of the terms are yet to be attested. It is highly probable that the English term "nondual" was also informed by early translations of the Upanishads in Western languages other than English from [[w:1775|1775]]. The term "nondualism" and the term "advaita" from which it originates are [[w:polyvalent|polyvalent]] terms. The English word's etymology is the [[w:Latin|Latin]] ''duo'' meaning "two" prefixed with "non-" meaning "not". Language is functional and creative, purposeful. There are no true synonyms in any natural language. If you are unsure what that means contemplate it. I tender there is sound wisdom in that point, sage. "Nondual" is neither especially aesthetic nor creative. That is clearly evident yes? Hence, I declare it to be functional. New terms enter a language for many reasons; for example, technological change and cultural appropriation and necessity, amongst others. New terms are often memes that are valuable such as loan words. I venture why would the English language have even had the construction or necessity for the term "nondual" when there are so many other attested, established, beautiful terms already in the historical lexicon. Neither "nondual" nor its derivatives are available in my Shorter Oxford on Historical Principles; hence, they are new words. I love a sound point of entry. ==What is nondual exactly?== Pritscher (2001: p.16) attributes a salient view on nondual realization to [[w:David Loy|Loy]] (b.1947), an author of a work on comparative philosophy of nondual theologies i.e. Loy (1988)<ref>Loy, David (1988). ''Nonduality: A Study in Comparative Philosophy'', New Haven, Conn: Yale University Press.</ref>: <blockquote>"According to David Loy, when you realize that the nature of your mind and the [U]niverse are nondual, you are enlightened."<ref>Pritscher, Conrad P. (2001). ''Quantum learning beyond duality''. Volume 113 of Value inquiry book series. Rodopi. {{ISBN|9042013877}}, 9789042013872. Source: [http://books.google.com.au/books?id=1-McfrykKkoC&pg=PA17&dq=zen+nondual&hl=en&ei=KeHQS-2tCIqTkAX437SJDA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDkQ6AEwAA#v=onepage&q=zen%20nondual&f=false] (accessed: Friday April 23, 2010), p.16</ref></blockquote> The experience of a human being's outer world, that is the sum and suite of experience perceived 'outside' their body with the five senses informs ones interiority, one's mind. That said, similarly within the imaginal and cognitive realm of mind when we conjure or envision a visionary or imaginal experience with eyes shut and ears sealed to external stimuli, we may still employ all five senses, e.g. we remember or imagine an event that entails the five senses through the faculty of our mind. According to our faculty for example, we hear and perceive the imaginal as a 'real' experience in mind. Our mind doesn't discriminate either between an internal or external event: we salivate the same if we imagine a delicious meal as if we perceive one that is not imaginary. Interiority of thought and exteriority of thought in relation to the projection and reach of the sensory apparatus are conventions that each human being has by and large been calibrating since they first began to walk, see, touch, hear and taste/smell. The calibration becomes a living convention and we forget, never know or denigrate this knowledge and its wisdom and value. A wonderful aside is taste and smell, they are not different senses but by convention in English we construct them as different sensory domains. In reprise, one's interiority consciousness one's subjectivity is a reciprocity with that which is exterior to our imaginal realm, the ontological separation of interior and exterior though counterintuitive is an illusion of the convention of our body-senses-mind calibrated or embodied experience within the bodymind system. Our interiority structures the interpretation of our experience. Our exterior socialization and experience in the world colour our interiority. Our consciousness events are determined by our experience of the world, coloured and flavoured by our cognitive propensity and sensory facility. How similar is the cognitive propensity and sensory facility of the Human? The Mind and memory as a reservoir for objects of experience and reflection upon living embodied experience is our Universe, that is we have bodily calibrated conventions that are mental constructs, formations or ideations, that are qualified and constituted by subtle names and formative patterns. The Sanskritic tradition knows these as 'names-and-form' (namarupa). For the subjective human, our respective Universe is the Mind: in the most broadest denotation Mind and Universe mutually evoke and qualify. The Universe denotes the totality of the external world that may never be experienced in sum, an unknowable and an indefinable. Defining any object in a subjective or cavalier attempt at objective view is fraught with insurmountable obstruction. Defining anything definitively is futile. This is counter-intuitive as well but contemplate it. Where does something begin and end, what is its context what are its processes what is universal about it independent of our experience. An awareness of such complexity and unknowability and undefinability may truly awaken a person who has lost their sense of awe in their lived experience. The world is alive with mystery wherever we look but we close mystery by explaining it away, often very unsuccessfully. Our Universe is our experience and knowledge of it. Therefore, in no uncertain terms Mind and Universe thoroughly interpenetrate. Interpenetrate is a particularly Chinese flavoured nondual language. This interior-exterior imagination-actuality isn't some metaphysical hocus-pocus some wordplay with no value or referent in the world: make no mistake. Is it possible to have an interior thought of something that has not been established from the building blocks of external experience? If I change my understanding of a concept, that change might not necessarily change the materiality the physical fabric of my World or of THE World as such but it changes my response, interpretation, knowledge and value of that materiality and physicality. Due to our subjectivity if we change our perception of our experience, our experience changes necessarily. What we understand as material and physical substance is problematic from the perspective of Human consciousness, as our experience is fundamentally subjective. Our subjectivity is the great qualifier of all our experiences 'in' the World. Where the 'in' really happens in the mind. This nonduality of the perceiver and perceived of that which is being apprehended by the apprehending sentient being is key to many nondual traditions and it is through this nonduality that other nondualities are mediated. Indeed, it may be due to the very nature of the nonduality of perceiver-perceived that all other types of nondualities in our experience of the World are evident. Loy (1988: p.3) contrasts his view of the historicity of nonduality in some of its evocations in the experience of the peoples of [[w:The East|The East]] and [[w:The West|The West]] as follows: <blockquote>"...[the seed of nonduality] however often sown, has never found fertile soil [in the West], because it has been too antithetical to those other vigorous sprouts that have grown into modern science and technology. In the Eastern tradition...we encounter a different situation. There the seeds of seer-seen nonduality not only sprouted but matured into a variety (some might say a jungle) of impressive philosophical species. By no means do all these [Eastern] systems assert the nonduality of subject and object, but it is significant that three which do - Buddhism, Vedanta and Taoism - have probably been the most influential."<ref>Loy, David (1988). ''Nonduality: A Study in Comparative Philosophy'', New Haven, Conn: Yale University Press, p.3</ref></blockquote> Nelson (1951: p.51-52) cites Radhakrishnan's ''[[w:The Principal Upanishads (book)|The Principal Upanishads]]'' (1953) where [[w:Sarvepalli Radhakrishnan|Radhakrishnan]] renders a passage of the ''[[w:Bṛhadāraṇyaka Upaniṣad|Bṛhadāraṇyaka Upaniṣad]]'' (verse 1.4.16) which demonstrates a theme that one becomes [[w:Transpersonal Psychology|transpersonally]] identified with, or nondual to, or develops qualities associated with that to which one is engaged, worships or holds holy and though it is translated with a male pronominal it may be understood as not being gender-specific: <blockquote>"Now this self, verily, is the world of all beings. In so far as he makes offerings and sacrifices, he becomes the world of the gods. In so far as he learns (the Vedas), he becomes the world of the seers. In so far as he offers libations to the fathers and desires offspring, he becomes the world of the fathers. In so far as he gives shelter and food to men, he becomes the world of men. In so far as he gives grass and water to the animals, he becomes the world of animals. In so far as beasts and birds, even to the ants find a living in his houses he becomes their world. Verily, as one wishes non-injury for his own world, so all beings with non-injury for him who has this knowledge. This, indeed, is known and well investigated."<ref>Nelson, Lance E. (1998). ''Purifying the earthly body of God: religion and ecology in Hindu India''. SUNY series in religious studies. SUNY Press. {{ISBN|0791439232}}, 9780791439234. Source: [http://books.google.com.au/books?id=plJspJoTN4EC&pg=PA129&dq=mahanirvana+tantra&lr=&ei=_xngS4N_g4aVBNHdve4L&cd=11#v=onepage&q=mahanirvana%20tantra&f=false] (accessed: Tuesday May 4, 2010)</ref></blockquote> ==Transpersonal psychology== Theriault (2005) in a thesis explores comparative non-dual experience and the psycho-spiritual mechanisms that bring the awareness about.<ref>Theriault, Brian (2005). ''The Non-dual Experience: a Phenomenological Hermeneutic Investigation of the Seeker's Journey Towards Wholeness''. Thesis. University of Lethbridge. Source: [http://www.google.com.au/url?sa=t&source=web&ct=res&cd=5&ved=0CCoQFjAE&url=http%3A%2F%2Fwww.uleth.ca%2Fdspace%2Fbitstream%2F10133%2F290%2F3%2FMR17428.pdf&ei=brPiS4qOOY-gkQWO2dTsDw&usg=AFQjCNElBAh8_CXbheEab69g9FP7KFdOeQ&sig2=vJNCEMK4mF4nHX5dgsrtkg] (accessed: Thursday May 6, 2010)</ref> Lewis (2007) in her thesis explores a number of specific women's experiences on their journey to wholeness and healthfulness in the nondual path of [[w:Tantra|Tantra]] post-sexual trauma and identifies common themes.<ref>Lewis, Lisa (2007). ''Tantric Transformations, a Non-dual Journey From Sexual Trauma to Wholeness: a Phenomenological Hermeneutics Approach''. Thesis. University of Lethbridge. Source: [http://www.google.com.au/url?sa=t&source=web&ct=res&cd=4&ved=0CCMQFjAD&url=http%3A%2F%2Fwww.uleth.ca%2Fdspace%2Fbitstream%2F10133%2F556%2F1%2Flewis%2C%2520lisa.pdf&ei=brPiS4qOOY-gkQWO2dTsDw&usg=AFQjCNEq53oQzva5SUJuf3Z1a7RQqFZUtA&sig2=3ycRlAHORz7aqGGgBg_i3Q] (accessed: Friday March 7, 2010)</ref> ==Nondualism versus monism== The English usage of the term "monism" is first attested in the work of [[w:Christian_Wolff_(philosopher)|Wolff]] (1679 - 1754) as relates Hegeler (2009: p.165): <blockquote>The terms monism and dualism are not yet two centuries old; the former was invented by Christian Wolff as a contrast to- the latter, which, according to Eucken, appears first in Thomas Hyde's book "Historia Religionis Veterum Persarum," 170o, as a designation of Zoroaster's religion. In the same sense, dualism is used by Bayle and Leibnitz. But Wolff applies the term generally to any theory that reduces existence to two independent substances, while monism to him is that doctrine which takes the unity of existence for granted. Wolff rejects monism and classes himself among the dualists.<ref>Hegeler, Edward C. (2009). ''The Monist (V. 4), Volume 4''. General Books: illustrated edition. {{ISBN|1150168250}}, 9781150168253. Source: [http://books.google.com.au/books?id=L06RWfXpwHYC&pg=PA165&dq=Christian+Wolff+monism&hl=en&ei=t8ZbTMnFBpCUuAPskrztAQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDIQ6AEwAQ#v=onepage&q=Christian%20Wolff%20monism&f=false] (accessed: Friday August 6, 2010), p.165</ref></blockquote> So it is useful to note that the construct of the term "monism" was established at its outset in a relationship of mutual exclusion with [[w:Zoroastrianism|Zoroastrianism]] which was in turn understood as an example of "dualism". <blockquote>The Greek language has two words for "one": ''hen'' and ''monos''. ''Hen'' is the numerical "one" that precedes two, three, and so on; ''monos'' means one alone, unique, or the only one, as in the term "monotheism."<ref>Source: [http://books.google.com.au/books?id=AmKE2xIjOwcC&pg=PA33&lpg=PA33&dq=buddhanature+monad&source=bl&ots=R_S2XB2OHe&sig=JQWbxnPeQuAY4Jby_l57clVJhR0&hl=en&ei=X4f1S9GtM5Dc7AP16en6BQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBYQ6AEwAA#v=onepage&q&f=false] (accessed: Friday May 21, 2010)</ref></blockquote> The philosophical concept of [[w:monism|monism]] is similar to nondualism. Indeed, the terms are used as congruent by many scholars. Some forms of monism hold that all [[w:phenomena|phenomena]] are actually of the same [[w:Substance theory|substance]], Substance Monism, where the substance is the substratum. Other forms of monism including Attributive Monism and Idealism are similar and sometimes overlapping concepts to some nondualisms. Nondualism proper holds that different phenomena are inseparable or that there is no hard demarcation separating them in the final analysis, but not that they are the same or identifiable. The distinction between these two types of views is considered critical in [[w:Zen|Zen]], [[w:Madhyamika|Madhyamika]] and [[w:Dzogchen|Dzogchen]], all of which are nondualisms proper. Some later philosophical approaches also attempt to undermine traditional dichotomies, with the view they are fundamentally invalid or inaccurate. For example, one typical form of [[w:deconstruction|deconstruction]] is the critique of binary oppositions within a text while [[w:problematization|problematization]] questions the context or situation in which concepts such as dualisms occur. [[w:Alain Daniélou|Daniélou]] (1907 – 1994) opines that "nondualism" is "dangerous" as it "rests" on "monism": <blockquote>"The term "nondualism" has proved, in many instances, to be a dangerous one, since it can easily be thought to rest on a monistic concept. The Hindu philosophical schools which made an extensive use of this term opened the way for religious monism, which is always linked with a "humanism" that makes of man the center of the universe and of "god" the projection of the human ego into the cosmic sphere. Monism sporadically appears in Hinduism as an attempt to give a theological interpretation to the theory of the substrata.... Nondualism was, however, to remain a conception of philosophers. It never reached the field of common religion."<ref>Daniélou, Alain (1991). ''The myths and gods of India: the classic work on Hindu polytheism from the Princeton Bollingen series''. Inner Traditions / Bear & Company. {{ISBN|0892813547}}, 9780892813544. Source: [http://books.google.com.au/books?id=1HMXN9h6WX0C&pg=PA11&dq=monism+nondualism&hl=en&ei=gEzkS7LOEMyGkAW7_dz3Dw&sa=X&oi=book_result&ct=result&resnum=7&ved=0CFAQ6AEwBg#v=onepage&q=monism%20nondualism&f=false] (accessed: Saturday May 8, 2010), p.11</ref></blockquote> Though I can't speak for Daniélou and say definitively why this is "dangerous". I understand this argument would be the position of absolute [[w:Dualism|Dualists]], for example [[w:Madhvacharya|Madhvacharya]] (1238-1317), who are reputed to have held an inseparable distinction between a supreme creator and the created, and would find monism qua advaita abhorrent. My personal view of duality and nonduality as a nondual system and mutually iterating is informed by [[w:Ramanuja|Ramanuja]] (1017 – 1137) as not only is it the most practical way of being in the World but also is commensurate with my living experience and realization. I hold that pure Advaita is perceiving from the Absolute Truth whereas pure Dualism is perceiving from the Relative Truth. None are right nor wrong if their heart is filled with love as they can abide in Mystery. I don't necessarily embrace all the bells and whistles of any of these traditions as theology does not stay fixed in time. I should also state here that my direct experience of Absolute Truth is that which partakes of [[w:Dipolar theism|Dipolar theism]] and this is the rationale why we all incarnate so that the unmoving is moved through us. My understanding of "Dipolar theism" is that deity is the nonduality of Being-and-Becoming. "Us" is to be understood as any not in the position of the Absolute. That said another way, the manifold iterate the unmanifest; the unmoving enjoys the becoming of the play of diversity. There is a reciprocity between the Two Truths. ===Macranthropic Monism=== McEvilley defines and discusses Greek/Latin Macranthropos and Greek Pantheos as related conceptions. Macranthropos is a contraction of the Greek and Latin prefix "[[:wikt:macro-|macro-]]" meaning "large" and "inclusive" and the Ancient Greek "[[:wikt:anthropo-|anthrōpos]]" or "[[:wikt:ἄνθρωπος|ἄνθρωπος]]" meaning “man, woman, human being”). Pantheos is a contaction of Ancient Greek "[[:wikt:πᾶν|πᾶν]]" or "[[:wikt:pan-|pan]]" meaning "all" and Ancient Greek "[[:wikt:θεός|θεός]]" or "[[:wikt:theos|theos]]" meaning "deity, a god, a goddess, God". [[w:Pan_(god)|Pan]] is also the fecund Ancient Greek deity of Nature and it is from the construction Pantheos that English owes the term [[w:Pantheism|Pantheism]]. McEvilley constructs an adjectival declension of "macranthropos" as "macranthropic" and employs it to qualify monism. The first attested usage I could find on the Internet for a declension of "Macranthropos" was "[[w:Macranthropy|Macranthropy]]" in the revised and enlarged Bollinger edition(1964: p.408), an English rendering by Willard R. Trask of [[w:Mircea Eliade|Eliade]]'s (1951)<ref>Eliade, Mircea (1951). ''Le Chamanisme et les techniques archaiques de l'extase''. Paris: Librairie Payot, Paris.</ref> seminal work from the French and it is contextualized thus: <blockquote>"The homologizations between the human body and the cosmos, of course, go beyond the shamanic experience proper, but we see that both the ''vrātya'' and the ''muni'' acquire macranthropy during an ecstatic trance."<ref>Eliade,Mircea (1964). ''Shamanism: Archaic Techniques of Ecstasy''. Volume 76 of Bollingen series. Bollingen Foundation distributed by Pantheon Books.</ref></blockquote> To extrapolate further, Macranthropos is an inclusive [[w:anthropomorphism|anthropomorphism]] of all-that-is (and the is-not in the sense of space-as-container) as a "Cosmic Person" whereas Pantheos doesn't necessarily though may anthroporphize the totality of all-that-is (and the is-not in the sense of space-as-container), but this totality is understood as a unitive deity that may be intuited as being without human attributes and this abstraction is important. This is-not aspect as a compliment of all-that-is is implied in a conception of totality but is rarely explicitly stated; making this explicit clarifies the possibility of a substantive conception or material bias. This all-that-is & is-not substance as unity is the monad, the very stuff of "Substance Monism" (which includes non-stuff) which may be further abstracted or qualified with attributes according to human conception. In context, both Macranthropos and Pantheos are understood as '''the''' all inclusive deity. Pantheos with its makeup of "theos" denotes that it is worshipful and worthy of veneration which Macranthropos does not necessarily entail in its etymology. It should also be stated that [[w:personification|personification]] doesn't necessarily imply anthropomorphism (but may): it is in this rarefied denotation that Pantheos may be a personification but not an anthropomorphism. Some may start and make exception at this distinction and proffer how can there be personality without human characteristics? This is in part salient but also ignorant of the human capacity for [[w:metaphorical extension|metaphorical extension]]. Can we conceive of a [[w:possible world|possible world]] where personality is an attribute of the non-human? Yes, it is not a significant challenge. Indeed, it is a stock narrative device and ancient storytelling technique employed in [[w:Allegory|Allegory]]. It may be conjectured that Humanity through its humane faculty its empathy does indeed perceive personality or attribute personality to the non-human and this may be a natural psychological attribution if not vain fancy. But this may also be understood as an experience or perception not as attribution; hence taking place prior to conception or the birth or forming of concepts, indeed the subtle cognition of an experience or intuition given form in language or other signification such as iconography by way of the function of metaphorical extension. ===Substance Monism or Material Monism=== Substance Monism stated simply where the type of monism posited is that all that is and is not (which is understood to denote a complete set of possibility and actuality) is fundamentally the same stuff or non-stuff, the substrate. '''[[w:Material monism|Material monism]]''' is a [[w:Presocratic|Presocratic]] belief which provides an explanation of the physical world by saying that all of the world's objects are composed of a single element. Among the material monists were the three Milesian philosophers: Thales, who believed that everything was composed of water; Anaximander, who believed it was ''apeiron''; and Anaximenes, who believed it was air. ====Thales==== [[File:Thales.jpg|thumb|Thales]] '''[[w:Thales|Thales of Miletus]]''' (Ancient Greek: Θαλῆς) (circa 624-546 BCE) was a [[w:Pre-Socratic philosophy|pre-Socratic]] [[w:Greek philosophy|Greek philosopher]] from [[w:Miletus|Miletus]] in [[w:Asia Minor|Asia Minor]], and one of the [[w:Seven Sages of Greece|Seven Sages of Greece]]. Many, most notably [[w:Aristotle|Aristotle]], regard him as the first philosopher of the [[w:Greek philosophy|Classical Hellenic tradition]].<ref>[[w:Aristotle|Aristotle]], ''[[w:Metaphysics_(Aristotle)|Metaphysics Alpha]]'', 983b18.</ref> According to [[w:Bertrand Russell|Bertrand Russell]], "Western philosophy begins with Thales."<ref>Russell, Bertrand. "The History of Western Philosophy". 1945</ref> Thales attempted to explain natural phenomena without recourse to mythology and mythological thinking and was formative in this influence. Almost all of the other pre-Socratic philosophers follow the lead of Thales in this endeavour to proffer an explanation of ultimate substance, change, and the existence of the World -- without recourse to mythology and mythological thinking. Those philosophers were also influential, and eventually Thales' rejection of mythological explanations became an essential tenet for the scientific revolution. Thales was also the first to define general principles and set forth hypotheses, and as a result has been given the nomenclature the "Father of Science".<ref>Singer, C. ''A Short History of Science to the 19th century.'' Streeter Press, 2008. p. 35.</ref><ref>Needham, C.W. ''Cerebral Logic: Solving the Problem of Mind and Brain.'' Loose Leaf, 1978. p. 75.</ref> ====Anaximander==== From Wikipedia and needs to be rewritten: <blockquote>The '''apeiron''' is central to the [[w:cosmology|cosmological]] theory created by [[w:Anaximander|Anaximander]] in the [[w:6th century BC|6th century BC]]. Anaximander's work is mostly lost. From the few existing fragments, we learn that he believed the beginning or first principle (''[[w:arche|arche]]'') is an endless, indefinite mass (''apeiron''), subject to neither old age nor decay, which perpetually yields fresh materials from which everything which we can perceive is derived.<ref>Aristotle, ''Phys.'' Γ5, 204b, 23sq.<DK12,A16.>, Hippolytus, ''Haer.'' I 6, 1 sq. <DK 12 A11,B2.></ref> Apeiron generated the [[w:opposites|opposites]], hot-cold, wet-dry etc., which acted on the creation of the world. Everything is generated from apeiron and then it is destroyed there according to necessity.<ref>Simplicius, ''in Phys.'', p. 24, 13sq.<DK 12 A9,B1.>, p. 150, 24sq.<DK 12 A9.></ref> He believed that infinite worlds are generated from apeiron and then they are destroyed there again.<ref>Aetius I 3,3<Pseudo-Plutarch; DK 12 A14.></ref></blockquote> ===Attributive Monism=== ==Nondualism versus solipsism== :I :Know you appear :Vivid at my side, :Denying you sprang out of my head, :Claiming you feel :Love fiery enough to prove flesh real, :Though it's quite clear :All you beauty, all your wit, is a gift, my dear, :From me. :[http://www.angelfire.com/tn/plath/soliloquy.html ~extract of Sylvia Plath's 'Soliloquy of the Solipsist'] Nondualism superficially resembles [[w:solipsism|solipsism]], but from a nondual perspective solipsism mistakenly fails to consider [[w:subjectivity|subjectivity]] itself. Upon careful examination of the [[w:Philosophy of language#References|referent]] of [[w:I (pronoun)|"I,"]] i.e. one's status as a separate observer of the [[w:Phenomena (philosophy)|perceptual field]], one finds that one must be in as much doubt about ''it'', too, as solipsists are about the existence of [[w:Problem of other minds|other minds]] and the rest of "the external world." (One way to see this is to consider that, due to the conundrum posed by one's own subjectivity becoming a perceptual object to itself, there is no way to validate one's "[[w:Self-concept|self-existence]]" except through the eyes of others—the independent existence of which is already solipsistically suspect!) Nondualism ultimately suggests that the [[w:Philosophy of language#References|referent]] of "I" is in fact an artificial construct (merely the border separating "inner" from "outer," in a sense), the transcendence of which constitutes [[w:Nirvana|enlightenment]]. ==Metaphors for nondualisms== <blockquote>"Buddhism has refined various methods to observe consciousness from the first person perspective for two thousand years. Therefore it is meaningful to bring the explanation models of Tibetan Buddhism into a cross cultural dialogue."<ref>Burchardi, Anne (2007). ''The Self reflexive awareness in Tibetan Buddhist Epistemology: How can the classical Buddhist science of consciousness enter into a cross cultural dialog with the science of cognition? University of Copenhagen.'' Source: [http://www.google.com.au/url?sa=t&source=web&ct=res&cd=2&ved=0CCQQFjAB&url=http%3A%2F%2Fwww.rel.su.se%2FPapers%2FBurchardi.pdf&ei=W_TiS8fxMtCHkQXBrYH_Dw&usg=AFQjCNFlYRGkjjCwcLWvwUINa3rQAIRMUQ&sig2=If8hB68wMxt26Dy9hGWmmA] (accessed: Friday May 7, 2010)</ref></blockquote> *[[w:Indra's net|Jewel Net of Indra]], ''[[w:Avatamsaka Sutra|Avatamsaka Sutra]]'' *The World Text, ''[[w:Avatamsaka Sutra|Avatamsaka Sutra]]'' NB: Northrop Frye *[[w:Blind men and an elephant|Blind men and an elephant]] *[[w:Eclipse|Eclipse]]<ref>Mandair, Arvind (2005). "The Politics of Nonduality: Reassessing the Work of Transcendence in Modern Sikh Theology". ''Journal of the American Academy of Religion''. Volume 74, Number 3, pp.646-673</ref> *[[w:Hermaphrodite|Hermaphrodite]], e.g. [[w:Ardhanārīśvara|Ardhanārīśvara]] *[[w:Mirror|Mirror]] and [[w:reflections|reflections]], as a metaphor for the continuum of the subject-object in the mirror-the-mind and the interiority of perception and its illusion of projected exteriority *[[w:Great Rite|Great Rite]] *[[w:Sacred marriage|Sacred marriage]] *[[w:Marriage|Marriage]] *[[w:Sexual union|Sexual union]], as well as [[w:orgasm|orgasm]] *Water-and-wave, ''[[w:Awakening of Mahayana Faith|Awakening of Mahayana Faith]]'' *Nonduality of rays-of-the-sun or sunrays from the [[w:Sun|Sun]], ''[[w:Lankavatara Sutra|Lankavatara Sutra]]'' *A lamp that self-illuminates as it illumines, for [[w:apperception|apperception]] or reflexive awareness *A lamp and its light, ''[[w:Platform Sutra|Platform Sutra]]'' a metaphor for [[w:Essence-Function|Essence-Function]] where Essence is lamp and Function is light<ref>Lai, Whalen (1979). "Ch'an Metaphors: waves, water, mirror, lamp". ''Philosophy East & West''; Vol. 29, no.3, July, 1979, pp.245-253. Source: [http://www.thezensite.com/ZenEssays/HistoricalZen/ChanMetaphors.htm] (accessed: Saturday May 8, 2010)</ref> ==Simulacra and Simulation== [[w:Jean Baudrillard|Baudrillard]] (1929 – 2007) ''Simulacres et Simulation'' (1985) seminal work in the French was rendered into English by Glaser as ''[[w:Simulacra and Simulation|Simulacra and Simulation]]'' (1996) & an Internet copy may be viewed at Scribd [http://www.scribd.com/doc/36147060/Jean-Baudrillard-Simulacra-Simulation Glaser's translation dated 1996]. It is through his work that Simulacrum takes on a different meaning and especially that used in iconography by Beer (1999) which is important. This is very thematic as it links with representationalism and facticity. Facticity is the unknowable first of which the simulacrum is facsimile without original: that holy grail objective truth that the subjective never accesses, except as fabled by some nondual traditions by 'direct perception' (S: pratyakṣa) through divine grace. It should be stated here that there are different types of direct perception and this will be discussed as this is key to some aspect of nondual experience. I want to mention phantasmagoria and sky-flowers here and the City of Gandharvas and [[w:Tulpa|Tulpa]] and the famous analogies of the Buddhadharma.... Beer (1999: p.11) employs the term '[[w:simulacrum|simulacrum]]' to denote the formation of a sign or iconographic image whether [[w:iconic|iconic]] or [[w:aniconic|aniconic]] in the landscape or greater field of [[w:Thanka|Thanka]] Art and Tantric Buddhist [[w:iconography|iconography]]. For example, an iconographic [[w:Representation (arts)|representation]] of a cloud formation sheltering a deity in a thanka or covering the auspice of a [[w:sacred mountain|sacred mountain]] in the natural environment may be discerned as a simulacrum of an 'auspicious canopy' (Sanskrit: [[w:Chhatra|Chhatra]]) of the [[w:Ashtamangala|Ashtamangala]].<ref>Beer, Robert (1999). ''The Encyclopedia of Tibetan Symbols and Motifs'', (Hardcover). Shambhala Publications. {{ISBN|157062416X}}, {{ISBN|978-1570624162}}, p.11</ref> [[w:Perceptions of religious imagery in natural phenomena|perceptions of religious imagery in natural phenomena]] approaches a [[w:cultural universal|cultural universal]] and may be proffered as evidence of the natural creative spiritual engagement of the experienced environment endemic to the [[w:human psychology|human psychology]]. Shakyamuni employed ten traditional similes in explanation of 'phenomena' (Sanskrit: ''dharmas'') these are known as the "Ten Similes of Illusory Phenomena" (Wylie: ''shes-bya sgyu-ma'i dpe-bcu'') and these inform the import of Baudrillard's "simulacrum": <blockquote> The ten similes which illustrate the illusory nature of all things are: illusion (''sgyu-ma''), mirage (''smig-rgyu''), dream (''rmi-lam''), reflected image (''gzugz-brnyan''), a celestial city (''dzi-za'i grong-khyer''), echo (''brag-ca''), reflection of the moon in water (''chu-zla''), bubble of water (''chu-bur''), optical illusion (''mig-yor''), and an intangible emanation (''sprul-pa'').<ref>Graham Coleman with Thupten Jinpa (editors). ''The Tibetan Book of the Dead'' [''English Title'']. ''The Great Liberation by Hearing in the Intermediate States'' [''Tibetan Title'']. Composed by [[w:Padmasambhava|Padma Sambhava]]. Revealed by [[w:Karma Lingpa|Karma Lingpa]]. Translated by Gyurme Dorje. Penguin Books. 2005. (The first complete translation). {{ISBN|978-0-140-45529-8}}. p.516</ref></blockquote> ===Sumer=== [[File:Ningizzida.jpg|thumb|left|[[w:Ningizzida|Ningishzida]]]] *[http://etcsl.orinst.ox.ac.uk/ The Electronic Text Corpus of Sumerian Literature] *http://www.sacred-texts.com/ane/sum/index.htm ===Enuma Elish=== *i have sourced the cuneiform and the transcription except neither of them are in unicode unfortunately... *http://books.google.com.au/books?id=xSyxDEGcAvcC&lpg=PA4&ots=F1jWKQ9Y2b&dq=enuma%20elish%20tablet%204&pg=PA5#v=onepage&q&f=false ===Egypt=== "What difference is there, then, between God and primigenial chaos?"<br > Umberto Eco, ''The Name of the Rose'' Egyptian iconography and art is mesmerizing isn't it? There is a unique quality to the visual experience of it that is capturing and defining but difficult to pinpoint. Egypt iconographic representationalism is patently different to other forms of human art and cultures and is particularly recognisable in its depiction of the human form, amongst other stock representations such as feathers and papyrus-reeds, etc. In general in the ancient Egyptian visual depiction of the human form there is a multidimentional collapse of perspectives. To contemporary understanding post Renaissance Europe informed by the representations of the human form in the Classical period of Greece and Rome, Ancient Egyptians had a particular sense of perspective of the human form. As a peoples they were magnificent engineers, with feats that persist in baffling our contemporary construction methodologies but a contemporary school child of primary school age with a particular flourish and artistic skill may represent perspective and dimensionality in visual representations with more apparent sophistication that these great ancient engineers. There is a progression of the general ''status quo'' of aptitude of the Human yes? The Egyptians ingeniously represented multidimensionality in a two-dimensional medium. A stepping stone of visual and representationalism problem-solving. Looking at the depictions of Egyptian numinous beings, we see both their shoulders and torso as though from a direct encounter, stated differently, that is a front-on view dimensionally collapsed upon legs and lower body as well as face in dimensional profile. Human perception and ability to represent that visual experience is constantly evolving. Visual representation is a metaphor for the development and iteration of the Human. We are a race in progress. The theology of the Human and their experience of Divinity and that which is Numinous is similarly a work in progress. Nonduality as a theological paradigm has become pervasive thoughout nominally non-theological dimensions. But from a nondual worldview all that is in and of the world ever-qualifies and mutually informs as we as a species refine our appreciation of manifold interconnectivities and subtle relationships of our lived experience. I don't really have any sense of historicity or progression with Ancient Egypt and its relationship with other cultures of the Ancient Near East. I am not an expert in anything in particular and have no special knowledges, this is just a play in the experience of the Human and my reflection upon matters of interest to me. If it is of value to someone, that is a miracle in and of itself. The rationale for evoking the experience of Egypt in the evolution of the experience of the Human and our theology I hope will be intuitively grasped by my audience. I will hint at it henceforth: the primordial waters '[[:wikt:mythologem|mythologem]]' to employ a term introduced into English by [[w:Károly Kerényi|Kerényi]] (1897 – 1973) is salient. This primordial waters motif is the mythological root of human mythological imagination that through the process of abstraction and metaphorical extension was reduced, distilled or refined to the concept of a 'material monism'. [[File:Neith.svg|thumb|left|Neith depicted with Egyptian goad in her right hand]] McEvilley identifies the ''[http://www.sacred-texts.com/pag/ppr/ppr16.htm Hymn of Amun-Ra]'' as the first documented origin of the idea of monism. Now I don't know whether this Hymn of Amon-Ra is at Hibis Temple definitively or even whether it is the first documented example of monism. No, the metalink above is not necessarily the Hymn of Amon-Ra cited by McEvilley. In context, the Hymn to which McEvilley makes reference is cited drawn from [[w:James B. Pritchard|Pritchard]] (1909 – 1997) in ''[[w:Ancient Near Eastern Texts Relating to the Old Testament|ANET]]'' (1955: pp.365-367) which in context appears to be drawn from the great temple of [[w:Karnak|Karnak]] but this is open as the context is ambiguous. There is a large Precinct of Amun-Re at Karnak but it is yet to be determined if the Hymn to which McEvilley makes reference is from Karnak or Khageh Oasis. This will require further direct investigation of ANET. The Egyptian texts in ANET were rendered in English from Hieroglyphics by [[w:John A. Wilson (Egyptologist)|Wilson]] (1899 - 1976) and not by Pritchard, but McEvilley did not give direct credit. Hibis Temple in [[w:Khargeh Oasis|Khargeh Oasis]] preserves the longest monumental hymns to Amun-Re ever carved in hieroglyphs. These religious texts, inscribed during the reign of Darius I, drew upon a large variety of New Kingdom sources, and later they served as sources for the Graeco-Roman hymns at Esna Temple. As such, the hymns to Amun-Re from Hibis are excellently suited for studying Egyptian theology during the Persian Period, on the eve of the supposed "new theology" created by the Graeco-Roman priesthood.<ref>Klotz, David (2006). ''Adoration of the Ram: five hymns to Amun-Re from Hibis temple''. Volume 6 of Yale Egyptological studies. Yale Egyptological Seminar. {{ISBN|0974002526}}, 9780974002521</ref> Why do I feel I need to include Amun-Re? Well Amun may be understood as the archetypal origin of the motif of Nirguna Brahman or Dharmakaya and Ra as the archetypal origin of the motif of Saguna Brahman and Sambhogakaya. This is in the typological and generic sense of first documented auspice for this dichotomy of sacred thought. I am not saying that they directly influenced each other. But for any scholar to argue that pervasion of influence is not a possibility would not only be unsound but untenable. [[File:Heh as chaos.svg|thumb|left|Nun]] The Ancient Egyptians envisaged the oceanic abyss of the "[[w:Nu_(mythology)|Nun]]" ("The Inert One") as enveloping a sphere in which life is encapsulated, representing the deepest mystery of their cosmogony.<ref>McBride, Daniel R. (2003). ''The Oxford Essential Guide to Egyptian Mythology''. Berkley. {{ISBN|0-425-19096-X}} </ref> In Ancient Egyptian creation accounts the original mound of land issues forth from the primordial waters of the Nun.<ref>Silverman, David P. (2003). Ancient Egypt. USA: Oxford University Press. {{ISBN|019521952X}}, p.120</ref> The Nun is the source of all that appears in a differentiated world, encompassing all aspects of divine and earthly existence. In the [[w:Ennead|Ennead]] cosmogony Nun is perceived as transcendent at the point of creation alongside Atum the creator god.<ref>McBride, Daniel R. (2003). ''The Oxford Essential Guide to Egyptian Mythology''. Berkley. {{ISBN|0-425-19096-X}} </ref> In [[w:Egyptian mythology|Egyptian mythology]], '''Nu''' ("Watery One") or '''Nun''' ("The Inert One") is also the deification of the primordial watery abyss. In the [[w:Ogdoad|Ogdoad]] [[w:cosmogony|cosmogony]], the name ''nu'' means "abyss". Nu was shown usually as male but also had aspects that could be represented as female or male. '''Naunet''' (also spelt '''Nunet''') is the female aspect, which is the name ''Nu'' with a female gender ending. The male aspect, '''Nun''', is written with a male gender ending. As with the primordial concepts of the Ogdoad, Nu's male aspect was depicted as a frog, or a frog-headed man. In Ancient Egyptian iconography, Nun also appears as a bearded man, with blue-green skin, representing water. Naunet is represented as a snake or snake-headed woman. The consort of Nun was sometimes understood to be Neith. Neith had a more rich and divergent attribution and iconography than Nun, but also shared in the personification of the primordial waters originating in the [[w:Ogdoad|Ogdoad]] theology, in this capacity Neith like Nun had no gender. *Beauford excursion to secure ANET texts as they are not available on the Internet. ===Akkadian Empire=== [[File:Orientmitja2300aC.png|thumb|A map of c2300 BCE locating Akkad and Egypt visually, unfortunately legend does not identify rationale for colour-usage.]] I didn't even know there was such an empire! It is included as two terms from the Akkadian language were identified by McEvilley as being in the Rig Veda and one of them is regarding "primordial waters". I don't really have any sense of historicity or progression or relationship of the Akkadian Empire with Egypt as yet... but that will come. I don't yet know how the Akkadian terms temporally related to the Egyptian iconography and hieroglyphics and the greater [[w:Religions of the Ancient Near East|Ancient Near Eastern religion's]] archetypal mythos of the "primordial waters". Now this is all just disembodied unless I affirm that on Earth life is understood by Science as originating in cellular form in water so this may be a profound intuition of our ancient kin that we have vindicated. [[w:Genetic memory (psychology)|Ancestral memory]]? [[w:Cellular memory|Cellular memory]]? Cellular memory unlike genetic memory has received rather fanciful treatment in fiction and sensationalist media but is also a useful concept as evidenced through its citation in a recent reputable work on the science of memory (2007).<ref>Henry L. Roediger, Yadin Dudai, Susan M. Fitzpatrick (2007). ''Science of memory: concepts.'' Science of Memory. USA: Oxford University Press. {{ISBN|0195310446}}, 9780195310443, Source: [http://books.google.com.au/books?id=SRZBDcwNOdAC&pg=PA212&dq=Cellular+memory&hl=en&ei=V17xS7mqDo2A7QPkq_WlBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CFMQ6AEwCA#v=onepage&q=Cellular%20memory&f=false] (accessed: Tuesday May 18, 2010), p.212</ref> Or just a bloody good guess? Humans are mostly salt water, so we just adapted to carrying the primordial waters with us upon land. This isn't really so far flung when we remember that in general the [[w:Human genome|Human genome]] bifurcation that makes a human unique is structured that certain chromosomes from the mother are matrilineal and certain chromosome from the father are patrilineal and both lineages go right back to the first progenitors. The most popular [[w:Genealogical DNA|Human ancestry tests]] are Y chromosome ([[w:Y-DNA|Y-DNA]]) testing and mitochondrial DNA ([[w:mtDNA|mtDNA]]) testing which ascertains direct-line paternal and maternal ancestry, to the [[w:Y-chromosomal Adam|Y-chromosomal Adam]] and [[w:Mitochondrial Eve|Mitochondrial Eve]], respectively. Refer ''[[w:The Seven Daughters of Eve|The Seven Daughters of Eve]]'' (2001). I make this divergence as an aside to affirm a magnificent aspect of human genetics that appears was intuited by the Ancient Near East creation myths and it is poignant to remind the reader and myself that we carry ancient code directly in our genetic constitution. The following is extracted from the Wikipedia article verbatim and needs to be rewritten and specified: The '''[[w:Akkadian Empire|Akkadian Empire]]''' (2334 to 2083 BCE) was an empire centered in the city of '''Akkad''' ([[w:Sumerian language|Sumerian]]: ''Agade'' , [[w:Arabic language|Arabic]]: أكد, [[w:Assyrian Neo-Aramaic|Assyrian]]: ܐܵܟܟܵܐܕ , [http://www.premiumwanadoo.com/cuneiform.languages/en_ecriture.htm Hittite] KUR A.GA.DÈKI "land of Akkad"; Biblical ''Accad'') and its surrounding region ([[w:Akkadian language|Akkadian]] URU Akkad KI)<ref>borrowed from Sumerian URU (uru(2)(ki), iri, rí; iri11: city, town, village, district [URU archaic frequency: 101; concatenation of 5 sign variants; UNUG archaic frequency: 206; concatenates 3 sign variants].) meaning city and Ki meaning place is a Sumerian - Akkadian determinative Idiom uru-bar-ra: outside the city, outskirts of the city, the countryside ('city' + 'outside' + nominative).uru-kúr(-ra):(in) a foreign city ('city' + 'strange' + locative).uru-šà-ga: the interior city (contrasts to uru-bar-ra)('city' + 'inside' + nominative).zag-uru: outskirts of the city ('edge, limit' + 'city').(see Hallorans Sumerian Lexicon)</ref> in [[w:Ancient Iraq|Ancient Iraq]],<ref>Peter Roger, Stuart Moorey, ''Ancient Iraq: (Assyria and Babylonia)'', Ashmolean Museum (1976).</ref><ref name="WebsterNinthNewCollege">Mish, Frederick C., Editor in Chief. “Akkad” ''[[w:Webster's Dictionary#The Collegiate Dictionary|Webster’s Ninth New Collegiate Dictionary]]''. ninth ed. Springfield, MA: [[w:Merriam-Webster|Merriam-Webster]] Inc., 1985. {{ISBN|0-87779-508-8}}, {{ISBN|0-87779-509-6}} (indexed), and {{ISBN|0-87779-510-X}} (deluxe).</ref> (''Mesopotamia''). The Akkadian state was the predecessor of the ethnic Akkadian states of [[w:Babylonia|Babylonia]] and [[w:Assyria|Assyria]]; formed following centuries of Akkadian cultural synergy with [[w:Sumer|Sumerians]], it reached the height of its power between the 24th and 22nd centuries BC following the conquests of king [[w:Sargon of Akkad|Sargon of Akkad]], and is sometimes regarded as the first manifestation of an empire in history.<ref>Liverani, Mario, ''Akkad: The First World Empire'' (1993)</ref> ==Nondual awareness== There are different types of nondual awareness. This is a playful gathering of some of them and they will be thematically discussed and contrasted in due course. [http://books.google.com.au/books?id=urkGraRPUmIC&lpg=PA63&ots=dCEzXlGZma&dq=Trik%C4%81laj%C3%B1a&pg=PA63#v=onepage&q&f=false Trikālajña] or "knower of the three times [past, present and future]": :Chalcas the wise, the Grecian Priest and Guide, :That sacred Seer whose comprehensive View :The past, the present, and the future knew. ::[[w:Alexander Pope|Pope's]] ''The Iliad of Homer'' The 'knower of the three times' along with the 'knower of the three worlds' are epithets of many saintly people and divinities, including Shakyamuni and is also poetically attributed by Pope to Chalcas, a sage and seer in his licentious creative construction and poetic adaptation of Homer's ''Illiad''. An English rendering of the ''Mahabharata'' relates the story of [[w:Janaka|Janaka]]: <blockquote>"Mention is made of a verse sung (of old) by Janaka who was freed from the pairs of opposites, liberated from desire and enjoyments, and observant of the religion of Moksha. That verse runs thus: 'My treasures are immense, yet I have nothing! If again the whole of Mithila were burnt and reduced to ashes, nothing of mine will be burnt!' As a person on the hill-top looketh down upon men on the plain below, so he that has got up on the top of the mansion of knowledge, seeth people grieving for things that do not call for grief. He, however, that is of foolish understanding, does not see this. He who, casting his eyes on visible things, really seeth them, is said to have eyes and understanding. The faculty called understanding is so called because of the knowledge and comprehension it gives of unknown and incomprehensible things. He who is acquainted with the words of persons that are learned, that are of cleansed souls, and that have attained to a state of Brahma, succeeds in obtaining great honours. When one seeth creatures of infinite diversity to be all one and the same and to be but diversified emanations from the same essence, one is then said to have attained Brahma."<ref>Source: [http://www.sacred-texts.com/hin/m12/m12a017.htm#fr_51] (accessed: Sunday May 30, 2010)</ref></blockquote> Craig, ''et.al''. (1998: p.476) convey a '[[w:Stream of consciousness (psychology)|stream of consciousness]]' or '[[w:mindstream|mindstream]]' as a procession of [[:wikt:mote|mote]] events of consciousness (C) with algebraic notation C₁, C₂ and C₃ thus to demonstrate the immediacy of nondual awareness: <blockquote>That nondual awareness is the only possible ''self''-awareness is defended by a ''reductio'' argument. If a further awareness C₂, having C₁ as content, is required for self-awareness, then since there would be no awareness of C₂ without awareness C₃, ''ad infinitum'', there could be no self-awareness, that is, unless the self is to be understood as limited to past awareness only. For self-awareness to be an immediate awareness, self-awareness has to be nondual.<ref>Craig, Edward (general editor) (1998). ''Routledge encyclopedia of philosophy: Luther to Nifo, Volume 6''. Taylor & Francis. {{ISBN|0415073103}}, 9780415073103. Source: [http://books.google.com.au/books?id=G3UBxqkkCX8C&pg=PA476&dq=ontolog++nondual&hl=en&ei=kufPS-X2J5GTkAWVuq2ACg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CD4Q6AEwAQ#v=onepage&q&f=false] (accessed: Thursday April 22, 2010), p.476</ref></blockquote> [[w:Nisargadatta Maharaj|Nisargadatta]] (1897 – 1981) is reported by Powell (1994, 2006: p.97) stating thus: <blockquote>...When a stage is reached that one feels deeply that whatever is being done is happening and one has not got anything to do with it, then it becomes such a deep conviction that whatever is happening is not happening really. And that whatever seems to be happening is also an illusion. That may be final. In other words, totally apart from whatever seems to be happening, when one stops thinking that one is living, and gets the feeling that one is being lived, that whatever one is doing one is not doing but one is made to do, then that is a sort of criterion.<ref>Nisargadatta Maharaj (principal dialogian) & Powell, Robert (editor) (1994, 2006). ''The Ultimate Medicine'', Source: [http://books.google.com.au/books?id=yIH3FPOcvKEC&dq=the+ultimate+medicine&printsec=frontcover&source=bn&hl=en&ei=zvHQS8H_D4GOkQXinLGYDA&sa=X&oi=book_result&ct=result&resnum=4&ved=0CBQQ6AEwAw#v=snippet&q=concept%20again.&f=false] (accessed: Friday April 23, 2010), p.97</ref></blockquote> ==Dualism== [[File:National_Park_Service_9-11_Statue_of_Liberty_and_WTC_fire.jpg|thumb]] Auden's poem ''[[w:September 1, 1939|September 1, 1939]]'' written on the date of its title covers many themes that I hold to be dualistic: war, division, hate. For sin to function is necessitated separation, compartmentalization and duplicity. Dualism also isn't all negative and malign, it is the way things appear but is not how things are. Duality as different to Monism or the 'One' or the 'Singular' in practice is often a gloss for Plurality, Variegatedness and Multiplicity. We tend to perceive objects and we reify these which we objectify from the position of our subjectivity. This subject and object dichotomy is constructed in the English language in grammar and is therefore inherited from the languages that historically influenced English. I question if this subject-object dualism is really natural and a human cultural universal but proffer that its origin is founded in language. Any language that tends to have the format of a Subject, Object Verb creates such a distinction. As an aside 'action' (S: karman; T:las), 'agent' (S: kartṛ; T: byed pa po) and 'activity' (S: kriyā; T: bya ba) entered Sanskritic philosophy from Sanskrit Grammar (refer [[w:Paul Williams (British professor)|Williams]], 1998, [http://books.google.com.au/books?id=b742C2o4pSwC&lpg=PP1&pg=PA38#v=onepage&q&f=false, p.38] {{ISBN|8120817141}}). *Dualism isn't the right place for this threefoldness but it is also appropriate to be mentioned here as well and then pointing to the timelessness of ritual, for example. These threefold Sanskrit terms are evident in this article: [http://ccbs.ntu.edu.tw/FULLTEXT/JR-PHIL/ew95314.htm The Mythico-Ritual Syntax of Omnipotence. By Lawrence, David Philosophy East & West. V. 48:4 (1998.10) pp. 592-622] Traditionally, Madhavacharya held a Dualist or [[w:Dvaita|Dvaita]] position and we cannot really understand the historicaly positions of Advaita and [[w:Vishishtadvaita|Vishishtadvaita]] without a historical understanding of the position of which he is an exemplar. In this article as Monism and Dualism have become considerably polyvalent and also to mean more as the semantic field of Nondualism has expanded though the exploration of this article. I will have to meditate more on how to convey this. It conveys all binaries or bifurcations that constitute the discrete positions or possibilities within a universal set, eg. Good and Evil. Is the binary of Good and Evil culturally relative? Is it a characteristic of living that Humans and other creatures with such facility and are so positioned within the predation cycle, that must use tooth, nail and knife and this eclipses kiss, caress and spoon? That is somewhat cheesy but I am going to leave it for now. I have talked about Capitalism elsewhere. The image of September 11 holds a certain salience with the duality and the liberal-democratic values enshrined by the [[w:Statue of Liberty|Statue of Liberty]] which is a representation of the goddess [[w:Liberty (goddess)|Liberty]] and together this image is a powerful and complex statement upon [[w:Liberal democracy|Liberal democracy]]. The proper nomenclature for the Statue of Liberty is Liberty Enlightening the World (French: la Liberté éclairant le monde): I pose Liberty Enlightening the World? The binary established is Liberal democracy versus Terrorism, but they should be understood as mutually conditioning and informing. Pointedly, war as constructed opposition with requisite tension in duality is a study in nonduality as it progressively yields through victory and its converse and the resolution of opposition and the transformation it yields. The battle of war with its interpenetration is a metaphor of nonduality, a very model of flux. ===Samkhya=== [[File:James Ballantyne (1808-1877).jpg|thumb|James R Ballantyne]] White (1998: p.17) identifies sources of Samkhya or Indian dualism within the Rgvedic hymn ''[[w:Purusha_sukta|Puruṣasūkta]]'' (10.90) the title of which has been rendered into English as 'Hymn of the Man' and in the mythopoeic narrative of [[w:Prajapati|Prajāpati]] within the [[w:Brhadaranyaka Upanisad|Bṛhadāraṇyaka Upanishad]]: <blockquote>"This dualistic approach, which finds early expression in the Rgvedic "Hymn of the Man" (10.90), is restated time and again in later texts, sometimes taking on sexual valences (to describe a universe in which all is ultimately two), such as in a ''Bṛhadāraṇyaka Upanishad'' myth which depicts Prajāpati as splitting into male and female halves to incestuously reintegrate "himself" through all manner of human and animal forms. This is the mythic foundation of Sāṃkhya, literally the "enumerating" philosophy, the earliest of the Indian Philosophical systems."<ref>White, David Gordon (1998). ''The Alchemical Body: Siddha Traditions in Medieval India.'' Illustrated edition, reprint. University of Chicago Press. {{ISBN|0226894991}}, 9780226894997</ref></blockquote> ''Encyclopædia Britannica Online Library Edition'' (May 2010) states that "Samkhya" is: <blockquote>"one of the six orthodox systems (darshans) of Indian philosophy... Samkhya adopts a consistent dualism of the orders of matter (prakriti) and soul, or self (purusha). The two are originally separate, but in the course of evolution purusha mistakenly identifies itself with aspects of prakriti. Right knowledge consists of the ability of purusha to distinguish itself from prakriti."<ref>"Samkhya". (2010). In ''Encyclopædia Britannica''. Retrieved May 30, 2010, from ''Encyclopædia Britannica Online Library Edition'': http://www.library.eb.com.au/eb/article-9065215</ref></blockquote> Samkhya ''qua'' doctrine may be traced to the older Upanishads, the ''Samkha Darshanam'' attributed to Kapila, the no longer extant ''Samkhya Sutra'', ''[[w:Samkhyakarika|Sāṁkhya Kārikā]]'' and the ''[[W:Samkhyapravachana Sutra|Sāṁkhyapravacana Sūtra]]'' amongst other primary resources. Samkhya ''qua'' tenet system was subsequent to many of its doctrines. [[w:Kapila|Kapila]], the legendary seer, is considered the founder of the system of Samkhya. Samkhya doctrines were formative for the later philosophies of the Sanatana Dharma and the Dharmic Traditions and World religions and philosophies that have responded to them particularly non-dual positions as is the import of this extended dalliance. The Samkhya may be identified as important for the general psycho-spiritual and theological development of the Indian peoples and as such important for global Systems Theology. Textual support apart from the older Upanishads and that which is attributed, associated or inspired by Kapila is the ''[[w:Mahabharata|Mahabharata]]'' (specifically the [http://www.sacred-texts.com/hin/mbs/mbs12012.htm Mokshadharma sections that is Book 12, Chapter 48 (or XLVIII)] and the ''[[w:Bhagavad Gita|Bhagavad Gita]]'') and the Samkhyakarika of Ishvarakrishna composed prior to 500 CE.<ref>Michaels, Axel (author) & Harshav, Barbara (author, translator) (2004). ''Hinduism: past and present.'' Edition: illustrated. Princeton University Press. {{ISBN|0691089523}}, 9780691089522. Source: [http://books.google.com.au/books?id=jID3TuoiOMQC&dq=Hinduism:+Past+and+Present.&printsec=frontcover&source=bn&hl=en&ei=iGcATPSoO8-GkAWzyODxDQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0CCoQ6AEwAw#v=onepage&q&f=false] (accessed: Saturday May 29, 2010), p.264</ref> Samkhya as a Dvaita or Dualist system may be founded on the hard split between [[w:Purusha|Purusha]] and [[w:Prakriti|Prakriti]]. These are technical terms in this tradition and we are going to need to identify the understanding of them in this tradition as different traditions understand their relationship and even what they constitute very differently. Samkhya is a dualistic philosophy and postulates two eternal realities: Purusha, the witnessing consciousness, and Prakriti, the root cause of the material substance and the natural world, composed of the three gunas. Purusha is often glossed "Soul" and Prakriti is often glossed "Nature". This additional conceptual overlay is helpful but also in itself problematic. The exact nature and constitution of the Puruṣa in the Sāṃkhya philosophical view is important to understand, [[w:Surendranath Dasgupta|Dasgupta]] (1992: p.239) holds that it is without attributes but its nature is "[[w:Cit_(consciousness)|cit]]" and contrasts it usefully with both the general Jaina understanding (in general [[w:Jiva|Jīva]], but Puruṣa amongst other terms are also employed) and the general understanding of the Vedānta (in general [[w:Ātman (Hinduism)|Ātman]], but Puruṣa amongst other terms are also employed): <blockquote>"Unlike the Jaina soul possessing ''anantajñāna'', ''anantadarśana'', ''anantasukha'', and ''anantavīryya'', the Sāṃkhya soul [puruṣa] is described as being devoid of any and every characteristic; but its nature is absolute pure consciousness (cit). The Sāṃkhya view differs from the Vedānta, first in this that it does not consider the soul to be of the nature of pure intelligence and and bliss (Ananda). Bliss with Sāṃkhya is but another name for pleasure and as such it belongs to prakṛti and does not constitute the nature of soul; secondly, according to Vedānta the individual souls (jīva) are but illusory manifestations of one soul or pure consciousness the Brahman, but according to Sāṃkhya they are all real and many."<ref>Dasgupta, Surendranath (1992). ''A history of Indian philosophy, Volume 1.'' Second edition. Motilal Banarsidass Publishers. {{ISBN|8120804120}}, 9788120804128. Source: [http://books.google.com.au/books?id=PoaMFmS1_lEC&pg=PA238&dq=purusha+in+samkhya&hl=en&ei=JZEBTLC1HM6HkQXF2KTxDQ&sa=X&oi=book_result&ct=result&resnum=7&ved=0CEUQ6AEwBg#v=onepage&q&f=false] (accessed: Sunday May 30, 2010)</ref></blockquote> In this quotation I intuit that Dasgupta in making reference to a (as different to THE) Vedanta view of the 'soul' (jivātman) in its fullness as Brahman ''qua'' "[[w:Satcitananda|Saccidānanda]]". Therefore, Dasgupta may be interpreted as glossing "saccit" (that is sat & cit) as "pure intelligence" which is unconventional but not untoward. The 28th aphorism of the ''Samkhya Darshanam'' has a clear statement as to the separation and division of the internal and the external, as rendered by [[w:James R. Ballantyne|Ballantyne]] (1813 – 1864) in 1885: <poem> :[http://en.wikiversity.org/wiki/User:B9_hummingbird_hovering/Blog/Personal_Weblog/Nonduality/Samkhya_Darshanam na bāhyābhyantarayoruparajyoparañjakabhāvo'pi deśavyavadhānācchrughnasthapāṭaliputrasthayoriva 28] </poem> <blockquote>[http://www.sacred-texts.com/hin/sak/sak1.htm "Also <nowiki>[</nowiki>in my opinion, as well as in yours, apparently<nowiki>]</nowiki>, between the external and the internal there is not the relation of influenced and influencer; because there is a local separation; as there is between him that stays at Srughna and him that stays at Páṭaliputra." Aphorism 28]</blockquote> *http://www.sacred-texts.com/hin/sak/ (this is the English translation of the [http://www.kanchiforum.org/redirect.php?url=http%3A%2F%2Fis1.mum.edu%2Fvedicreserve%2Fdarshanas%2Fsamkhya.pdf%0D Samkhya Darshanam (I have this saved in PDFs)] [[User:B9 hummingbird hovering/Blog/Personal Weblog/Nonduality/Samkhya Darshanam|Samkhya Darshanam (my working copy in Unicode with conversion errors, check with PDF when refining)]] *http://www.hss.iitb.ac.in/courses/hs467/sankhya.html The Mokshadharma section (Mokshadharma Parva) of the Mahabharata is the second part of Book 12 which has three parts on total: where each of these three parts has numerous chapters. Book 12 is an interpolation into the older parts of the Mahabharata. Section 48 (XLVIII) of the first part of this Book 12 includes the following about the Sankhyas, the followers of the Samkhya Darshan: <blockquote>"Thou always conscious and present in self, the Sankhyas still describe thee as existing in the three states of wakefulness, dream, and sound sleep. They further speak of thee as possessed of sixteen attributes and representing the number seventeen. Salutations to thy form as conceived by the Sankhyas!"<ref>Source: [http://www.sacred-texts.com/hin/m12/m12a047.htm#fr_14] (accessed: Sunday May 30, 2010)</ref></blockquote> ===Challenges to Cartesian dualism=== Brown (2006: p.19) charts the lineage of philosophers, namely [[w:Nietzsche|Nietzsche]] (1844 – 1900), [[w:Edmund Husserl|Husserl]] (1859 – 1938), [[w:Martin Heidegger|Heidegger]] (1889 – 1976), [[w:Jean-Paul Sartre|Sartre]] (1905 – 1980), [[w:Maurice Merleau-Ponty|Merleau-Ponty]] (1908 – 1961), and [[w:Emmanuel Levinas|Levinas]] (1906 -- 1995) who challenged the entrenched [[w:Cartesian dualism|Cartesian dualism]] of a [[w:Mind-body dichotomy|hard split between "body" and "mind"]] and hence, embraced different views of nondual '[[w:bodymind|bodymind]]' or body-mind [[w:continuum|continuum]] thus: <blockquote>"Like the writings of Nietzsche, the writings of phenomenologists Edmund Husserl, Martin Heidegger, Jean-Paul Sartre, Maurice Merleau-Ponty, and Emmanual Levinas have been recognized by many as providing alternatives to a Cartesian-dualist and Enlightenment-subjectivity worldview. If Nietzsche's response to Cartesian dualism, enlightenment subjectivity (i.e., Kant), reductive materialism (i.e., Marx), and reductive idealism (i.e., Hegel) is not the only nineteenth-century response, it is one of the most effective."<ref>Brown, Kristen (2006). ''Nietzsche and embodiment: discerning bodies and non-dualism''. SUNY series in contemporary continental philosophy. Annotated edition SUNY Press. {{ISBN|0791466523}}, 9780791466520. Source: [http://books.google.com.au/books?id=uVgPHRI3om8C&pg=PA23&dq=Nietzsche+nondual&ei=VtPiS-ShF4nclQTA4JXJCQ&cd=2#v=onepage&q=non-dual&f=false] (accessed: Friday May 7, 2010), p.19</ref></blockquote> Philosopher and Buddhist, [[w:Herbert V. Günther|Günther]] (1917 - 2006), stated: <blockquote> "What we call 'body' and 'mind' are mere abstractions from an identity experience that cannot be reduced to the one or the other abstraction, nor can it be hypostatized into some sort of thing without falsifying its very being."<ref>Mipham, Lama (Tarthang Tulku, trans.) (1973). ''Calm and Clear''. Emeryville, CA: Dharma Publ. (NB: with forward by Herbert V. Günther), pp.15-16</ref> </blockquote> ==Nondual religious and spiritual traditions and teachings== I feel it important to define and unpack "spiritual". I choose to employ the term "spiritual" as "religious" is too loaded with ties to a particular spiritual tradition. Unfortunately, the term "spiritual" tends to denote something immaterial due to its etymology. Within this article it must be understood that spirituality is always embodied in the experience of the human within the Human Condition and is therefore subjective, but not immaterial. The term "spiritual" is employed to denote that [[w:Mysticism|mystic]] impulse, mystic propensity and mystic predisposition as qualities of certain individual's experience and way of being in the World. Consequently, this work is informed by [[w:Evelyn Underhill|Underhill]] (1875–1941) who published her seminal work (1911) foregrounding the focus of mysticism scholarship as [[w:Praxis (process)|praxis]] and from the experience of praxis or 'practice' rather than mere theory.<ref>Underhill, Evelyn (1911). ''Mysticism: A Study of the Nature and Development of Man's Spiritual Consciousness'' (1911); 12th edition reprinted by Dutton 1961; reprint 1999 ({{ISBN|1-85168-196-5}}); see also [http://www.archive.org/details/mysticismstudyin00undeuoft online edition @ Internet Archive].</ref> ==Myths are not other than science== [[w:Mary Midgley|Midgley]] (2004: p.1) challenges the separation of '[[w:science|science]]' and '[[w:myth|myth]]': <blockquote>We are accustomed to think myths as the opposite of science. But in fact they are a central part of it: the part that decides its significance in our lives. So we very much need to understand them. Myths are not lies. Nor are they detached stories. They are imaginative patterns, networks of powerful symbols that suggest particular ways of interpreting the world. They shape its meaning. For instance, machine imagery, which began to pervade our thought in the seventeenth century, is still potent today. We still often tend to see ourselves, and the living things around us, as pieces of clockwork: items of a kind that we ourselves could make, and might decide to remake if it suits us better. hence the confident language of 'genetic engineering' and 'the building-blocks of life'.<ref>Midgley, Mary (2004). ''The myths we live by''. Routledge. {{ISBN|0415340772}}, 9780415340779. Source: [http://books.google.com.au/books?id=5BRU-0JuK84C&printsec=frontcover&dq=myths+that+we+live+by&hl=en&ei=uWTxS72DAoLs7APl74WSBg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwAA#v=onepage&q&f=false] (accessed: Tuesday May 18, 2010)</ref></blockquote> The auspice Science now subsumes innumerable disciplines, sub-disciplines and interdisciplinary fields of human inquiry that have markedly divergent, shared and discipline specific technical lexicons within their specialty that often make the knowledges enshrined inaccessible to unrelated disciplines let alone the lay, non-academic or general populace. Therefore, the necessity of popular science. Popular science may be understood as functional myths as may hypotheses. Darwin was reputed to be a gifted storyteller if not a fanciful scally-wag or even an abject liar when young. Not different to a number of children in truth. But this skill has been noted to have assisted in the perpetuation of the mythos of his scientific hypotheses and contributing to their longevity. Find source. A good story or yarn a resonant myth a powerful pitch is also mandatory to secure research funding is fundamental to garner confidence in value of any prospective project. Nonduality also partakes of mythic discourse... not just the Pharoah is divine, the narrative of divinity is increasingly opening to others, and this has become an ever-growing theme in nondual traditions: we too may partake of divinity and it is not something Other than us reserved to an elite or a privileged. There is a Scientific discipline that studies holiness. Actually, there are numerous intersecting and cross-fertilizing disciplines that do so. One really has to wonder why quality information the discourses of true value are not made available to all people. [[w:Gaia hypothesis|Gaia hypothesis]] may be understood as a mythic narrative of nonduality of Humanity with their lived environment informed by such disciplines as [[w:Deep Ecology|Deep Ecology]]. Midgley (2004) reminded me of the value of myths and also their danger, respect and honour a myth if it is determined to be of value but do not be caught by it. Do not be caught by the metaphor. Herein I tender resides value and wisdom, value in understanding the shared discourse of nondualty and myths. Nonduality is a powerful conceptual tool that embraces causality. '''Exercise''': Pick two arbitrary things or concepts far apart in time, discipline and area of knowledge and then do various kinds of brain storming to chart their relationship, to find ways they interpenetrate, inform and intersect. Put various terms in an Internet Browser and document what is returned and employ that to iterate search parameters and as a springboard for further brainstorming. A study in nonduality. Often if we identify how thing are different that also helps us identify how they are similar as a byproduct. Dualities entail one-another. Entailment theory may be informed by concepts such as apoha and the colour wheel and human perception. [[w:Opponent process|Opponent process]] isn't really what I was looking for. There is an optical illusion. You stare at a simple icon outline of a sun-yellow buddha for a long time and memorize the colour and quickly look away or shut your eyes, I forget which but then you see its 'colour compliment' in the eye of the mind, a violet buddha. Refer Wolfe (1997).<ref>Wolfe, Wynn (1997). Thirty Seconds to a Blue Buddha: Simultaneous Contrast. (From Sunrise magazine, April/May 1997. Copyright © 1997 by Theosophical University Press). Source: [http://www.theosophy-nw.org/theosnw/arts/ar-wolf.htm] (accessed: Tuesday May 18, 2010)</ref> Wolfe mythically evokes the the effect of "simultaneous contrast" of complementary colour in Human perception and embeds a quotation from [[w:Johannes Itten|Itten]] (1888 – 1967) of the famed [[w:Bauhaus|bauhaus school]]: <blockquote>Not only are the rays that carry and give color not colored, but every single color perceived in nature hides within itself the diametrically opposite color in the color wheel; hence, simultaneous contrast. Johannes Itten, famed Bauhaus teacher, tells us: "Simultaneous contrast results from the fact that for any given color the eye simultaneously requires the complementary color and generates it spontaneously. The simultaneously generated complementary occurs as a sensation in the eye of the beholder. It is not objectively present. It cannot be photographed."</blockquote> ==Morality== :[http://en.wikisource.org/wiki/Bible_(Wikisource)/Titus "To the sanctified, all things are sanctified, however for those living in iniquity and unbelief nothing is sanctified, as even their frame of mind and conscience is unclean." Titus 1.15] <br > :What blurt is this about virtue and about vice?<br > :[[s:Leaves_of_Grass/Book_III#22|Section 22, Line 464]], ''Song of Myself'', Walt Whitman :"There is a curious turning point where a bad reputation becomes a good one, because it has been especially naughty." :Beauford Anton Stenberg aka B9HH (NB: I remember there being a quote with this sentiment by Oscar Wilde but I haven't yet been able to source it so wrote my own.) Nondual traditions have been referred to as morally ambiguous. A nondual stance would be that moral 'rules' are for moral 'infants'. The idea that there is some certitude in a fixed moral code is fallacy. A cursory survey of history and different peoples will substantiate my point. Who determines moral code and why and by what mechanism is it perpetuated and transmitted through time? These are important questions in relation to nonduality. Importantly, morality is culturally constructed. Some say a moral code is sanctioned by Godhead and divinity but then there are markedly different moralities so sanctioned. Indeed, there are many moralities. What happens when different moralities collide? Moral, immoral and amoral need to be extrapolated and defined. That said, some people hold that nonduality requires that there is no Absolute Goodness or Absolute Evil. But there is evil in the world let us be under no delusion. There is also goodness. I tender that most people experience evil in some form in their lifetime and how this experience is understood and integrated is key to their spiritual success or failure. Can there be spiritual failure? Well that necessitates a rationale for why we are in the field of experience and why there is even a field of experience in the first place. The terror I have talked of earlier that must be visited upon other living entities by humans to continue living when done in full conscience, mercy and heartfulness is other than evil. Well, that is how I perceive it. This to me is a presentiment of the salience of saying grace and prayer and thanksgiving. I have seen a cat torment a mouse and wondered whether that was evil. But then, not all cats torment their prey. Well I pray that mice are not so tortured. What humans do to other species in animal testing is inhumane indeed inhuman. In regards to grappling with evil I have meditated on the cases of young children with seemingly all the graces of a loving and the auspice of stable homelife and honourable parenting that have lured other children to their violent death. How is the murder of a child by a child to be understood morally? Who is culpable? Child, parent, society, victim, situation or a complex? Most spiritual personages I am aware of have had a nemesis, great loss, suffering or pain which has quickened their sanctity. I want to mention the ''[[w:The Screwtape Letters|The Screwtape Letters]]'' here and the Book of Job and [[w:Simone Weil|Simone Weil]]. They are unrelated and I don't even yet know what I want to say about them but they will be discussed in morality and nonduality. <blockquote>"There are two equal and opposite errors into which our race can fall about the devils. One is to disbelieve in their existence. The other is to believe, and to feel an excessive and unhealthy interest in them. They themselves are equally pleased by both errors and hail a materialist or a magician with the same delight." C. S. Lewis ''The Screwtape Letters'', extract</blockquote> I find it interesting that Lewis in the process of writing The Screwtape Letters underwent a transition from an atheist or agnostic to a theist. I hold that neither a theist nor atheist stance is inherently moral or amoral or even immoral. I find the function of spirituality and its affect on morality very interesting. In Wilde I find a very interesting case of complex morality especially given his same-sex sexuality and historical love-affair that due to societal mores and double-standards brought about his defamation and the end and ruination to a glorious talent. As an allegory The Picture of Dorian Gray is very powerful and has many teachings but I find the skin-deep of beauty and that appearances are not always as they seem to be key. Morality similarly is often about appearances rather than reality. Inner morality is more important than outer morality and therefore intentionality is key. [http://ebooks.adelaide.edu.au/w/wilde/oscar/house/chapter4.html The Star-Child] I find an important teaching on duress and suffering and how all favour and being spoiled harden the heart. I find the sacrifice in [http://www.artpassions.net/wilde/nightingale.html The Nightingale and the Rose] a beautiful tale of sacred morality on the behalf of the nightingale and it also conveys how transient goodness is and how persistent that which is other. The [[w:The Pearl (novel)|The Pearl (novel)]] conveys a teaching similar to the ruby and ivory ankus in the King's Ankusha of Kipling that I mentioned at the outset. I have resolved to tell stories here as there is a truth that is conveyed in narrative that is never really imparted when directly conveyed. Sexuality has often been demonized in western cultures, especially certain forms of sexuality. I tender that if you want to control a person and successfully curtail their sexuality then you can easily control anything else about them. Social control and sexual control have a shared and protracted history. When birth control was not generally available (even though many women in different cultures practices indigenous and folk medicine that fulfilled this function), the economics of mouths to feed offset by the input of another set of helping-hands as a possibility in the future, sexual control is easily understood as a social necessity. I would also like to declare that monasticism and continence of the sexual impulse is unnatural. Though, sexuality as a sacrifice of going with out through continence or engaging in the sexual acts as sacred are both manifestations of honouring sexuality as a sacred rite. Lust, may be understood as an unhealthy sexuality but defining exactly what this is is problematic as people have different sexual requirements. Lust as one of the Seven Deadly Sins has a significant history within the cultures of certain forms of Christianity and hence pervades their literature and cultural forms and social mores. Therefore, it is a legacy to the modern world as English is not only the lingua franca but is approaching a common language for the contemporary global community. 'Lust' was one of the first English glosses for [[w:Taṇhā|tṛṣṇā]] (Sanskrit), one of the three fires of the Buddhadharma attributed to Shakyamuni and these are also known as one of the three poisons or klesha. Karma has entered popular spirituality throughout the world from the Dharmic Traditions. I don't know if I 'believe' in karma. Karma has been and continues to be a justification for disparity and inequity that is culturally constructed and socially perpetuated. Just as it was done in Mother India with the caste system. Now the caste system wasn't all demonic in that it also was the medium for the transmission of skills as well as a social support system for the infirm, unwell, aged and those children without parent and guardian. I find a valuable analogue to Karma in the history of the Caucasian peoples to be the memes of [[w:Orlog|Orlog]] and [[w:Wyrd|Wyrd]]. Cammell's 1968 film [[w:Performance (film)|Performance (film)]] had Jagger in character intone "Nothing is real, everything is permitted" as the wisdom of the Old Man of the Mountain, as Hassan-i-Sabah.<ref>Source: [http://home.comcast.net/~flickhead/DonaldCammell.html]</ref> The Old Man of the Mountain may have been a title of the head of the assassins as [[w:Rashid ad-Din Sinan|Rashid ad-Din Sinan]] is known with this appellation. Marco Polo mentions the Old Man of the Mountain ([http://en.wikisource.org/wiki/The_Travels_of_Marco_Polo/Book_1/Chapter_24 and the assassin's creed upon consumption of hashish?]) is similarly found rendered into English in the novel [[w:Alamut (1938 novel)|Alamut (1938 novel)]] thus: <blockquote>"There's a strange double edge to the maxim that nothing is real and everything is permitted, as I just showed you with the pathetic example of my son. For those who by nature aren't meant for it, all it means is a heap of empty words. But if someone is born for it, it can become the north star of his life. The Carmatians and Druzes, to which Hakim the First belonged, recognized nine grades that their novices had to fight their way through. Their dais courted new adherents with tales of Ali's family and the coming of the Mahdi. Most of these converts were satisfied with simple legends like those. The more ambitious ones pressed the dais for more answers and were told that the Koran is a kind of wondrous metaphor for higher mysteries. Those who still weren't satisfied had their faith in the Koran and Islam undermined by their teachers. If somebody wanted to press even further, he learned that all faiths are equal in their accuracy or inaccuracy. Until, finally, a small, elite handful was ready to learn the highest truth of all, based on the negation of all doctrines and traditions. That grade required the greatest courage and strength from a man. Because it meant that he would spend his whole life without any firm ground beneath his feet and with no support. So there's no need to worry about our principle losing its effectiveness, even if a lot of people find out about it. Most of them won't understand it anyway."<ref>Source: [http://books.google.com.au/books?id=SVVIrDkeGJsC&lpg=PP3&dq=Alamut&pg=PA168#v=onepage&q=%22everything%20is%20permitted%22&f=false] </ref></blockquote> Some people, actually many, may balk at my employ of assassin wisdom in a spiritual context. But the Assassin like Shaolin Monks and the Bushido Code of the Samurai whilst not necessarily nondual paths, hold the nondual truth that fighting and violence may be compassionate and just and contextually appropriate when left with no other recourse. Moreover, Padma Translation Committee's rendering of an embedded quotation of one of the famed "Twelve Vajra Laughs" (drawn from the [[w:Piled Gems|''Pile of Jewels Tantra''; Wylie: ''Rin-po-che spungs-pa' rgyud'']] which is numbered as one of the [[w:seventeen tantras|seventeen tantras]]) cited in the ''[[w:Nelug Dzo|Nelug Dzö]]'' one of [[w:Longchenpa|Longchenpa]]'s "[[w:Seven Treasuries|Seven Treasures]]" (Wylie: ''mDzod bdun'') ''is'' clearly an example of the technical [[w:twilight language|twilight language]] of ''Atiyoga'' and the pedigree of the [[w:upaya|skillful]] doctrine of the mindstream: <blockquote> Listen further, O Vajra of Speech! Behold the nature of phenomena, empty and all-pervasive timeless awareness. How marvelous — it is unborn and abides timelessly, coemergent with being itself. Even if a person were to seize a sharp weapon and slay all beings at once, that person's mindstream would still be free of benefit or harm. Ha! Ha!<ref>Rabjam, Longchen & The Padma Translation Committee (1998). ''The Precious Treasury of The Way of Abiding'' (Wylie: ''Gnas lugs mdzod''). Junction City, California, USA: Padma Publishing. {{ISBN|1-881847-09-8}} (hardback). p.95</ref> </blockquote> [[w:Svecchacara|Svecchācāra]] ([[w:IAST|IAST]]; Sanskrit: स्वेच्छाचार) is a Sanskrit word and important in the [[w:Nath|Nath]] Sampradaya.<ref>International Nath Order Wiki (August 2009). ''Svecchachara''. Source: [http://www.nathorder.org/wiki/Svecchachara] (accessed: Thursday March 11, 2010)</ref> 'Svecchācāra' means: acting as one likes, arbitrariness, acting without restrain.<ref>Source: [http://books.google.com.au/books?id=ht0Dazob1yAC&pg=PA817&lpg=PA817&dq=svecchAcAr&source=bl&ots=EsJKQQXQjz&sig=9w0AAIUWvq-MYmnCfrHRS0YbiF0&hl=en&ei=D42YS4fGK82LkAXe8IGNAQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAoQ6AEwAA#v=onepage&q=svecchAcAr&f=false] (accessed: Thursday March 11, 2010)</ref> [[w:John Woodroffe|Woodroffe]] (1951: p.440) defines 'svecchācāra' and associates it to notions of [[w:Antinomianism|Antinomianism]] and that it is evident in the [[w:Upanishads|Upanishads]] and the [[w:Tantras|Tantras]]: <blockquote>"Lastly, the doctrine that the illuminate knower of Brahman (Brahmajnani) is above both good (Dharma) and evil (Adharma) should be noted. Such an one is a Svechacari whose way is Svechacara or "do as you will". Similar doctrine and practices in Europe are there called Antinomianism. The doctrine is not peculiar to the Tantras. It is to be found in the Upanishads, and is in fact a very commonly held doctrine in India."<ref>Woodroffe, John George (1951). ''Shakti and Shâkta: essays and addresses on the Shâkta Tantrashâstra''. Forgotten Books. {{ISBN|1-60620-145-X}}, 9781606201459. Source: [http://books.google.com.au/books?id=3e3_GVggCgUC&pg=PA449&lpg=PA449&dq=Svecch%C4%81c%C4%81ra&source=bl&ots=LbR1RGGxJU&sig=z44EAVrzP1QYBp-QzOyes3wEsDM&hl=en&ei=PnCYS-HBA9GLkAXjnvGKAQ&sa=X&oi=book_result&ct=result&resnum=6&ved=0CBcQ6AEwBQ#v=onepage&q=svecchacara&f=false] (accessed: Thursday March 11, 2010), p.440</ref></blockquote> Woodroffe (1951: pp.440-441) also goes on to state that: <blockquote>"In Svecchacara there is theoretical freedom, but it is not consciously availed of to do what is known to be wrong without fall and pollution."<ref>Woodroffe, John George (1951). ''Shakti and Shâkta: essays and addresses on the Shâkta Tantrashâstra''. Forgotten Books. {{ISBN|1-60620-145-X}}, 9781606201459. Source: [http://books.google.com.au/books?id=3e3_GVggCgUC&pg=PA449&lpg=PA449&dq=Svecch%C4%81c%C4%81ra&source=bl&ots=LbR1RGGxJU&sig=z44EAVrzP1QYBp-QzOyes3wEsDM&hl=en&ei=PnCYS-HBA9GLkAXjnvGKAQ&sa=X&oi=book_result&ct=result&resnum=6&ved=0CBcQ6AEwBQ#v=onepage&q=svecchacara&f=false] (accessed: Thursday March 11, 2010), pp.440-441</ref></blockquote> Svecchācāra is important in the Nath Sampradaya evocation of their realized ideal, of the [[w:Avadhuta|Avadhuta]]; as [[:simple:Shri Gurudev Mahendranath|Mahendranath]] states: <blockquote>"Sveccha means one's own wish or free will. Svecchachara means a way of life where one acts as one wishes and does what is right in one's own eyes. Doing one's own Will. The concluding Sanskrit expression in the Avadhoota Upanishad is "Svecchachara Paro." <br > The term "Paro" means a mysterious or secret pattern to that action done by one's own Will. In other words, we do our Will but with discretion, not making it too obvious, nor to harm or hurt other people. Yet this is also a typical Nathism; a complete reversal of Vedic morals and philosophy."<ref>International Nath Order Wiki (August 2009). ''Svecchachara''. Source: [http://www.nathorder.org/wiki/Svecchachara] (accessed: Thursday March 11, 2010)</ref><ref>Mahendranath, Gurudev (1999, 2002). ''Ecstasy, Equipoise, and Eternity . Source'': [http://www.mahendranath.org/twiyoga1/twiyoga1.html] (accessed: Thursday March 11, 2010)</ref></blockquote> Mahendranath and [[w:Aleister Crowley|Aleister Crowley]] were intimates. "Do what thou wilt shall be the whole of the Law" (AL I:40) This term is employed in the closure of the ''[[w:Avadhuta Upanishad|Avadhuta Upanishad]]''.<ref>International Nath Order Wiki (August 2009). ''Svecchachara''. Source: [http://www.nathorder.org/wiki/Svecchachara] (accessed: Thursday March 11, 2010)</ref> The term 'svecchācāra' also appears nine times in the Mahanirvana Tantra first translated into the English from Sanskrit by Woodroffe (1913).<ref>Source: [http://muktalib5.org/DL_CATALOG/TEXTS/ETEXTS/mahaanirvaanatantramHK.txt] (accessed: Thursday March 11, 2010) NB: search using 'svecchAcAr'</ref> Svechchhachara, “following one's own [true] will” is also evident in the Kali Tantra 8.19. *[[w:Victorian morality|Victorian morality]] and sexuality *[[w:Songs of Innocence and of Experience|Songs of Innocence and of Experience]] *Mahasiddha *Avadhuta Shashibhusan Dasgupta (1976: p.63) to his third edition (1969) reprint (1976) of his seminal text on five [[w:sahaja|sahaja]] traditions entitled ''Obscure Religious Cults'' first published in 1946, holds that: <blockquote>In the post-Upaniṣadic period a free spirit of religion, leaning mainly to the subjective side, characterises the early epic literature of India, particularly the ''Mahābhārata''. There are stories in the ''Mahābhārata'', where the teachings of true religion are being received from people belonging to the lowest class of the social order. In the ''Anuśāsanika-parva'' of the ''Mahābhārata'' [Chapter 108], where Bhīṣma is explaining to Yudiṣṭhira the really sacred places of pilgrimage, we find that the mind with the transparent water of purity and truth, when associated with the lake of patience, is the best of all places of pilgrimage. He whose body is washed with water, cannot be said to be the really cleansed one; he, who has controlled all his senses, is the really cleansed one, and he is pure within as well as without. To dive into the water of the bliss of Brahma-knowledge in the lake of the pure heart of all bathing, and it is only he, whom the wise recognise as the real pilgrim.<ref>Dasgupta, Shashibhusan (1946, 1969 third edition, 1976 reprint). ''Obscure Religious Cults''. Firma KLM Private Limited: Calcutta, India. Sarasvati Printing Press.</ref></blockquote> [http://www.sacred-texts.com/hin/m13/m13b073.htm|SECTION CVIII = Chapter 108] ==Tradition: an uncommon or a common trade== What is being traded in tradition? Individuality, pregnant possibility, ill-informed constructs, knowhow and/or wisdom? This isn't the right spot for this but something important came to mind, and it needed to be captured. I need to mention the discourse of the Great Tradition of those with 'prestige' and the Little Tradition of the 'common' folk ([[w:Robert Redfield|Robert Redfield]], ''Peasant Society and Culture'', 1956: p.70): <blockquote>In a civilization there is a great tradition of the reflective few, and there is a little tradition of the largely unreflective many. The great tradition is cultivated in schools or temples; the little tradition works itself out and keeps itself going in the lives of the unlettered in their village communities. The tradition of the philosopher, theologian, and literary man is a tradition consciously cultivated and handed down; that of the little people is for the most part taken for granted and not submitted to much scrutiny or considered refinement and improvement.</blockquote> This overly simplistic diachronic view of Redfield is of its time and problematic but it inadvertently highlighted the importance of prestige in subsequent discourse and this was of profound importance. The little tradition became one of prestige in its own right, as the hegemony of standard speech variety over dialect became moot. Refer: [http://books.google.com.au/books?id=49GVZGD8d4oC&pg=PA7&lpg=PA7&dq=redfield+great+tradition+little+tradition&source=bl&ots=nBkKSSH9vm&sig=sCT7yyvuCJ3mdzCDPv05fiEms2g&hl=en&ei=nZDxS9W8K5Hg7AOh4fWOBg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBoQ6AEwATgU#v=onepage&q=redfield%20great%20tradition%20little%20tradition&f=false] Anthropology and the varieties of experience. We chunk the auspice term of "Sanatana Dharma", "Dzogchen", "Christianity" for example around like that actually has some quantifiable denotation and a shared consensus at that. Remember the problematics of 'entities'? Is a spiritual tradition invested in its mythos,its transmission lineage, its iconography and semiology, its theology and philosophy, its technical lexicon, in the soil of its point of origin, in its textual tradition, in its ritual, in the florescence of its culture, in the cultural tokens and ornaments of mind and what appears to those said minds which the foolish objectify? Is there ever a Golden Age? Is a neo-Tradition really new? What is new about the New Age except on the large poor scholarship? Most New Age artifacts are really repackaged ancient knowledge unattributed at worst, or vain fancy at best. All of this play shares in the discourse of nonduality as there is nothing in truth new under the Sun except maybe a new narrative archetype and a new technology or a newborn. That said, is an aggregated recombination ever really new? We are the stuff of the cauldron of timelessness ([[w:Brian Swimme|Swimme]] & [[w:Thomas Berry|Berry]], ''The Universe Story'', 1992). [[w:Johannes_Itten|Itten]] (1974: p.11) one of the former teachers of the bauhaus school holds that: <blockquote>Learning from books and teachers is like traveling by carriage, so we are told in the Veda. The thought goes on, "But the carriage will serve only while one is on the highroad. He who reaches the end of the highroad will leave the carriage and walk afoot."<ref>Itten, Johannes (1974). ''The art of color: the subjective experience and objective rationale of color''. Second revised edition, illustrated. John Wiley and Sons. {{ISBN|0471289280}}, 9780471289289. Source: [http://books.google.com.au/books?id=D-skaDZAumIC&printsec=frontcover&dq=Johannes+Itten,+The+Art+of+Color,&ei=eoDxS5quD5iUlAS3_vCjCg&cd=1#v=onepage&q=87&f=false] (accessed: Tuesday May 18, 2010)</ref></blockquote> Philip Almond (''The British Discovery of Buddhism'', 1988: p.13): <blockquote>Buddhism, by 1860, had come to exist, not in the Orient but in the Oriental libraries and institutes of the West, in its texts and manuscripts, at desks of the Western savants who interpreted it. It had become a textual object, defined, classified, and interpreted through its own textuality.... By the middle of the century, the Buddhism that existed "out there" was beginning to be judged by a West that alone knew what Buddhism was, is, and ought to be.<ref>Source: [http://books.google.com.au/books?id=0sL0pUS3yBAC&pg=PA166&lpg=PA166&dq=P%C4%81%C5%9Ba+iconography&source=bl&ots=lZXiyVmbK4&sig=ek9g9mfVPLeL_PcFBa8QDtUwDxA&hl=en&ei=E7bhSaGSNJnW7AOYvuWIDA&sa=X&oi=book_result&ct=result&resnum=4#PPA14,M1] </ref> </blockquote> Berthrong (1994: p.133) conveys what I understand as the importance of an inclusive theology that includes many voices to inform our "emerging world of global thought" which he calls a "pluralistic vision of theology": <blockquote>For any major religious tradition there are many lineages, many stories, many theologies and philosophies. From inside the pluralist vision of theology, the sooner that we learn to do theology honouring this pluralism, the better off we will be in terms of the emerging world of global thought.<ref>Berthrong, John H. (1994). ''All under heaven: transforming paradigms in Confucian-Christian dialogue''. SUNY Press. {{ISBN|079141857X}}, 9780791418574. Source: [http://books.google.com.au/books?id=rfPexVxxOVUC&printsec=frontcover&dq=John+Berthrong&source=gbs_book_similarbooks#v=onepage&q&f=false] (accessed: Tuesday May 18, 2010)</ref></blockquote> ==Integration== I just realized that now I am no longer required to keep apples with apples but may integrate thematically according to my premise. Whatever that may be. But there is something key, very visible, very humane, very real and very wonderful in the example of the divine stigmata. Now to convey something salient in relation to the compassionate manifestation of the stigmata embodied in sainthood through a personal anecdote that is so very unrelated but so very apt. I came to this realization by the way of a small child at an empowerment who I through a serendipitous happenstance encountered as I was seated, tying my shoelace in the liminality of a doorway. His eyes for no reason that I knew just started to well with tears as his eyes searching met mine and I responded in kind, an empathic bridging, a transpersonal very humane compassion. An open channel. [[Image:Padre-Pio-young.jpg|thumb|A young [[w:Padre Pio|Padre Pio]] adorned with the divine stigmata.]] The God of traditional [[w:Christianity|Christianity]] is defined by one exegete, namely [[w:Thomas Aquinas|Aquinas]] (ca. 1225 – 1274) in his ''[[w:Summa Theologica|Summa Theologica]]'' which may be respectfully rendered into English with no disservice as omniscient, omnipotent and omnipresent. The [[w:devil|devil]] or adversary is an opposing character, but as a general convention is given as subordinate to God. In one simplistic estimation, the Christian faith thus considers God and the Adversary to be two distinct, opposing "personalities", though unequal in power and prestige. The Adversary becomes the constructed nexus of all that is wicked by consensus and personally by a given individual. This partakes of the discourse of [[w:psychological projection|psychological projection]], the [[w:postmodernist|postmodernist]] construction of [[w:Other|The Other]], attribution of falsehood and the Jungian [[w:Shadow_(psychology)|Shadow]]. Traditional Christianity, with its emphasis on the struggle between good and evil, is decidedly incompatible with nondualistic thinking. Some mystical varieties of Christianity are non-dual, as are some varieties of Judaism. Both are Abrahamic Traditions. Now why the stigmata? Well this is first documented with St Francis whom I have always loved. What is the psycho-spiritual mechanism embodied here? What is bloody going on? I feel there is a salient discourse that may be brought into play from Vajrayana. The [[w:generation stage|generation stage]] and the [[w:completion stage|completion stage]]. This characterization is rather minimal and not truly representative but employed for a purpose. The generation stage is where a sadhaka creates a charged nexus of holiness thoughout diverse sensory, creative and experiential channels whether imaginal or 'internal' that is through creative internalized imaginative visualization within the mind or 'external' through the generation of an 'external' mandala outside the mind *chuckle* for example. The sadhaka generates an arbitrary loaded 'prestige' construction of holiness. Vajrayana Buddhists qualify this construct as being nondual to Shunyata and then merge the plasticity the Prajna of their psyche with it. My personal practice has been the construction of little crystal mandala or medicine wheels and though I have worked with archetypal personalities such as deities through such conduits, my favoured have always been natural phenomena such as flora and fauna and the Classical Elements which I gloss 'processes'. Indeed, where I have worked with Dharmic and non-Dharmic deities, my key point of entry in beginning working with them has been their divine accoutrement, tools or weapons and their mount or steed for example. Though this sounds iconic, my preferred mode is aniconic. This is a positive spin on the terror and openness of [[w:Existentialism|Existentialism]] and the [[w:Human Condition|Human Condition]] and that we are most effective and spiritually mature when we choose what gives our life and the experience of living meaning. Ferrer & Sherman (2008: pp.156-157): <blockquote>A participatory spiritual universalism does not establish any a priori hierarchy of positive attributes of the mystery: Nondual insights are not necessarily higher than dual, nor are dual higher than nondual. Personal enactions are not necessarily higher than impersonal, nor are impersonal higher than personal. And so forth. Since the generative dimension of the mystery is undetermined and dynamic, spiritual qualitative distinctions cannot be made by matching our insights and conceptualizations with any pregiven or fixed features. By contrast, I suggest that qualitative distinctions among spiritual enactions can be made by not only evaluating their emancipatory power for self, relationship, and world, but also discriminating how grounded in or coherent with the ongoing unfolding of the mystery they are. Moreover, because of their unique psychospiritual dispositions, individuals and cultures may emancipate themselves better through different enaction of the mystery, and this not only paves the way for a more constructive and enriching interreligious dialogue, but also opens up the creative range of valid spiritual choices potentially available to us as individuals.<ref>Jorge N. & Sherman, Jacob H. (2008). ''The participatory turn: spirituality, mysticism, religious studies''. SUNY Press. {{ISBN|0791476014}}, 9780791476017, pp.156-157</ref></blockquote> ==Two Truths== *excellent source on two truths, comparative treathment of buddhadharama and advaita vedanta (p.177) by key scholar.... I don't necessarily agree but they will be good as grounding [http://books.google.com.au/books?id=d7ryQuk_inUC&pg=PA276&dq=Santanantara&hl=en&ei=ig7rS-3cKNCIkAWgksWdDg&sa=X&oi=book_result&ct=result&resnum=4&ved=0CD0Q6AEwAzgK#v=onepage&q=Santanantara&f=false] My first exposure to a Two Truths doctrine of sorts was in the ''[[w:The Birth of Tragedy|The Birth of Tragedy]]'' rendered into English from the German by I remember not who: the Two Truths of sorts being the Dionysian and Apollonian. I have found the discourse and the meme of the 'Two Truths' (satyadvaya) to be both of value as well as obstructive and obscuring. The meme is evident in varieties of the Sanatana Dharma and varieties of the Buddhadharma and numerous other traditions as well I intuit: <blockquote>"Though the notion of two truths (satyadvaya) is implicit in Buddhism from the beginning, as it is in Vedanta and, indeed, in any philosophy or religion that holds to a norm distinct from the everyday, Madhyamika alone makes the distinction into its crucial thought."<ref>Source: [http://books.google.com.au/books?id=p1bASTAOhjoC&pg=PA121&lpg=PA121&dq=satyadvaya&source=bl&ots=LAkTvcOqSV&sig=5lIiukrkC2EVau8m_w4IKJu9g_I&hl=en&ei=H_LnS8bRMM2OkQXEg93XBg&sa=X&oi=book_result&ct=result&resnum=10&ved=0CEAQ6AEwCQ#v=onepage&q=satyadvaya&f=false] p.121</ref> </blockquote> The [[w:Nyingma|Nyingma]], a development of the Madhyamaka amongst other accretions, innovations and consolidations, integrate the two truths as a living experience as reputably did [[w:Ramanuja|Ramanuja]]. They may have identified the interpenetration or coalescence of the Two Truths around the same time. I have found it powerful and useful to embrace the Two Truths as two perspectives of '''one''' Truth. That way the nonduality and interpenetration of Relative and Absolute is very present and not Other. Relative is dual vision, dual experience, duality and an experience of dualities. Some will make the sound charge that in this world we cannot escape dualities as they constitute the field of experience. I have had the personal experience in states of consciousness, of a boundless unity. Which is why I can't seem to live life blinded by dualities as I perceive do so many of my 'peers'. I am not interested in winning, fame or materiality and I am also somewhat impractical from a relativistic view as a direct result of my experience of nonduality. This isn't to say that I don't experience heat and cold, night and day. These states reveal to me that there is diversity and duality in experience. But my personal experience also has demonstrated that there are mechanisms or filters of experience that may alter and render void the duality of experience. Be it boon, bane, vain fancy or grace, I cherish the many different states of consciousness with which I have been blessed and as a result nothing is as important to me as being engaged in spirituality as a lived experience to honour my realization. I find it a useful tool, given the nomenclature 'nonduality', to gloss the process of living and abiding in the Absolute, 'yoga' (communion): a dynamic engaged darshan as lived experience in the World and throughout all states of being or consciousness in a play of equalibrium to dualities, resolving their bitter-sweet into the one-taste of equanimity and surrender. [[w:Robert Thurman|Thurman]] (1976, 1991: p.3) in his introduction to his rendering of the nondual Mahayana text ''[[w:Vimalakīrti Nirdeśa Sūtra|Vimalakīrti Nirdeśa Sūtra]]'' states as he perceives it that: <blockquote>There are basically two kinds of theory about ultimate reality: nihilism and "absolutism." Of course, intellectual history abounds in theories vastly different in detail, but all share one or the other of these basic postulates about reality. They either deny it altogether or they posit some sort of ultimate entity, substratum, or superstratum that serves as foundation, essence, container, or whatever, of the immediate reality. And this absolutism, while appearing to affirm something ultimately, actually negates the immediate reality in favour of the hypostatized ultimate reality. For if "God," "Brahman," the "universe," "the void," "nirvana," "pure mind," "the Tao," "pure being," and so forth, make an ultimate reality beyond the imperfection of our world, the spiritual man must naturally strive to escape this imperfection to reach his ultimate and eternal well-being. However widely the absolutes posited may differ, they all impel us in practice to negate our immediate reality.<ref>Source: [http://books.google.com.au/books?id=n4K7p6XSpSYC&printsec=frontcover&source=gbs_v2_summary_r&cad=0#v=onepage&q&f=false] (accessed: Monday May 10, 2010)</ref></blockquote> The gendered bias is somewhat forgiven in the quotation given the initial date of publication but the summary is sound in respect of two kinds of "absoluteness" posited: first, a nomenclature for an ineffable as a positive value, eg. Tao, Brahaman etc.; second, an absence as anti-nominal attributed a nomenclature "nihilism". Well the two types or kinds of absoluteness sound reasonable to me given the representative sample of the discourse I have studied, which is of course partial. Any sample can never be representative. That said, there is always an aspect to any discourse to which we may never be privy: a book we never read, an article to which we never have access, a point we never understand or a key piece of information we never even know we don't understand, or the representation to which we are exposed has no real correlation with what it purports to represent or significantly misrepresents. To call nihilism a kind of absoluteness is a bit curious but I hold that holding no position is still the stance of a position. Somewhat like a zero as a placemarker in the decimal system. Nagarjuna historically contented with exactly that point and (he asserted that) he held no assertions, (he opined) no opinions and (held the position of) no position or view. There is a pervasive theme apparent to me in many varieties of nondual tradition that I have ascertained through cultural traces, cultural artifacts and representations: that a true and full awareness of our present reality, is that to which we are to awaken and not to some otherness, some other reality at the end of the breathless body in this life. I have experienced other realities in this life so, I know that 'ordinary reality' is really a value judgment and partakes of consensus reality discourse as well as the prestige of the 'common ground' of experience. Such assertions as to common ground is very tenuous and unprovable, but even as each of us if we have legs that move stand on one Earth, you hopefully discern the point. I know that truth is only invested in my experience of the world as Subjectivity and I have long-since ceased seceding my power and authority to Other, to do so would be in my estimation a duality. This is not to render anybody impotent, but, truth is nowhere other than where I am. Reader, you too should embrace this if you are ready for the fullness of it. Not my truth but the orientation of your own Subjectivity. I research in order to qualify my experience and to define my spiritual knowing and as I have a calling to do so, not out of the drive or need of a seeker. There is nothing to seek. Though in my heart of hearts I do hope there is a purpose to this whole play of living and there is somehow justice and a reckoning for the disparity I perceive in my human world. I have experienced the surety that this is so but this may be just the particular endochrinal chemical alterations worked upon my bodymind and that was in the sealed samadhi of One. Back to critiquing Thurman. He is perpetuating constructions of Otherness of "God", "Brahman" the "Universe", "The Void", "Nirvana", "Pure Mind", "pure being". A few of these terms are numbered amongst the technical lexicon of Buddhadharma varieties but there are discourses where they are all placed in awakened experience in body in this reality so Thurman's bias to his Buddhadharma tradition and his perception of that tradition is to be declared. Well this is my reading. There is an untenable fallacy in positing the fullness of the experience of these ineffables that are unknowable outside of direct experience that are given a nomenclature in their native traditions and then that nomenclature appropriated by a colonized and colonial, partiality. Especially when they are not the prestige lexicon favoured by the writer who writes with agenda. In regards to a denotation of "God", "Brahman" the "Universe", "The Void", "Nirvana", "Pure Mind", "pure being", how has Thurman determined and established the signified of the said signifier? I respect Thurman, but all this is too open. Maybe I have too many critical tools and uncertainty is too apparent to me. I have contemplated at different times throughout my life each term Thurman has isolated as a gloss of the Absolute. I am confident his meaning of them, my meaning of them and yours dear reader are shared though very different. What is individuality in one analysis if not a colonization of the individual mind and experience of a person with cultural constructions and cultural tokens that are not native to the individual and are annexations from without? Memes, the discourse of memes. [[w:James Legge|Legge]] (p.47) rendered an extract of the purport to the commentary of [[w:I_Ching_hexagram_01#Hexagram_1|Hexagram 1]] "Heaven" ([http://www.afpc.asso.fr/wengu/wg/wengu.php?no=1&l=Yijing 乾 Ch'ien: 'The Creative']), ''[[w:I Ching|I Ching]]'', thus: <blockquote>"The great man is he who is in harmony in his attributes with heaven and earth; in his brightness with the sun and moon; in his orderly procedure with the four seasons; and in his relation to what is fortunate and what is calamitous with the spiritual agents."<ref>Source: [http://books.google.com.au/books?id=YhwagzNUydsC&printsec=frontcover&source=gbs_v2_summary_r&cad=0#v=onepage&q=great%20man%20is%20he%20who%20is%20in%20harmony&f=false] p.47</ref></blockquote> ==Ancient Traditions== [[File:Amun.svg|thumb|right|Amun the ineffable]] Studying [[w:Thomas McEvilley|McEvilley]] (2002) has in a way conveyed the possibility of how much significant cultural exchange and influence may have happened between Egyptian, Persian, Mesopotamian, Hellenic, Roman and Indian cultures, amongst other ancient peoples not herewith mentioned. People move around and their ideas, technologies and innovations travel like wildfire especially through the [[w:trade wind|watery winds]] of trade, currents of ocean and the [[w:silk road|earthy cloven trail]] and blistered foot. Ideas, technologies and innovations may also take place concurrently and without diffusion. McEvilley's work is the fruit of 30 years research. There are problems with it, but that said, there is no work without flaw and no work on the historicity of global spirituality could be informed without being aware of the matters to which McEvilley is conversant. In charting nonduality, I wouldn't have anticipated that I would include Egypt, Sumer and other non-Indian ancient cultures but following McEvilley (2002), I intuit that would be not only naive but grave omission. As I am going to chart nonduality, I have to also chart Monism as "monism" was how the term "advaita" was first translated into English. ===Classical Traditions=== Michaelson (2009: p.130) identifies what he perceives to be the origins of nondualism proper founded in the [[w:Neoplatonism|Neoplatonism]] of [[w:Plotinus|Plotinus]] within [[w:Ancient Greece|Ancient Greece]] and employs the ambiguous binary construction of "[[w:the West|the West]]" [as different to 'the East', refer [[w:Edward Said|Saïd]]'s utilization of the [[w:discourse|discourse]] of '[[w:Other|The Other]]' in ''[[w:Orientalism (book)|Orientalism]]'' (1978)]: <blockquote>"Conceptions of nonduality evolve historically. As a philosophical notion, it is most clearly found for the first time in the West in the second century C.E, in the Neoplatonism of Plotinus and his followers."<ref>Michaelson, Jay (2009). ''Everything Is God: The Radical Path of Nondual Judaism''. Shambhala Publications. {{ISBN|1590306716}}, 9781590306710. Source: [http://books.google.com.au/books?id=jmrgoeaQvlwC&pg=PA130&dq=i+and+thou+martin+buber+nondual&hl=en&ei=0I3iS_jIFdGgkQX50dHtDw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDYQ6AEwAA#v=onepage&q=i%20and%20thou%20martin%20buber%20nondual&f=false] (accessed: Thursday May 6, 2010), p.130</ref></blockquote> ===Advaita=== [[Image:Ramana 3 sw.jpg|thumb|right|Ramana Maharshi]] [[w:Advaita|Advaita]] (Sanskrit ''a'', not; ''dvaita'', dual) is a nondual tradition from [[India]], with [[w:Advaita Vedanta|Advaita Vedanta]], a branch of [[w:Hinduism|Hinduism]], as its philosophical arm. Advaita may be rendered in English as 'nondual', 'not-two' or 'peerless' and though there are monist themes in the most recent sections of the ancient ''[[w:Rig Veda|Rig Veda]]'' ([[w:Mandala 1|Mandala 1]] and [[w:Mandala 10|Mandala 10]]), that is, the sections that were finalized or [[w:Interpolation (manuscripts)|interpolated]] last; nonduality finds its first sophisticated exposition in the "[[w:Tat Tvam Asi|Tat Tvam Asi]]" of the venerable ''[[w:Chandogya Upanishad|Chandogya Upanishad]]'' (6.8.7)<ref>Raphael, Edwin (1992). ''The pathway of non-duality, Advaitavada: an approach to some key-points of Gaudapada's Asparśavāda and Śaṁkara's Advaita Vedanta by means of a series of questions answered by an Asparśin.'' Iia: Philosophy Series. Motilal Banarsidass. {{ISBN|8120809297}}, 9788120809291. Source: [http://books.google.com.au/books?id=P_u-4_kMfKwC&pg=PA94&lpg=PA94&dq=nonduality+chandogya&source=bl&ots=9XRtOCNSYG&sig=v6vGOC-sX7SK32sPy8PN_SQp5xc&hl=en&ei=65vVS7D5JZWXkQWegZWfDA&sa=X&oi=book_result&ct=result&resnum=9&ved=0CC0Q6AEwCA#v=onepage&q&f=false] (accessed: Tuesday April 27, 2010), p.Back Cover</ref>, an [[w:upanishad|upanishad]] favoured by subsequent proponents of Advaita Vedanta. [[w:Gaudapada|Gauḍapāda]] (c.600 CE) furthered this philosophical theory that was later consolidated by Sri [[w:Adi Shankara|Shankara]]charya in the 8th century CE. Most [[w:smartha|smartha]]s are adherents to this theory of nonduality. Further to this, Craig, ''et.al''. (1998: p.476) hold that the nonduality of the Advaita Vedantins is of the identity of [[w:Brahman|Brahman]] and the [[w:Ātman (Hinduism)|Atman]] where the identity is "objectless consciousness, as awareness nondualistically self-aware": <blockquote>Advaita Vedānta is a scripturally derived philosophy centred on the proposition, first found in early Upaniṣads (800-300 BC), that Brahman - the Absolute, the supreme reality - and the self (ātman) are identical. The identity is understood as an objectless consciousness, as awareness nondualistically self-aware. Arguments in support of the view that nondual awareness is the sole reality are developed by classical and modern Advaitins, from Gauḍapāda (c.600 AD) and Śaṅkara (c.700 AD), in hundreds of texts. Some of these are suggested in Upaniṣads.<ref>Craig, Edward (general editor) (1998). ''Routledge encyclopedia of philosophy: Luther to Nifo, Volume 6''. Taylor & Francis. {{ISBN|0415073103}}, 9780415073103. Source: [http://books.google.com.au/books?id=G3UBxqkkCX8C&pg=PA476&dq=ontolog++nondual&hl=en&ei=kufPS-X2J5GTkAWVuq2ACg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CD4Q6AEwAQ#v=onepage&q&f=false] (accessed: Thursday April 22, 2010), p.476</ref></blockquote> ===Vaishnava traditions of Advaita=== [[File:Chaitanya_sankirtan.jpg|thumb|Chaitanya and the Avadhuta Nityananda depicted leading sankirtana in medieval India]] [[w:Chaitanya Mahaprabhu|Chaitanya]] (c.1485-c.1533), a mystic saint of medieval Bengal in India, disseminated a form of monotheistic devotion to [[w:Krishna|Krishna]] that also suggested an admixture of monistic theism. For Chaitanya, Krishna is the sole supreme entity in the Universe, and all other conceptions of god are manifestations of Him, including the ineffable [[w:Brahman|Brahman]]. Advaita does not equal Advaita Vedanta. Advaita Vedanta is for the most part Shaivite or is often identified as having [[w:Shaivite|Shaivite]] leanings and is a specific tradition within the Sanatana Dharma but many other traditions within the Sanatana Dharma have Advaita or nondual aspects, cults, teachings and texts, the Vaishnavas and their [[w:Avadhuta|Avadhuta]] literature e.g. [[w:Uddhava Gita|Uddhava Gita]] and [[w:Hamsa Gita|Hamsa Gita]], the teachings of [[w:Ramanuja|Ramanuja]], etc. [[w:Vaishnava-Sahajiya|Vaishnava-Sahajiya]] cult, etc. <blockquote> "Modern-day Orissa was a place of rich ferment of varied religious and philosophical traditions between the 15th and the 17th century AD. The Buddhist Tantric tradition which prevailed between the 8th–11th century AD had a powerful influence on later Vaisnavism. Later Nāthism became popular on the soil. At various points of time Saivism and Sāktism also left their impressions on the Orissan land. Fourteenth to 16th century was a critical period in Orissan literary history where the language of literary Oriya matured. So also did its religious and spiritual tradition, growing into a rich amalgam of diverse faiths and spirits unified primarily by the cult of Lord Jagannātha at Puri. The poetry that emerged showed diverse influences and experiments in terms of both thoughts as well as forms. They also showed refinement in terms of assimilation of various religious and philosophical beliefs and resulted in a fairly evolved form of Vaisnavism that was unique to Orissa."<ref>Pattanayak, D.P.; Mohanty, J.M.; Chaturvedi, Namrata; Patnaik, Tandra (2009). 'Oriya Medieval Bhakti Poetry'. ''Muse India Literary eJournal'': May-June 2009. Source: [http://www.scribd.com/doc/18685740/ORIYA-MEDIEVAL-BHAKTI-POETRY] (accessed: Friday August 6, 2010)</ref> </blockquote> Simhadeba (1987: p.317) affirms that the first record of [[w:Jagannatha|Jagannatha]] in literature is by the [[w:Mahasiddha|Mahasiddha]] [[w:Indrabhuti|Indrabhuti]], the [[w:Vajrayana|Vajrayana]] adept of [[w:Buddhadharma|Buddhadharma]], in his famed work, the ''Jnana Siddhi'': <blockquote>"N. K. Sahu emphasises Indrabhuti, the Raja of Sambhal (Sambalpur), who has been assigned to have ruled during the dark period of the history of western Orissa by the author to have been the propounder of Vajrayana Buddhism. He was the first Siddha to identify Buddha with Jagannath and he was the worshipper of Jagannatha whom he prays at several places of his famous work Jnana Siddhi. In Sanskrit literature Jagannath as a deity is unknown before Jnana Siddhi, Jagannath mentioned in the Shantiparva of the Mahabharata of Bangabasi edition and even in early publication of Bangabasi edition is an interpolation as this is not found in the Poona edition of Kumbakonam edition ...".<ref>Simhadeba, Jitāmitra Prasāda (1987). ''Cultural profile of south Kōśala: from early period till the rise of the Nāgas and the Chauhans in 14th century A.D''. Gyan Books. {{ISBN|8121200954}}, 9788121200950. Source: [http://books.google.com.au/books?id=xEtVtSED0u0C&dq=jagannath+indrabhuti&source=gbs_navlinks_s] (accessed: Friday March 5, 2010)</ref></blockquote> *http://books.google.com.au/books?id=7LFzfbhmJcMC&pg=PA54&lpg=PA54&dq=orissa+vaishnava&source=bl&ots=UJ2v06DT5D&sig=KRXh2NtyzsaY7-CGVPdsICdjSL4&hl=en&ei=pQ5cTPHDG5aMvAO5k5maBQ&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCEQ6AEwAg#v=onepage&q=orissa%20vaishnava&f=false ====Ramanuja (c. 1017 – c. 1137 CE)==== [http://www.archive.org/search.php?query=subject%3A%22Ra%CC%84ma%CC%84nuja%2C+1017-1137%22 There are three texts on Ramanuja @ the Internet Archive] *svayamprakasata *[http://books.google.com/books?id=un1i5icdpdoC&pg=PA192&lpg=PA192&dq=luminosity+jivatman&source=bl&ots=bER7tppzD1&sig=edO7N7Gvrhn2SEN7z-tCHYPuTg8&hl=en&ei=-41WTaqXKY_GvQPPpYzgBA&sa=X&oi=book_result&ct=result&resnum=7&ved=0CDoQ6AEwBg#v=onepage&q=luminosity%20jivatman&f=false Jivatman as self-luminous p.193] [http://www.iep.utm.edu/ramanuja/ "According to Ramanuja, Brahman is the Self of all. However, this is not because our individual personhood is identical with the personhood of Brahman, but because we, along with all individuals, constitute modes or qualities of the body of Brahman. Thus, Brahman stands to all others as the soul or mind stands to its body. The metaphysical model that Ramanuja thus argues for is at once cosmological in nature, and organic. All individuals are Brahman by virtue of constituting its body, but all individuals retain an identity in contradistinction to other parts of Brahman, particularly the soul of Brahman."] ====Bhagavata Purana==== [http://www.sjcme.edu/profiles/faculty/Daniel-Sheridan Daniel P. Sheridan] has documented the Advaita tradition of the ''[[w:Bhagavata Purana|Bhagavata Purana]]''. Sheridan (1986: p.136) introduces the nondual or advaitic themes to the Bhagavata Purana thus: <blockquote> The ''Bhāgavata'', according to its own account, is a treatise on the meaning of Brahman (''āśraya''), which is the support and refuge of the devotees. All its other topics and characteristics are devoted to explaining the ramifications of Brahman. Brahman is above all the Supreme Deity, Bhagavān Kṛṣṇa. Yet in the very first verse of the ''Bhāgavata'', Brahman, the highest reality, is connected with the universe: "Him from whom is the creation, etc. of the universe." In the first canto Sūta declares the non-dual nature of this absolute reality: "Those who possess the knowledge of the Truth call the knowledge of non-duality as the Truth; it is called Brahman, the Highest Self and Bhagavān. The ''Bhāgavata'' thus clearly appeals to the non-dualist tradition of Vedānta as the framework for its assertions about the non-dual nature of the Absolute, who is identified with Bhagavān Kṛṣṇa.<ref>Sheridan, Daniel P. (1986). ''The Advaitic theism of the Bhāgavata purāṇa''. Motilal Banarsidass Publishers. {{ISBN|8120801792}}, 9788120801790. Source: [http://books.google.com.au/books?id=qrtYYTjYFY8C&pg=PA2&lpg=PA2&dq=sheridan+purana&source=bl&ots=NLlEhUvl4y&sig=ERngT8k07g_vWHRaDKYO4e_HNVs&hl=en&ei=5OdbTP79OouWvAPOnuWZBQ&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCMQ6AEwAg#v=onepage&q=sheridan%20purana&f=false] (accessed: Friday August 6, 2010)</ref> </blockquote> Śrīmad Bhāgavatam 3.27.11 :mukta-liṅgaṃ sad-ābhāsam :asati pratipadyate :sato bandhum asac-cakṣuḥ :sarvānusyūtam advayam [http://books.google.com.au/books?id=qrtYYTjYFY8C&printsec=frontcover&source=gbs_atb#v=onepage&q&f=false Sheridan renders this verse into English and discusses it contextually according to his experience.] ===Celtic Traditions=== ====Welsh==== I read ''[[w:The_Owl_Service|The Owl Service]]'' (1967), I happened upon it for 50 cents or thereodd and purchased it and from so doing with the serendipity of happenstance I found the following which smells nondual yet I will need to go deeper. My intuition never fails me in matters of research. I have learnt to trust in the process and get out of the way and let it unfold. <poem> Not of mother and father Did my Creator create me But of nine-formed virtues, Of the fruit of fruits, Of the fruit of the primordial God, Of primroses and blossoms, Of the flower, wood and tree. ''[[w:Cad Goddeu|Kat Godeu]]'' , Beauford source who rendered in English... </poem> Graves, following Nash, accepted that the poem is a composite of several different sections, among which he named a Hanes Taliesin (History of Taliesin) and a Hanes Blodeuwedd (History of [[w:Blodeuwedd|Blodeuwedd]]). The following is how the passage is rendered in the [[w:William Forbes Skene|Skene]] version, drawn from [[w:Book of Taliesin|Book of Taliesin]] within the [[w:Four Ancient Books of Wales|Four Ancient Books of Wales]] (1868)([http://www.sacred-texts.com/neu/celt/fab/index.htm complete @ Sacred Texts] though Mary Jones of the Celtic Literature Collective has provided a lovely free site with valuable annotations) : <poem> Not of mother and father, When I was made, Did my Creator create me.9 Of nine-formed faculties, Of the fruit of fruits, Of the fruit of the primordial God, Of primroses and blossoms of time hill, Of the flowers of trees and shrubs. Of earth, of an earthly course, When I was formed. Of the flower of nettles, Of the water of the ninth wave.<ref>http://www.maryjones.us/ctexts/t08.html</ref> </poem> Philip Shallcrass in 'A Priest of the Goddess' within Pearson & Samuel (1998) states: <blockquote>"Having conceived of a supreme, non-dual reality and dubbed it Brahman, Hindu philosophers realised it was not accessible to the human mind and personified it as Ishvara, Creator and Lord of the Universe, viewing all other gods as manifestations of aspects of Ishvara. The name I give to non-dual, ultimate reality is Celi, a Welsh word meaning 'creator', or 'deity'. I see each of us as containing a spark of this essential spirit, a spark that connects us with all the rest of creation, for every other being also contains this same undifferentiated spiritual essence. And here I refer not only to human beings, but to all our relations, animal, vegetable, mineral and spiritual. In common with many Pagans, I am a pantheist, seeing spirit in all things."<ref>Source: [http://books.google.com/books?id=rLYCOOUlsBcC&pg=PA159&lpg=PA159&dq=welsh+nondual+druid&source=bl&ots=mvv0vMgoeJ&sig=gUwTbrXOce7lHfC8nPASujSuR5A&hl=en&ei=JFIbTZcpx8JxgNPFiQo&sa=X&oi=book_result&ct=result&resnum=5&sqi=2&ved=0CDAQ6AEwBA#v=onepage&q&f=false] p.159</ref></blockquote> ===Orphic Mysteries=== *[[w:Thomas Taylor (neoplatonist)|Taylor]] (1758 - 1835) *[[w:Ivan Mortimer Linforth|Linforth]] (1879-1976) *[[w:Eric Robertson Dodds|Dodds]] (1893 – 1979) *[[w:W. K. C. Guthrie|Guthrie]] (1906 – 1981) I have secured primary resources as well as English translations. *[[User:B9_hummingbird hovering/Blog/Personal_Weblog/Nonduality/Dreams_(Orphic Hymn)]] Today I have remembered the primacy of poetics and I am contemplating having Nonduality and poetry as a theme. '''[[w:Orphism (religion)|Orphism]]''' (more rarely '''Orphicism''') ([[W:Ancient Greek|Ancient Greek]], "''Ορφικά''") is the name given to a set of religious<ref>Sexuality in Greek and Roman Culture (Ancient Cultures) by Marilyn B. Skinner,2005,page 135,"... of life, there was no coherent religious movement properly termed "Orphism" (Dodds 1957: 147-9; West 1983: 2-3). Even if there were, ..."</ref> *http://books.google.com.au/books?id=fkwkgUEZjrkC&pg=PA29&dq=orphic&ei=9l3sS62aGILMlAT5koiYCA&cd=10#v=onepage&q=orphic&f=false *http://books.google.com.au/books?id=-C6wNyrxUO8C&printsec=frontcover&dq=orphic&ei=E1_sS7HfCJTukgTdpNCdCA&cd=1#v=onepage&q&f=false *http://www.sacred-texts.com/cla/hoo/index.htm *http://books.google.com.au/books?id=Ba8wAAAAYAAJ&printsec=frontcover&dq=thomas+taylor+hymns+orpheus&source=bl&ots=Lw0vsPGSVO&sig=bZgfRuZEGLFgYDXem28qIhqDbtU&hl=en&ei=q1_sS7Vgj-zsA-6W9ZgG&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBwQ6AEwAQ#v=onepage&q&f=false *http://books.google.com/books?id=zGCqj2VkEoMC&printsec=frontcover&source=gbs_v2_summary_r&cad=0#v=onepage&q&f=false *http://books.google.com.au/books?id=qn4L6iboYesC&dq=Dodds+1957+greeks&source=gbs_navlinks_s *http://books.google.com.au/books?id=AGZssi0h0LgC&dq=linforth+orpheus&source=gbs_navlinks_s ===Yantra, Mandala, Rangoli=== [[File:Sri_Yantra_256bw.gif|thumb|left|[[w:Sri_Yantra|Sri Yantra]]]] <blockquote>"You have noticed that everything an Indian does is in a circle, and that is because the Power of the World always works in circles, and everything tries to be round. In the old days when we were a strong and happy people, all our power came to us from the sacred hoop of the nation, and so long as the hoop was unbroken, the people flourished. The flowering tree was the living center of the hoop, and the circle of the four quarters nourished it. The east gave peace and light, the south gave warmth, the west gave rain, and the north with its cold and mighty wind gave strength and endurance. This knowledge came to us from the outer world with our religion. Everything the Power of the World does is done in a circle. The sky is round, and I have heard that the earth is round like a ball, and so are all the stars. The wind, in its greatest power, whirls. Birds make their nests in circles, for theirs is the same religion as ours. The sun comes forth and goes down again in a circle. The moon does the same, and both are round. Even the seasons form a great circle in their changing, and always come back again to where they were. The life of a man is a circle from childhood to childhood, and so it is in everything where power moves. Our tepees were round like the nests of birds, and these were always set in a circle, the nation's hoop, a nest of many nests, where the Great Spirit meant for us to hatch our children."<ref>Source: [http://www.firstpeople.us/articles/Black-Elk-Speaks/Black-Elk-Speaks-At-First-Cure.html] (accessed: Friday May 28, 2010)</ref></blockquote> [[w:Krishna|Krishna]] said in the [[w:Bhagavad Gita|Bhagavad Gita]], Chapter 12, Verse 5, that it is much more difficult to focus on God as the unmanifested than God with form, due to human beings having the need to perceive via the senses.<ref>http://www.bhagavad-gita.org/Gita/verse-12-04.html</ref> [[w:Yantra|Yantra]] function as revelatory conduits of cosmic truths. Yantra, as instrument and spiritual technology, may be appropriately envisioned as prototypical and esoteric [[w:concept map|concept map]]ping machines or conceptual looms. Certain yantra are held to embody the energetic signatures of, for example, the Universe, consciousness, ''[[w:ishta-devata|shta-devata]]''. Though often rendered in two dimensions through art, yantra are conceived and conceptualised by practitioners as multi-dimensional [[w:sacred architecture|sacred architecture]] and in this quality are identical with their correlate the ''[[w:mandala|mandala]]''. [[w:Meditation|Meditation]] and [[w:trance|trance]] induction that generates the yantra of the [[w:subtle body|subtle body]] in the complementary modes of the ''[[w:Generation Stage|utpatti-krama]]'' and ''[[w:Completion stage|saṃpanna-krama]]'' are invested in the various lineages of tantric transmission as exterior and interior sacred architecture that potentiate the accretion and manifestation of ''[[w:siddhi|siddhi]]''. Khanna (2003: p.21) in linking [[w:Mantra|Mantra]], Yantra, Ishta-devata, and [[w:thoughtform|thoughtform]]s states: <blockquote> Mantras, the Sanskrit syllables inscribed on yantras, are essentially 'thought forms' representing divinities or cosmic powers, which exert their influence by means of sound-vibrations.<ref>Khanna, Madhu (2003). ''Yantra: The Tantric Symbol of Cosmic Unity. '' Inner Traditions. {{ISBN|0892811323}} & {{ISBN|978-0892811328}} </ref> </blockquote> Yantra is an [[w:aniconic|aniconic]] ''[[w:temenos|temenos]]'' or [[w:Tabernacle (disambiguation)|tabernacle]] of ''[[w:Deva (Hinduism)|deva]]'', ''[[w:asura|asura]]'', ''[[w:genius loci|genius loci]]'' or other archetypal entity. Yantra are [[w:theurgy|theurgical]] device that engender ''[[w:entelecheia|entelecheia]]''. Yantra are realised by ''[[w:sadhu|sadhu]]''s through ''[[w:darshana|darshana]]'' and ''[[w:samyama|samyama]]''. Yantra, or other permutations and cognate phenomena such as Mandala, [[w:Rangoli|Rangoli]], [[w:Kolam|Kolam]], [[w:Rangavalli|Rangavalli]] and other [[w:sacred geometry|sacred geometrical traditions]], are endemic throughout [[w:Dharmic Traditions|Dharmic Traditions]]. Some [[w:Tantra|Hindu esoteric practitioners]] employ yantra, mantra and other items of the ''[[w:saṃdhyā-bhāṣā|saṃdhyā-bhāṣā]]'' (Bucknell, ''et al.''; 1986: p.ix) in their ''[[w:sadhana|sadhana]]'', ''[[w:Puja (Hinduism)|puja]]'' and ''[[w:yajna|yajna]]'' <ref>Bucknell, ''et al.''; 1986: p. ix.</ref>. ===Alvars=== [[w:Nathamuni|Nathamuni]] (c.900-950)<ref>Bartley, C. J. (2002). ''The theology of Rāmānuja: realism and religion''. Routledge. {{ISBN|0700714596}}, 9780700714599. Source: [http://books.google.com.au/books?id=Og7MoH69mPUC&printsec=frontcover&dq=R%C4%81m%C4%81nuja&hl=en&ei=B92hS4aCCNGIkAWdlonYCA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDMQ6AEwAA#v=onepage&q=&f=false] (accessed: Thursday March 18, 2010), p.1</ref> was a [[w:Vaishnava|Vaishnava]] scholar who founded the [[w:Sri Vaishnava|Sri Vaishnava]] tradition. He is notable for collecting and codifying the 4,000 verses of the [[w:Alvars|Alvars]] known as the "[[w:Divya Prabandha|Naalayira Divya Prabhandham]]" in the [[w:Tamil Language|Tamil Language]]. Nathamuni systematized the Sri Vaishnava theology in Sanskrit after collecting the Tamil verses of the Alvars. He was engaged in significant active dialectic with Buddhadharma proponents. In the following Werner (1993: p.86) identifes the dynamic between the personal deity and the bhakta or devotee which are are ostensibly a dualism as there is required a separation to have a relationship though the mystery of the loving reciprocity unites and binds them in nonduality according to their devotion: <blockquote>Thus the supreme ideal of ''bhakti'' is a state of reciprocal love between Kṛṣṇa, who is the Lord himself, and the devotee who is in His power, a state in which each member of the loving relationship feels spiritually deficient and void without the other and find his fulfilment only in the other.<ref>Werner, Karel (1993). ''Love divine: studies in bhakti and devotional mysticism''. Issue 3 of Durham indological series. Routledge. {{ISBN|0700702350}}, 9780700702350. Source: [http://books.google.com.au/books?id=pYahlaJCLnYC&pg=PA213&dq=alvars+nondual&hl=en&ei=IIvwS4r9CYzk7AOg0eySBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CE4Q6AEwCA#v=onepage&q&f=false] (accessed: Monday May 17, 2010)</ref></blockquote> 'I am Thine', 'Thou art mine' and 'Thou I am', these are the developmental modes of devotion according to [[w:Madhusūdana Sarasvatī|Madhusūdana]], an authority on both nondualism and bhakti. It should be stated that this closely charts the mysticism of [[w:Martin Buber|Buber]] in ''[[w:I and Thou|I and Thou]]''. *http://books.google.com.au/books?id=8TwHhuZrZ-wC&pg=PA11&dq=alvars&hl=en&ei=vYjwS-jvE4vY7AOP88iLBg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CCwQ6AEwAQ#v=onepage&q=alvars&f=false * ===Yoga=== Whicher (2003) in Whicher and Carpenter (2003: pp.51-52) challenges the "dualistic" historical paradigm of Yoga scholarship founded in a separation of "[[w:puruṣa|puruṣa]]" and "[[w:prakṛti|prakṛti]]" thus: <blockquote>"It is often said [by Western scholarship] that, like classical Sāṃkha, Patañjali's yoga is a dualistic system, understood in terms of ''puruṣa'' and ''prakṛti''. Yet, I submit, yoga scholarship has not clarified what "dualistic" means or why yoga had to be "dualistic". Even in avowedly non-dualistic systems of thought such as Advaita Vedanta we can find numerous examples of basically dualistic modes of description and explanation."<ref>Whicher, Ian (2003). "The Integration of Spirit (Puruṣa) and Matter (Prakṛti) in the ''Yoga Sutra''"; cited in Whicher, Ian & Carpenter, David (2003). ''Yoga: the Indian tradition.'' Routledge. {{ISBN|0700712887}}, 9780700712885. Source: [http://books.google.com.au/books?id=HlPta3a38P0C&pg=PA5&dq=yoga+nondual&hl=en&ei=ffDjS4jgF4yOkQX03eT5Dw&sa=X&oi=book_result&ct=result&resnum=6&ved=0CFAQ6AEwBTgU#v=onepage&q&f=false] (accessed: Friday May 7, 2010), pp.51-52</ref></blockquote> Rājarshi (2001: p.45) conveys his estimation of the historical synthesis of the School of [[w:Yoga|Yoga]] (one of the six [[w:Āstika|Āstika]] schools of Sanatana Dharma) which he holds introduces the principle of "[[w:Isvara|Isvara]]" as [[w:Saguna Brahman|Saguna Brahman]], to reconcile the extreme views of [[w:Vedanta|Vedanta]]'s "advandva" and [[w:Sankya|Sankya]]'s "dvandva": <blockquote>"Introducing the special tattva (principle) called Ishvara by yoga philosophy is a bold attempt to bring reconciliation between the transcendental, nondual monism of vedanta and the pluralistic, dualistic, atheism of sankhya. The composite system of yoga philosophy brings the two doctrines of vedanta and sankya closer to each other and makes them understood as the presentation of the same reality from two different points of view. The nondual approach of vedanta presents the principle of ''advandva'' (nonduality of the highest truth at the transcendental level.) The dualistic approach of sankhya presents truth of the same reality but at a lower empirical level, rationally analyzing the principle of ''dvandva'' (duality or pairs of opposites). Whereas, yoga philosophy presents the synthesis of vedanta and sankhya, reconciling at once monism and dualism, the supermundane and the empirical."<ref>Swami Rājarshi Muni (2001). ''Yoga: the ultimate spiritual path''. Second edition, illustrated. Llewellyn Worldwide. {{ISBN|1567184413}}, 9781567184419. Source: [http://books.google.com.au/books?id=3XXxnxn8K8sC&pg=PA38&dq=anumana+three+types&hl=en&ei=CpTjS6urHpKekQXNmPQM&sa=X&oi=book_result&ct=result&resnum=5&ved=0CEIQ6AEwBDgK#v=onepage&q=anumana%20three%20types&f=false] (accessed: Friday May 7, 2010), p.45</ref></blockquote> ===Sikh dharma=== <blockquote> "Modern scholarship has demonstrated how Sikhism has--through the Sant tradition--been influenced by the devotionalism of Hindu (Vaiṣṇava) Bhakti, the hath-yoga of the Nāth tradition, and the mysticism of Sufism."<ref>Nayar, Kamala Elizabeth & Sandhu, Jaswinder Singh (2007). ''The socially involved renunciate: Guru Nanak's Discourse to the Nāth yogis''. SUNY Press: illustrated edition. {{ISBN|0791472132}}, 9780791472132. Source: [http://books.google.com.au/books?id=WTfKwGV6mBkC&pg=PA105&dq=sikhism+nonduality&hl=en&ei=AiZcTOTcLZCmvQPbuZj5AQ&sa=X&oi=book_result&ct=result&resnum=8&ved=0CFIQ6AEwBw#v=onepage&q&f=false] (accessed: Saturday August 7, 2010), p.105</ref> </blockquote> [[w:Sikhism|Sikhism]] is a monotheistic religion which holds the view of non-dualism. A principle cause of suffering in Sikhism is the ego (''ahankar'' in [[w:Punjabi language|Punjabi]]), the delusion of identifying oneself as an individual separate from the surroundings. From the ego arises the desires, pride, emotional attachments, anger, lust, etc., thus putting humans on the path of destruction. According to Sikhism the true nature of all humans is the same as God, and everything that originates with God. The goal of a Sikh is to conquer the ego and realize your true nature or self, which is the same as God's. An important nondual text which is held to be a guru by Sikhs is the ''[[w:Guru_Granth_Sahib|Guru Granth Sahib]]''. *Mandair, Arvind (2005). 'The Politics of Nonuality: Reassessing the Work of Transcendence in Modern Sikh Theology' in ''Journal of the American Academy of Religion'', September 2006, Vol. 74, No.3, pp. 646-673. *[[w:Harbhajan Singh Yogi|Yogi Bhajan]] ===Sant Mat=== '''[[w:Sant Mat|Sant Mat]]''' (Devanagari: संत मत) was a loosely associated group of teachers that became prominent in the northern part of the Indian sub-continent from about the 13th century. Theologically, their teachings are distinguished by an inward, loving devotion to a divine principle, and socially by an [[w:egalitarian|egalitarianism]] opposed to the qualitative distinctions of the Hindu [[w:caste|caste]] system, and to those between [[w:Hindus|Hindus]] and [[w:Muslims|Muslims]].<ref name=Woodhead>Woodhead, Linda & Fletcher, Paul. ''Religion in the Modern World: Traditions and Transformations'' (2001) pp.71-2. Routledge (UK) {{ISBN|0-415-21784-9}}"</ref> The sant lineage can be divided into two main groups: The northern group of Sants from the provinces of the [[w:Punjab region|Punjab]], ([[w:Rajasthan|Rajasthan]] and [[w:Uttar Pradesh|Uttar Pradesh]]), who expressed themselves mainly in [[w:vernacular|vernacular]] Hindi, and the southern group, whose language is archaic [[w:Marathi|Marathi]], represented by [[w:Namdev|Namdev]] and other Sants of [[w:Maharashtra|Maharashtra]].<ref name=Woodhead /> Schomer & McLeod (p.216-217): <blockquote>"In the Maharashtrian tradition, the Sants are not only thought of and referred to as Vaishnava bhaktas or Bhagavatas, but are specifically identified as Varkaris, the devotees of Lord Vitthala of Pandharpur. According to R.D. Ranade, the Sants of Maharashtra all belong to the Vitthala-''sampradāy'': "not that the followers of other sampradayas are not Sants, but the followers of the Varkari Sampradaya are Sants ''par excellence''." Furthermore, though the Sant tradition as a whole is sometimes thought of as propounding ''nirguṇa'' bhakti, i.e. bhakti having as its object the quality-less, invisible, all-pervading supreme Reality, this formulation is more appropriate to the northern Sants than to the Sants of Maharashtra. The Varkaris are to a certain extent ''saguṇa'' bhaktas in that they are devoted to at least one visible ''mūrti'', that of Vitthala, considered a ''svarūpa'' ('spontaneous manifestation') of the Godhead. It is indeed the popular cult of Vitthala that gives the Maharashtrian Sant tradition as a whole its characteristic Vaishnava flavor. The popular cult of Vitthala as a young cowherd boy merges into the cult of Krishna as cowherd, and Vitthala himself is identified with Krishna-Gopal. Ultimately, however, it is not the Vitthala icon that grants salvation, but devotion to the Name of God, the invisible, all-pervading Godhead. It is this ardent devotion to the divine Name which is the rallying point of all the Sants, both northern and southern. This is well exemplified in two ''sākhīs'' attributed to Kabir in the ''Ādi Granth'' collection of the 'sayings' (bānī) of the 'Bhagats'. The ''sākhīs'' present a brief dialogue between two Maharashtrian Sants -- Namdev (Nama), the tailor (''śimpi'') and his contemporary Trilochan: <poem> O Nama, ''māyā'' has deceived you said his friend Trilochan: Why do you keep printing cloth instead of meditating on Ram? Said Nama: O Trilochan, with your mouth, invoke Ram, With your hands and feet, do all your work, keeping your soul fixed on Niranjan. </poem> Such an utterance, put by Kabir in the mouth of his illustrious predecessor Namdev, points to the convergence of the spiritual attitudes of the northern Sants and the Sants of Maharashtra. Whatever their particular religious tradition, the Sants are seekers of the 'Pure' (''nirañjan''), the Absolute, a Godhead which transcends their own traditional allegiances. This spiritual attitude tends to blur not only the distinction between ''nirguṇa'' and ''sirguṇa'', but also the traditional distinction between Shaivism and Vaishnavism. In Maharashtra, as well as Gujarat and later on, in Karnataka, it is possible to follow step by step the gradual merging of the Shaiva faith into the nonsectarian Vaishnava bhakti of the Sants."<ref>Schomer, Karine & McLeod, W. H. (1987). The Sants: studies in a devotional tradition of India. Berkeley religious studies series. Motilal Banarsidass Publishers. {{ISBN|8120802772}}, 9788120802773 Source: [http://books.google.com/books?id=OkKhOivXrhgC&pg=PA217&lpg=PA217&dq=Obscure+Religious+Cults+by+Shashibhushan+Dasgupta&source=bl&ots=wKtu2AFx6-&sig=YTkMZ8GSKCXTihNN5GYDyj0c0qg&hl=en&ei=QDohTb6VJISkugPp3tH0DQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBEQ6AEwADgK#v=onepage&q&f=false] (accessed: Monday January 3, 2011)</ref></blockquote> *[[s:One_Hundred_Poems_by_Kabir|'One Hundred Poems' by Kabir translated by Rabindranath Tagore]] ===Kashmiri Shaivism=== *[http://www.scribd.com/doc/22588991/Tantric-Yoga-A-study-of-the-Vedic-precursors-historical-evolution-literatures-cultures-doctrines-and-practices-of-the-eleventh-century-Kasmiri-S Hartzell, James Francis (1997). Tantric Yoga: A Study of the Vedic Precursors, Historical Evolution, Literatures, Cultures, Doctrines, and Practices of the 11th Century Kaśmīri Śaivite and Buddhist Unexcelled Tantric Yogas. Thesis: Columbia University.] *[[w:Kashmir Shaivism|Kashmir Shaivism]] ====Abhinavagupta==== '''[[w:Abhinavagupta|Abhinavagupta]]''' ([[w:Kashmiri_language|Kashmiri_language]]: अभिनवगुप्त) (approx. 950 - 1020 AD<ref>Triadic Heart of Shiva, Paul E. Muller-Ortega, page 12</ref><ref>Introduction to the Tantrāloka, Navjivan Rastogi, page 27</ref>) was one of [[w:India|India]]'s greatest [[w:Indian philosophy|philosophers]], [[w:Mysticism|mystics]] and [[w:Aesthetics|aestheticians]]. He was also considered an important [[w:Music of India|musician]], [[w:Indian poetry|poet]], [[w:Theatre in India|dramatist]], [[w:exegesis|exegete]], [[w:theology|theologian]], and [[w:Indian logic|logician]]<ref name="Re-accessing Abhinavagupta page 4">Re-accessing Abhinavagupta, Navjivan Rastogi, page 4</ref><ref>Key to the Vedas, Nathalia Mikhailova, page 169</ref> - a [[w:polymath|polymath]]ic personality who exercised strong influences on Indian culture.<ref>The Pratyabhijñā Philosophy, Ganesh Vasudeo Tagare, page 12</ref><ref>Companion to Tantra, S.C. Banerji, page 89</ref> He was born in the [[w:Kashmir region|Valley of Kashmir]]<ref>Doctrine of Divine Recognition, K. C. Pandey, page V</ref> in a family of scholars and mystics and studied all the schools of philosophy and art of his time under the guidance of as many as fifteen (or more) teachers and [[w:guru|guru]]s.<ref>Introduction to the Tantrāloka, Navjivan Rastogi, page 35</ref> In his long life he completed over 35 works, the largest and most famous of which is [[w:Tantraloka|Tantrāloka]], an encyclopedic treatise on all the philosophical and practical aspects of [[w:Trika|Trika]] and [[w:Kaula|Kaula]] (known today as [[w:Kashmir Shaivism|Kashmir Shaivism]]). Another one of his very important contributions was in the field of philosophy of aesthetics with his famous [[w:Abhinavabharati|Abhinavabhāratī]] commentary of [[w:Natya Shastra|Nāṭyaśāstra]] of [[w:Bharata Muni|Bharata Muni]].<ref name="ReferenceA">Luce dei Tantra, Tantrāloka, Abhinavagupta, Raniero Gnoli, page LXXVII</ref> *http://ccbs.ntu.edu.tw/FULLTEXT/JR-PHIL/ew95314.htm *http://books.google.com.au/books?id=kQf2m8VaC_oC&lpg=PA272&dq=Introduction%20to%20the%20Tantr%C4%81loka&pg=PA225#v=onepage&q&f=false *http://books.google.com.au/books?id=EYWmc0gfYyIC&dq=Tantr%C4%81loka&lr=&source=gbs_navlinks_s *http://books.google.com.au/books?id=hMCFQ39cHNkC&printsec=frontcover&dq=kashmir%20shaivism&source=gbs_slider_thumb#v=onepage&q&f=false === Jain dharma === *[[w:Digambara|Digambara]] *Khajuraho Group of Monuments *Khajaraho thru [[w:Alain Daniélou|Daniélou]] (1907 – 1994) and the photography of his lover that brought the temples to the World ==== Buddha dharma ==== Though popular [[w:discourse|discourse]] both [[w:etic|etic]] and [[w:emic|emic]] as well as the discourse of [[w:scholarship|scholarship]] with which it intersects, employ the term "Buddhism" for the Buddhadharma (and often employ the term uncritically), it is salient to be mindful that the Buddhadharma is not a [[w:monolithic|monolithic]] tradition<ref>Robinson, B.A. (2008).'' A brief discussion of Buddhist traditions: East & West.'' Ontario Consultants on Religious Tolerance. Source: [http://www.religioustolerance.org/buddhism2.htm] (accessed: Thursday April 22, 2010)</ref> but a continuum of a number of sub-traditions and [[w:Praxis (process)|praxis]]-[[w:Lineage (Buddhism)|lineages]] (or [[w:sadhana|sadhana]]-lineages), many of which tout a number of nondualities proper in various sub-traditions and '[[w:Yana (Buddhism)|vehicles]]' (Sanskrit: yana); refer [[w:B. Alan Wallace|Wallace]] (2007: pp.106-107).<ref>Wallace, B. Alan (c2007). "Buddhist Nontheism, Polytheism, and Monotheism." Cited in: Wallace, B. Alan (2007). ''Contemplative science: where Buddhism and neuroscience converge.''The Columbia series in science and religion. Columbia University Press. {{ISBN|0231138342}}, 9780231138345. Source: [http://books.google.com.au/books?id=9efzkyBVzsQC&pg=PA107&dq=buddhism+monolithic+tradition&ei=_lXQS-LsO5LgkwT39_zuCA&cd=8#v=onepage&q=buddhism%20monolithic%20tradition&f=false] (accessed: Friday April 23, 2010), pp.106-107</ref> ====Nonduality as Shunyata and Prajna==== Huntington & Wangchen (1995: p.119) hold where 'emptiness' is a gloss of [[w:Shunyata|Shunyata]] (Sanskrit) and 'wisdom' is a gloss of [[w:Prajna|Prajna]] (Sanskrit): <blockquote>With the actualization of emptiness, manifest in wisdom as an effect, the bodhisattva gains access to the nondualistic knowledge of a buddha. It may be that this concept seems particularly abstruse because it is associated not so much with a way of ''knowing'' as with a way of ''being'', for we have seen the justification underlying claims to knowledge of this type is necessarily immersed in a certain form of life...a kind of nondualistic knowledge is present wherever a particular epistemic act is embedded in an intuitive awareness of the unique context through which two apparently discrete phenomena are intimately related, as is usually the case, for example, when we speak of a cause and its effect.<ref>Huntington, C. W. & Wangchen, Namgyal (1995). ''The Emptiness of Emptiness: An Introduction to Early Madhyamika.'' National Foreign Language Center Technical Reports. University of Hawaii Press. {{ISBN|0824817125}}, 9780824817121. Source: [http://books.google.com.au/books?id=lt1iMCbdXaIC&printsec=frontcover&source=gbs_navlinks_s#v=onepage&q&f=false] (accessed: Thursday April 22, 2010), p.119</ref> </blockquote> Further to the coincidence or nonduality of Shunyata and Prajna within the 'Pure-and-perfect-Mind' ({{bo|w=byang chub sems}}<ref>Dharma Dictionary (2008). "byang chub sems". Source: [http://rywiki.tsadra.org/index.php/byang_chub_sems] (accessed: Friday April 23, 2010)</ref>; Sanskrit: [[w:Bodhicitta|Bodhicitta]]), [[w:Herbert V. Guenther|Günther]] & [[w:Chögyam Trungpa|Trungpa]] (1975: p.30) state that: <blockquote>We cannot predicate anything of prajna except to say that when it is properly prajna it must be as open as that which it perceives. In this sense we might say that subjective and objective poles, (prajna and shunyata) coincide. With this understanding, rather than saying that prajna is shunyata, we can try to describe the experience by saying that it has gone beyond the dualism of subject and object.<ref>Guenther, Herbert V. & Trungpa, Chögyam (1975). ''The dawn of tantra.'' Routledge. {{ISBN|0877730598}}, 9780877730590. Source: [http://books.google.com.au/books?id=mUs9AAAAIAAJ&pg=PA30&dq=shunyata+prajna&hl=en&ei=YkfQS6vZDc6GkAXCmN0O&sa=X&oi=book_result&ct=result&resnum=2&ved=0CEEQ6AEwAQ#v=onepage&q=shunyata%20prajna&f=false] (accessed: Thursday April 22, 2010)</ref></blockquote> =====Buddhadharma (general)===== All schools of Buddhism teach No-Self ([[w:Pali|Pali]] [[w:anatta|anatta]], Sanskrit anatman). Non-Self in Buddhism is the Non-Duality of Subject and Object, which is very explicitly stated by the Buddha in verses such as “In seeing, there is just seeing. No seer and nothing seen. In hearing, there is just hearing. No hearer and nothing heard.” (Bahiya Sutta, [[w:Udana|Udana]] 1.10). Non-Duality in Buddhism does not constitute merging with a supreme Brahman, but realising that the duality of a self/subject/agent/watcher/doer in relation to the object/world is an illusion.<ref>{{cite web|url=http://ccbs.ntu.edu.tw/FULLTEXT/JR-AN/26715.htm|title=Enlightenment in Buddhism and Advaita Vedanta: Are Nirvana and Moksha the Same?|accessdate=2009-12-20}}</ref> Within the Mahayana presentation, the [[w:two truths|two truths]] may also refer to specific perceived phenomenon instead of categorizing teachings. Conventional truths would be the appearances of mistaken awareness - the awareness itself when mistaken - together with the objects that appear to it or alternatively put the appearance that includes a duality of apprehender and apprehended and objects perceived within that. Ultimate truths, then, are phenomenon free from the duality of apprehender and apprehended.<ref name="LevinsonAug06">Levinson, Jules (August 2006) ''[http://www.berotsana.org/pdf/lotsawa_timesII_sc.pdf Lotsawa Times Volume II]''</ref> In the [[w:Mahayana|Mahayana]] [[w:Buddhism|Buddhist]] canon, the [[w:Diamond Sutra|Diamond Sutra]] presents an accessible nondual view of "self" and "beings", while the [[w:Heart Sutra|Heart Sutra]] asserts [[w:shunyata|shunyata]] &mdash; the "emptiness" of all "form" and simultaneously the "form" of all "emptiness". The [[w:Lotus Sutra|Lotus Sutra]]'s parable of the Burning House implies that all talk of Duality or Non-Duality by Buddhas and Bodhisattvas is merely Skillful Means (Sanskrit [[w:upaya|upaya]] kausala) meant to lead the deluded to a much higher truth. The fullest philosophical exposition is the [[w:Madhyamaka|Madhyamaka]]; by contrast many laconic pronouncements are delivered as [[w:koan|koan]]s. Advanced views and practices are found in the [[w:Mahamudra|Mahamudra]] and [[w:Maha Ati|Maha Ati]], which emphasize the vividness and spaciousness of nondual awareness. Mahayana Buddhism, in particular, tempers the view of nonduality (wisdom) with respect for the experience of duality (compassion) &mdash; ordinary dualistic experience, populated with selves and others (sentient beings), is tended with care, always "now". This approach is itself regarded as a means to disperse the confusions of duality (i.e. as a path). In [[w:Theravada|Theravada]], that respect is expressed cautiously as non-harming, while in the [[w:Vajrayana|Vajrayana]], it is expressed boldly as enjoyment (especially in tantra). Williams & Tribe (2000, 2002: p.73): <blockquote> It was the Buddha who declared that ''karman'' is intention, a mental event. In so doing, Gombrich comments, the Buddha 'turned the brahmin ideology upside down and ethicised the [U]niverse. I do not see how one could exaggerate the importance of the Buddha's ethicisation of the world, which I regard as a turning point in the history of civilisation' (Gombrich 1996:51). Thus the Buddha turned attention from physical acts cleansing the pollution resulting from 'bad karma' --such as acts of physical asceticism, or the Brahmanic actions of purification, which typically involve washing, or ingesting 'the five products of the cow' --to 'inner purification', mental training.<ref>Source: [http://books.google.com.au/books?id=IDgZXlzxzRsC&dq=Williams+%26+Tribe+buddhist+thought&printsec=frontcover&source=bl&ots=MrjxpJk8PD&sig=T1bJN1bLk2ND46kWgPp7nsJpJec&hl=en&ei=Mk0hSpDLLYmMkAWx3ZSJBQ&sa=X&oi=book_result&ct=result&resnum=1#PPA73,M1] (accessed: Sunday May 31, 2009)</ref> </blockquote> [[File:Kolam outside a house in tamil nadu.jpg|thumb|left|Buddhadharma or Hindudharma? Symbols are so slippery]] Now, this is wonderful but also a misrepresentation. The ritual specialist of the Brahmanas were accomplished in interior visualization. Case in point, as a matter of course it was mandatory that there be one Brahmana who in the interior of his mind, they were all men to my knowledge, I have not known of a women in such ritual. But then I am sure there were women's traditions but I have not been privy to them. Though I will mention [[w:Rangoli|Rangoli]] etc. Now back to the point. The one Brahamana would visualize perfectly the whole ritual in the interiority of his mind at the time the ritual was simultaneously being enacted in the world by others. It was a failsafe. Also, understood that the 'dreaming' Brahman was dreaming the action into being. Somewhat like the dreaming Lord dreams the world into being and all activities and sentient beings. But it is understood in the tradition that the real ritual is the interior offering. That is the salient point. Actually, that is not quite it. The interior offering seals the exterior offering. No, the interior offering makes the outer offering a sacrament. That's better. Need to find quote. One does wonder how there was synchronization between the visualizer and the enactors, but then mythic time is timeless so maybe absolute synchrony was not required. ====Textual example of a nondualism in the literature of the Buddhadharma: Vimalakīrti Nirdeśa Sūtra==== [[w:Vimalakirti Sutra|Vimalakīrti Nirdeśa Sūtra]] ([[w:IAST|IAST]]) (Wei-mo-chieh ching) This text has nondual themes which it gives the Sanskritic nomenclature "Reconciliation of Dichotomies" (yamakavyatyastahara). *http://www.westernbuddhistreview.com/vol3/nonduality.html *http://books.google.com.au/books?id=n4K7p6XSpSYC&pg=PA6&lpg=PA6&dq=Yamakavyatyastahara&source=bl&ots=v3N-tAEtjr&sig=WpyEpwSAjho3caQaDBafbwv8gTQ&hl=en&ei=4EepTLmiOMaecauMkZ0N&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBQQ6AEwAA#v=onepage&q=Yamakavyatyastahara&f=false *Richard B. Mather. (1968). 'Virmalakirti and Gentry Buddhism'. ''History of Religions''. The University of Chicago Press. @JSTOR http://www.jstor.org/pss/1061746 *[[w:Burton Watson|Burton Watson]] (b.1925) ===Korean Buddhism=== Park (1983: p.&nbsp;147) identifies [[w:essence-function|essence-function]] as an East Asian Buddhist strategy to convey [[w:nonduality|nonduality]]: <blockquote>Since the ''t'i-yung or'' "essence-function" construction is originally used by East Asian Buddhists to show a non-dualistic and non-discriminate nature in their enlightenment experience, it should not exclude any other frameworks such as ''neng-so'' or "subject-object" constructions. Nevertheless the essence-function construction must be distinguished from the subject-object construction from a scholastic perspective because the two are completely different from each other in terms of their way of thinking.<ref>Park, Sung-bae (1983). ''Buddhist Faith and Sudden Enlightenment.'' SUNY series in religious studies. SUNY Press. {{ISBN|0873956737}}, 9780873956734. Source: [http://books.google.com.au/books?id=_A2QS03MP5EC&printsec=frontcover&dq=Sung-bae+Park&source=bl&ots=kc-DMX8UBI&sig=VcgHGoIDY_r_XjWPrxx68TRO21k&hl=en&ei=QxG_S5jqA86TkAW_lrTWBQ&sa=X&oi=book_result&ct=result&resnum=7&ved=0CCAQ6AEwBg#v=onepage&q&f=false] (accessed: Friday April 9, 2010), p.147</ref></blockquote> Park (2009: p.11) holds that: <blockquote>"...the terms ''mom'' and ''momjit'' are familiar to all Koreans, and have their roots in ancient history. Although I translated them in the introduction as "essence" and "function", a more accurate definition (and the one the Korean populace is more familiar with) is "body" and "the body's functions". The implications of "essence/function" and "body/its functions" are similar, that is, both paradigms are used to point to a nondual relationship between the two concepts. There is a subtle but crucial difference, however, between the two models, "essence/function" and "body/its functions". The term essence/function (which is often translated by East Asian scholars into the Chinese term t'i-yung) has a rather abstract, philosophical tone, connoting an impression of being somewhat removed from the nitty-gritty details of everyday life. My primary interest, however, is in the human being's personal understanding and experience of nonduality."<ref>Park, Sung-bae (2009). ''One Korean's approach to Buddhism: the mom/momjit paradigm''. SUNY series in Korean studies: SUNY Press. {{ISBN|0791476979}}, 9780791476970. Source: [http://books.google.com.au/books?id=TEAq0ldHjVYC&pg=PA11&dq=essence-function+nondual&hl=en&ei=wibkS7vzAY2gkQWNh6X7Dw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDcQ6AEwAA#v=onepage&q=essence-function%20nondual&f=false] (accessed: Saturday May 8, 2010), p.11</ref></blockquote> *I love the following Korean Tradition book on interpenetration just from the tutular icon NB: Avatamsaka, Korean Tradition, entwine with poetry metaphors and Frye, Frye entwines with Blake *http://books.google.com.au/books?id=FPeNprjfSzcC&pg=PA91&dq=The+World+Text+flower+garland&ei=8o_tS7vrEo2SkASEo4yMDA&cd=4#v=onepage&q=The%20World%20Text%20flower%20garland&f=false ====Vajrayana==== =====Yab-yum===== [[w:Rita M. Gross|Gross]] (2009: p.207) a leading [[w:Feminist theology|Feminist theologian]] identifies the nondual import of [[w:yab-yum|yab-yum]] iconography where His ever-so-skillful 'method' ([[w:upaya|upaya]]) really enjoys Her ever-so-spacious 'wisdom' ([[w:prajna|prajna]]), a wisdom where wisdom-in-[[w:reciprocity|reciprocity]] enjoys method; where His-Her enjoining is coincident in 'great bliss' (mahasukha): <blockquote>...a vital point must be made, especially given that the ''yab-yum'' image is always said to be an image in which the partners are in sexual union...[t]hough it may seem paradoxical and difficult to understand, this image, nevertheless, is not literally about sex, as in sexual intercourse. It is about nonduality, which is visually represented by the ''yab-yum'' icon.<ref>Gross, Rita M. (2009).'' A garland of feminist reflections: forty years of religious exploration.'' University of California Press. {{ISBN|0520255852}}, 9780520255852. Source: [http://books.google.com.au/books?id=_sYJrwQTfngC&pg=PA207&dq=Yab-Yum+nondual&hl=en&ei=V5TQS4WtGc2IkAXTjd0N&sa=X&oi=book_result&ct=result&resnum=2&ved=0CD4Q6AEwAQ#v=onepage&q&f=false] (accessed: Friday April 23, 2010), p.207</ref></blockquote> ====Indigenous Americans==== Burrus & Keller (2006: p.71-72) in their work of transdisciplinary theological colloquia, convey the casestudies of Indigenous Americans which sing-a-song of nondual gender and nondual biological sexual designation and the natural spectrum of possibility: <blockquote>However objective it may seem, even the scientific framework for defining the "two sexes" is a cultural construction. As Judith Butler has shown, the dominant American ideology of the body affirms the existence of two sexes, two genders, and two basic sexualities that are treated as naturally distinct. But biological sex is not ideologically independent of the other terms; our culture defines our genetics, object-oriented genital joining, and other gender practices in binary fashion in order to identify us dualistically as either male/masculine or female/feminine (where "normal" males and females are heterosexual). Violations of these norms are deemed unnatural. So doctors have tended to define genetic sex dualistically, as XX or XY, and to label violations of the genetic dualism (such as XXY and XO people), including "mismatches" between genetics, hormones, and appearance, as "diseased." But as Anne Fausto-Sterling describes, there is a spectrum of such deviations, naturally occurring bodies with non-dual genital combinations and diverse physicals expressions. Hidden among the males and females living in America are so-called "true hermaphrodites," who possess both ova and testes, "genetically male" (XY) people with Androgen Insensitivity Syndrome who look like and are usually raised as women, "genotypically female" (XX) children whose genitalia are virilized at puberty, and "genotypically male" (XY) children who are anatomically female or androgynous at birth but at puberty develop testes, a fused scrotum, and secondary male sex characteristics.<ref>Burrus, Virginia & Keller, Catherine (2006). ''Toward a theology of eros: transfiguring passion at the limits of discipline.'' Transdisciplinary theological colloquia. Fordham University Press. {{ISBN|0823226360}}, 9780823226368. Source: [http://books.google.com.au/books?id=CJWsZxyVzFgC&pg=PA72&dq=navajo+nondual&hl=en&ei=lsDQS72KN4GUkAXsyeQE&sa=X&oi=book_result&ct=result&resnum=2&ved=0CD4Q6AEwAQ#v=onepage&q&f=false] (accessed: Friday April 23, 2010), pp.71-72</ref></blockquote> Though the inclusion of nondual bodies, genders and sexual designations and other biological florescence, are by definition qualified for inclusion in this article and such inclusion is [[:wikt:rarefied|rarefied]], especially when understood as as embodying a syncretic and wholistic ideal, a "a one-sex/body, multi-gender model that reflected ancient gender norms" and which is metaphorically apt in many spiritual nondual traditions as Burrus & Keller (2006: p.71) state: <blockquote>...the dominant ideology of the body in the premodern West was a one-sex/body, multi-gender model that reflected ancient gender norms for the distribution of power. Only with the rise of Western medicine and genetics has sex been conceived as dual and ontologically stable--male and female.<ref>Burrus, Virginia & Keller, Catherine (2006). ''Toward a theology of eros: transfiguring passion at the limits of discipline.'' Transdisciplinary theological colloquia. Fordham University Press. {{ISBN|0823226360}}, 9780823226368. Source: [http://books.google.com.au/books?id=CJWsZxyVzFgC&pg=PA72&dq=navajo+nondual&hl=en&ei=lsDQS72KN4GUkAXsyeQE&sa=X&oi=book_result&ct=result&resnum=2&ved=0CD4Q6AEwAQ#v=onepage&q&f=false] (accessed: Friday April 23, 2010), p.71</ref></blockquote> ===Dzogchen=== ====Introduction==== [[w:Dzogchen|Dzogchen]] is a relatively esoteric tradition to date but I have been engaged in bringing its traditionally closed discourse into the open in a way that is respectful of the tradition and its transmission restrictions. Times change and I hold that the discourses of nonduality are key for the growth and harmony of our global peoples. Dzogchen is concerned with the "natural state", and emphasizing direct experience. This tradition is found in the [[w:Nyingma|Nyingma]] tradition of [[w:Tibetan Buddhism|Tibetan Buddhism]], where it is classified as the highest of this lineage's [[w:nine yanas|nine yanas]], or vehicles of practice. Similar teachings are also found in the non-Buddhist [[w:Bön|Bön]] tradition, where it is also given the [[w:nomenclature|nomenclature]] "Dzogchen" and in one evocation the ninth in a nine vehicle system. The nine vehicles in both the Bonpo and Buddhadharma traditions are different but they mutially inform. In Dzogchen, for both the Bonpo and Nyingmapa, the primordial state, the state of nondual awareness, is called ''[[w:rigpa|rigpa]]''. The Dzogchen practitioner realizes that appearance and emptiness are inseparable. One must transcend dualistic thoughts to perceive the true nature of one's pure mind. This primordial nature is clear light, unproduced and unchanging, free from all defilements. One's ordinary mind is caught up in dualistic conceptions, but the pure mind is unafflicted by delusions. Through meditation, the Dzogchen practitioner experiences that thoughts have no substance. Mental phenomena arise and fall in the mind, but fundamentally they are empty. The practitioner then considers where the mind itself resides. The mind can not exist in the ever-changing external phenomena and through careful examination one realizes that the mind is emptiness. All dualistic conceptions disappear with this understanding.<ref>{{cite book|last=Powers|first=John|title=Introduction to Tibetan Buddhism|pages=334–342|publisher=Snow Lion Publications|year=1995}}</ref> *http://www.scribd.com/doc/38639434/Mahayoga-Texts-at-Tun-Huang =====Ground-of-being===== [[w:Ground of Being (Dzogchen)|'Ground [of Being]']] ({{bo|t=གཞི|w=gzhi}})<ref>Longchen Rabjam (author); Barron, Richard (translator, annotations) (1998). ''The Precious Treasury of the Way of Abiding''. (NB: English rendering of'' [[w:Nelug Dzö|:Nelug Dzö]]''). Padma Publishing, p.288</ref> (pronounced: zhi) is an essential [[w:cultural token|cultural token]] of the [[w:Dzogchen|Dzogchen]] tradition of both the [[w:Bonpo|Bonpo]]<ref>Rossi, Donatella (1999).'' The Philosophical View of the Great Perfection in the Tibetan Bon Religion''. Ithaca, New York: Snow Lion. {{ISBN|1-55939-129-4}}, p.52</ref> and the [[w:Nyingmapa|Nyingmapa]].<ref>Dudjom Rinpoche and Jikdrel Yeshe Dorje (1991). ''The Nyingma School of Tibetan Buddhism: its Fundamentals and History''. Two Volumes. Translated and edited by Gyurme Dorje with Matthew Kapstein. Wisdom Publications, Boston. {{ISBN|0-86171-087-8}}, p.354 Index of Technical Terms</ref> It is a seminal conceptual point and focus of [[w:praxis (process)|praxis]] foregrounded in the Dzogchen literature (particularly the [[w:Seventeen Tantras|Seventeen Tantras]]) and [[w:sadhana|sadhana]] (Sanskrit) [[w:Lineage (Buddhism)|lineages]] and may be apprised as a [[w:memetic|memetic]] conduit for the [[w:mindstream|continuum[-of-being<nowiki>]</nowiki>]] to enter into the concept-less Dzogchen [[w:nondual|nondual]] 'awareness', '[[w:rigpa|rigpa]]' (Wylie: rig pa; [[IAST]]: vidyā)<ref>Dudjom Rinpoche and Jikdrel Yeshe Dorje (1991). ''The Nyingma School of Tibetan Buddhism: its Fundamentals and History''. Two Volumes. Translated and edited by Gyurme Dorje with Matthew Kapstein. Wisdom Publications, Boston. {{ISBN|0-86171-087-8}}, p.336 Index of Technical Terms</ref>, Dzogchen-''as-process'' where the praxis albeit 'natural' (Wylie: lhan skyes; [[IAST]]: [[w:sahaja|sahaja]])<ref>Kvaerne, Per (1975). "On the Concept of Sahaja in Indian Buddhist Tantric Literature". First published in: ''Temenos XI'' (1975): 88-135. Cited in: Williams, Jane (2005). ''Buddhism: Critical Concepts in Religious Studies'', Volume 6. Routledge. {{ISBN|0415332265}}, 9780415332262. Source: [http://books.google.com.au/books?id=Ypsz9qEzZjwC&pg=PA162&lpg=PA162&dq=lhan-skyes+sahaja&source=bl&ots=8im5401SiF&sig=gPp8fLN-DbwBKIzf-mbLORGUq84&hl=en&ei=xra-S5T2LY6OkQWeitjdBQ&sa=X&oi=book_result&ct=result&resnum=9&ved=0CB4Q6AEwCDgK#v=onepage&q&f=false] (accessed: Friday April 9, 2010), p.162</ref> and 'effortless' (Wylie: lhun grub; [[IAST]]: anābhoga)<ref>Karmay, Samten Gyaltsen (2007). ''The Great Perfection (rDzogs chen): A Philosophical and Meditative Teaching of Tibetan Buddhism''. Second Edition. Volume II. Boston: Brill's Tibetan Studies Library. {{ISBN|9789004151420}}. Source: [http://books.google.com.au/books?id=vyS71NvZPXsC&pg=PA51&lpg=PA51&dq=lhun+grub+effortless&source=bl&ots=47lHFtiPZm&sig=LvvPK83yBZjFJvyUX-9mWA2Os78&hl=en&ei=xrK-S4yBJ4yTkAXe5PzTBQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0CBUQ6AEwAw#v=onepage&q=lhun%20grub%20effortless&f=false] (accessed: Friday April 9, 2010), p.51</ref> has the sense of 'spontaneity'.<ref>Keenan, John P. (1989). "Spontaneity in Western Martial Arts -- A Yogācāra Critique of ''Mushin'' (No-Mind)". Source: [http://www.nanzan-u.ac.jp/SHUBUNKEN/publications/jjrs/pdf/302.pdf] (accessed: Friday April 9, 2010), p.287</ref><ref>Muller-Ortega, Paul Eduardo (1989). ''The triadic Heart of Śiva: Kaula tantricism of Abhinavagupta in the non-dual Shaivism of Kashmir.'' SUNY series in the Shaiva traditions of Kashmir. SUNY Press. {{ISBN|0887067867}}, 9780887067860. Source: [http://books.google.com.au/books?id=ObmRkyVbgDUC&pg=PA52&lpg=PA52&dq=sahaja+spontaneity&source=bl&ots=87-AQPnlC4&sig=M6CuZPyDhStEQ3f6FnjmEMCPfwA&hl=en&ei=FsK-S-K6LcGLkAWyy7zmBQ&sa=X&oi=book_result&ct=result&resnum=10&ved=0CCkQ6AEwCQ#v=onepage&q=sahaja%20spontaneity&f=false] (accessed: Friday April 9, 2010), p.52</ref> The [[w:Gankyil|Gankyil]] is the polysemic teaching tool employed in the Dzogchen tradition to iconographically signify the triune of the Ground, a symbol of primordial [[w:nonduality|nonduality]]. [[Image:Sam Taeguk.jpg|thumb|right|180px|[[w:Gankyil|Gankyil]]]] The Gankyil is the polysemic teaching tool employed in the Dzogchen tradition to iconographically signify the triune of the Ground, a symbol of primordial nonduality. Throughout the [[w:Seventeen Tantras|Seventeen Tantras]], the principal tantras of the Nyingma Dzogchen doctrinal [[w:view (Buddhism)|view]] on the [[w:Sugatagarbha|Sugatagarbha]] ''[[:wikt:qua|qua]]'' 'Ground' ({{bo|w=gzhi}}), the triune of 'essence' ({{bo|w=ngo bo}}), 'nature' ({{bo|w=rang bzhin}}) and 'power' ({{bo|w=thugs rje}}) is foregrounded. Where '''''essence''''' is openness or emptiness ({{bo|w=ngo bo stong pa}}), '''''nature''''' is luminosity, lucidity or clarity (as in the [[w:luminous mind|luminous mind]] of the [[w:Five Pure Lights|Five Pure Lights]]) ({{bo|w=rang bzhin gsal ba}}) and '''''power''''' is universal compassionate energy ({{bo|w=thugs rje kun khyab}}), unobstructed ({{bo|w=ma 'gags pa}}).<ref>Petit, John Whitney (1999). ''Mipham's Beacon of Certainty: Illuminating the View of Dzochen, the Great Perfection''. Boston: Wisdom Publications (1999). {{ISBN|0861711572}}. Source: [http://books.google.com.au/books?id=lJcLSietnfcC&pg=PA78&lpg=PA78&dq=kun+khyab+unobstructed&source=bl&ots=V23Z0umnDC&sig=JhKDeOgxctVWz7yezP5rLZMWcG8&hl=en&ei=TCgFSqipCY6W6wOuh8mPAw&sa=X&oi=book_result&ct=result&resnum=1] (accessed: Saturday May 9, 2009), p.78-79</ref> Goodman & Davidson (1992: p.14) render the triune of the Ground as '[[:wikt:facticity|facticity]]' ({{bo|w=ngo bo}}), '[[:wikt:actuality|actuality]]' ({{bo|w=rang bzhin}}) and '[[:wikt:resonance|resonance]]' ({{bo|w=thugs rje}}) and in so doing place this esoteric cultural token of Dzogchen-as-praxis within the wider technical language of contemporary philosophical discourse in the English: <blockquote>"Process-oriented ''rdzogs-chen'' has as its pivot the notion of ''gzhi'' which means both ground (the static, sort of steady-state) and reason (the dynamic, the intensity with which the unfolding of the initial pure potential occurs). As such pure potential (''gzhi ka-dag chen-po'') it is discussed in terms of a triune dynamics, referred to as facticity (''ngo-bo''), actuality (''rang-bzhin''), and resonance (''thugs-rje''). This English rendering of highly technical terms constantly employed in the original Tibetan sources has been chosen in order to avoid any essentialist associations, so much more so as the texts themselves repeatedly state that ''ngo-bo'' (facticity) has nothing to do with nor can even be reduced to the (essentialist) categories of substance and quality; that ''rang-bzhin'' (actuality) remains open-dimensional, rather than being or turning into a rigid essence despite its being what it is; and that ''thugs-rje'' (resonance) is an atemporal sensitivity and response, rather than a distinct and narrowly circumscribed operation."<ref>Goodman, Steven D. & Davidson, Ronald M. (1992). ''Tibetan Buddhism: reason and revelation''. SUNY series in Buddhist studies. SUNY Press. {{ISBN|0791407853}}, {{ISBN|9780791407851}}. Source: [http://books.google.com.au/books?id=L_M8Aa9WVlIC&printsec=frontcover&dq=%22gzhi%20dus%20rig%20pa%22&source=gbs_slider_thumb#v=onepage&q&f=false] (accessed: Saturday May 1, 2010), p.14</ref></blockquote> In their annotations to this paragraph, Goodman & Davidson (1992: p.147) identify that they draw the sense of 'resonance' from the work of [[w:Erich Jantsch|Jantsch]] (1975)<ref> Jantsch, Erich (1975). ''Design for Evolution''. New York: Brazillier, pp.42 f.</ref> and further define thus: <blockquote>"...it is resonance (''thugs-rje'') with its fluctuations as high-level excitation (''rig-pa'') and low-level excitation (ma-rig-pa)--in cognitive terms: understanding (''rtogs'') and lack of understanding (''ma-rtogs'')--that stochastically determines the final outcome of the process."<ref>Goodman, Steven D. & Davidson, Ronald M. (1992). ''Tibetan Buddhism: reason and revelation''. SUNY series in Buddhist studies. SUNY Press. {{ISBN|0791407853}}, 9780791407851. Source: [http://books.google.com.au/books?id=L_M8Aa9WVlIC&printsec=frontcover&dq=%22gzhi%20dus%20rig%20pa%22&source=gbs_slider_thumb#v=onepage&q=facticity&f=false] (accessed: Saturday May 1, 2010), p.147</ref></blockquote> [[w:Herbert V. Günther|Günther]] (1984) provides a definition and discussion of facticity in relation to the Dzogchen Ground.<ref>Guenther, Herbert (1984).'' Matrix of Mystery: Scientific and Humanistic Aspects of rDzog-chen Thought''. Boulder, CO {{ISBN|0877732914}}</ref> I have yet not sourced a suitable definition of facticity in relation to my experience of the Essence of the Base. ===Ngakpa tradition=== Caplan (2009: p.163), with an indirect quotation, conveys her understanding of the view of a contemporary [[w:Ngakpa|Ngakpa]] who holds duality and nonduality to be nondual: <blockquote>"Ngakpa Chögyam, a Tibetan Buddhist teacher from Wales, offers a perspective on nonduality that includes all of life as a direct expression of the nondual core of truth. He explains that nonduality, or emptiness, has two facets: one is the empty, or nondual, and the other is form, or duality. Therefore, duality is not illusory but is instead one ''aspect'' of nonduality. Like the two sides of a coin, the formless reality has two dimensions -- one is form, the other is formless. When we perceive duality as separate from nonduality (or nonduality as separate from duality), we do not engage the world of manifestation from a perspective of oneness, and thereby we fall into an erroneous relationship with it. From this perspective it is not "life" or duality that is maya, or illusion; rather, it is our relationship to the world that is illusory."<ref>Caplan, Mariana (2009). ''Eyes Wide Open: Cultivating Discernment on the Spiritual Path''. Sounds True. {{ISBN|1591797322}}, 9781591797326. Source: [http://books.google.com.au/books?id=cDX8zNEbUXoC&pg=PA232&dq=transpersonal+psychology+nonduality&hl=en&ei=eDHgS7nnA4ugkQWj6snCCA&sa=X&oi=book_result&ct=result&resnum=10&ved=0CGcQ6AEwCQ#v=onepage&q=transpersonal%20psychology%20nonduality&f=false] (accessed: Wednesday May 5, 2010), p.163</ref></blockquote> ====Bonpo Dzogchen==== [[w:Svabhava|Svabhava]] (Sanskrit; Wylie: rang bzhin) is very important in the [[w:nontheistic|nontheistic]] [[w:theology|theology]] of the Bonpo Dzogchen 'Great Perfection' tradition where it is part of a technical language to render macrocosm and microcosm into [[w:nonduality|nonduality]], as Rossi (1999: p.58) states: <blockquote>"The View of the Great Perfection further acknowledges the ontological identity of the macrocosmic and microcosmic realities through the threefold axiom of Condition (ngang), Ultimate Nature (rang bzhin) and Identity (bdag nyid). The Condition (ngang) is the Basis of all (kun gzhi)--primordially pure (ka dag) and not generated by primary and instrumental causes. It is the origin of all phenomena. The Ultimate Nature (rang bzhin) is said to be unaltered (ma bcos pa), because the Basis [gzhi] is spontaneously accomplished (lhun grub) in terms of its innate potential (rtsal) for manifestation (rol pa). The non-duality between the Ultimate Nature (i.e., the unaltered appearance of all phenomena) and the Condition (i.e., the Basis of all [kun gzhi]) is called the Identity (bdag nyid). This ''unicum'' of primordial purity (ka dag) and spontaneous accomplishment (lhun grub) is the Way of Being (gnas lugs) of the Pure-and-Perfect-Mind [byang chub (kyi) sems]."<ref>Rossi, Donatella (1999). ''The Philosophical View of the Great Perfection in the Tibetan Bon Religion''. Ithaca, New York: Snow Lion. {{ISBN|1-55939-129-4}}, p.58</ref></blockquote> ===Zen=== [[File:DogenP2.JPG|thumb|[[w:Dogen|Dogen]]]] [[w:Zen|Zen]] is a non-dual tradition. It can be considered a religion, a philosophy, or simply a practice depending on one's perspective. It has also been described as a way of life, work, and an art form. Zen practitioners deny the usefulness of such labels, calling them, "The finger pointing at the moon." [[w:Tozan|Tozan]], one of the founders of [[w:Soto|Soto]] Zen in China, had a teaching known as the Five Ranks of the Real and the Ideal, which points out the necessity of not getting caught in the duality between Absolute and Relative/Samsara and Nirvana, and describes the stages of further transcendence into fully realising the Absolute in all activities. Nondual themes are very strong in the literary work of [[w:Dogen|Dogen]] (1200 - 1253).<ref>Foshay, Toby Avard (1994). "Denegation, Nonduality, and Language in Derrida and Dogen." ''Philosophy East and West.'' Volume 44, Number 3. July 1994, pp.543-558. Source: [http://ccbs.ntu.edu.tw/FULLTEXT/JR-PHIL/toby.htm] (accessed: Thursday May 6, 2010)</ref> *Lotus Sutra thru Dogen *http://www.thezensite.com/ZenEssays/DogenStudies/The_Lotus_Sutra_as_a_Source.html *Shobogenzo, and particularly Discourse of the Mountains etc... and Deep Ecology... theme of nonduality: subject as context *http://books.google.com.au/books?id=AmKE2xIjOwcC&pg=PA33&lpg=PA33&dq=buddhanature+monad&source=bl&ots=R_S2XB2OHe&sig=JQWbxnPeQuAY4Jby_l57clVJhR0&hl=en&ei=X4f1S9GtM5Dc7AP16en6BQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBYQ6AEwAA#v=onepage&q&f=false ==Indigenous traditions== ===Diné/Navajo=== [[Image:Catlin - Dance to the berdache.jpg|thumb|left|Detail of ''Dance to the Berdashe'', painted by [[w:George Catlin|George Catlin]]]] Burrus & Keller (2006: p.73) further to the greater cultural context of mainland America and the diverse [[w:two-spirit|two-spirit]] cultures of the Indigenous American peoples, convey the spiritual view of the [[w:Navajo_people|Diné or Navajo]] peoples in relation to the ideal that "all humans were spiritually androgynous": <blockquote>...eighteenth- and nineteenth-century Navajo had a three-sex, multigender system that included the ''nádleehí'', a "two-spirit" (bi-gender) person who had one of three anatomical birth-sexes (male, female, or androgynous), but was identified by a combination of masculine and feminine gender-attributes. Because Native Americans typically thought birth sex matured over time and defined gender primarily based on work preference, "two-spirit" people included non-dually sexed persons; born-males who adopted women's work, manners, and speech patterns; born-females who took up men's work and mannerisms; or those born either male or female who combined elements of women and men's cultural roles. Finally, the Navajo did not denounce the ''nádleehí'' as unnatural because gender or sex practices did not fit an individual's birth-sex; rather, they thought that all humans were spiritually androgynous, so they treated the ''nádleehí'' as a special but natural gender.<ref>Burrus, Virginia & Keller, Catherine (2006). ''Toward a theology of eros: transfiguring passion at the limits of discipline.'' Transdisciplinary theological colloquia. Fordham University Press. {{ISBN|0823226360}}, 9780823226368. Source: [http://books.google.com.au/books?id=CJWsZxyVzFgC&pg=PA72&dq=navajo+nondual&ei=A8rQS5TrGozolQTtwpD2CA&cd=2#v=onepage&q&f=false] (accessed: Friday April 23, 2010), p.73</ref></blockquote> ===South Pacific=== For e.g. Hawai'i, Samoa, Tahiti and New Zealand, etc. ====Hawai'i==== [[w:Ho'oponopono|Ho'oponopono]] *http://www.facebook.com/editprofile.php?sk=philosophy&success=1#!/notes/sage-mahosadha/hooponopono-practice-nonduality-and-my-new-years-gift-to-the-world/10150386731140457 *to affirm this edit as I hadn't logged in I had to enter the text "teamocean". sublime ==Will-o'-the-wisp of Wayfinding: Process Thought as orientation within Systems Theology== Tuesday, May 18, 2010: I awoke with the [[w:hypnagogia|hypnagogia]] of the names [[w:Alfred North Whitehead|Whitehead]] (1861 – 1947) and [[w:Károly Kerényi|Kerényi]] (1897 – 1973) resonant as sound-forms in my mind. I interpret this as direct instruction that they are to be included in this exploration. Both names are familiar to me by the way but I have not encountered them in any specifically nondualist mode of thought, nor am I familiar with either of their body of work nor inquiry to my knowledge so this is curious. I found the following quotation by Berthrong (1998: p.149) on Whitehead and nonduality which makes reference to a conclusion in the work of Loy (1988), a text to which I do no have reference so this is wonderful: <blockquote>One of the provocative features of Western process thought has been its apparent ability to link modern philosophy, science, and technology. What is more, it also claims that these linkages would include material taken from outside of the classical North Atlantic intellectual world. While there is nothing novel in trying to link philosophy and theology in the praxis of classical, medieval, or even early modern Western thought, there is something subversive about the mixture of non-Western and Western material in the eyes of many contemporary philosophers. But if Whitehead expresses a nondual vision of reality, such as discussed in great detail by David Loy (1988) in terms of Hindu, Buddhist, and Taoist Asian thought, then the conjunction of theology and philosophy is not hard to understand. Loy argues that all nondual philosophies are closely allied to soteriology because the typical nondual strategy is not only to thematize how we describe and perceive reality but to suggest that we can overcome the errors of dualistic thinking. Most nondual thinkers, according to Loy, develop methods for human beings to deconstruct their false perceptions of reality. Better thinking is one key to better living. Whitehead also argued that one task of philosophy was to help us live better and better still.<ref> H. (1998). ''Concerning creativity: a comparison of Chu Hsi, Whitehead, and Neville''. SUNY series in religious studies: SUNY Press. {{ISBN|0791439437}}, 9780791439432. Source: [http://books.google.com.au/books?id=BTI-muUjztEC&pg=PA149&lpg=PA149&dq=whitehead+nondual&source=bl&ots=EvJ3XBGRFA&sig=utH63NsPoAgjaXTLiH46OuSfAFg&hl=en&ei=yEnyS4GxIIvq7AOXqtGBDA&sa=X&oi=book_result&ct=result&resnum=10&ved=0CD8Q6AEwCQ#v=onepage&q=whitehead%20nondual&f=false] (Tuesday, May 18, 2010)</ref></blockquote> There was an uncited paragraph in ''[[w:Process_philosophy|Process_philosophy]]'' (May 18, 2010) thus: <blockquote>Whitehead enumerated three essential natures of God. The primordial nature of God consists of all potentialities of existence for actual occasions, which Whitehead dubbed eternal objects. God can offer possibilities by ordering the relevance of eternal objects. The consequent nature of God prehends everything that happens in reality. As such, God experiences all of reality in a sentient manner. The last nature is the superjective. This is the way in which God’s synthesis becomes a sense-datum for other actual entities. In some sense, God is prehended by existing actual entities.</blockquote> I don't understand these terminologies but I find it pointed given Valentinus' triune, Allogenes' triune, the Gankyil and now Whitehead's there is import here. A resolved system: a monad yet a trinity, as a triune. Different aspects of that which is aspectless: divisions within the indivisible. I need to find more of this triune of Whitehead in context so I understand his lexicon. Just as point of entry and as interpretation and attribution, I am going to venture an unfounded conjecture as to what this paragraph is pointing: *First, the primordial nature the Being, is the wellspring of the potentiality of all that was, is and will be, it is hidden and unknowable. *Second, the second nature is "consequent" upon the first, it is Becoming, that is it emanates from the First which is not lesser nor decreased from the emanation and there is no real division between the First and Second as they are monad. The Second constitutes what appears to our experience, indeed is the stuff of our experience-of-god. Experience or consciousness qua sentience is prehended by God as our sentience is of God though the First as hidden and unknowable is untouched by sentience. *Third, is the sentient-being who though Third is as synthesis of First and Second. Nonduality-in-communion apart fromwhich is the dual. *http://books.google.com.au/books?id=rfPexVxxOVUC&printsec=frontcover&dq=John+Berthrong&source=gbs_book_similarbooks#v=onepage&q&f=false Wilmot (1979: p.53) interprets and conveys his understanding of Whitehead's conception of an "entity" a category of which God in Whitehead's lexicon is reputed to partake and the discourse upon which Wilmot draws is ''[[w:Process_and_Reality|Process and Reality]]'': <blockquote>Inasmuch as God is an "actual entity" of a unique kind it will be well to review, however briefly, the structure of an actual entity. In his analysis of this category, Whitehead identifies three phases of the epochal process of becoming. In the first place, each actual entity arises out of the "objectifications" in the actual world relative to it as its "efficient" cause or causes, and, in this sense, "has the character 'given' for it by the past." But each specific process of concrescence arises by virtue of the "subjective aim" at "satisfaction" which it receives from God as the principle of concretion, and has the subjective character which constitutes its "final cause" or "lure." And thirdly, it has what Whitehead described as its "superjective" character, by which it achieving its "satisfaction" (i.e., completing its own process of becoming) it becomes new data, qualifying the transcendent creativity in successive concretions, and remains as an element in the content of creative purpose. This means that actualities beyond that satisfied superject will be enriched or impoverished accordingly.<ref>Wilmot, Laurence F. (1979). ''Whitehead and God: prolegomena to theological reconstruction''. Canadian electronic library. Wilfrid Laurier University Press. {{ISBN|088920070X}}, 9780889200708. Source: [http://books.google.com.au/books?id=FrIEEz1RKd4C&pg=PA53&lpg=PA53&dq=Process+and+Reality+whitehead&source=bl&ots=ryQetAvKJH&sig=K2IUaLkYe_Wts36vC6kljVdaTrw&hl=en&ei=CaLyS-qRMoH66QPQxbmODA&sa=X&oi=book_result&ct=result&resnum=11&ved=0CFYQ6AEwCg#v=onepage&q=Process%20and%20Reality%20whitehead&f=false] (accessed: Wednesday May 19, 2010), p.53</ref></blockquote> So in the first sentence of the paragraph abovecited, God is identified as an entity. But the entity-hood of God though being an "entity" is not being discussed as it is "unique". In this instance only "entity" is being discussed and though this has import for God there is an implication or entailment of a qualification that is not stated ''per se''. That is comprehended from the complexity of the embedded clauses within the first sentence: the relationship of which may be misapprehended if the reader is inattentive. Though God is an entity the entity-hood of God is "unique" in contradistinction to other entities and as such is "transcendent", "exclusive" but also "unitive" and "inclusive": which all partake of the semantic field of "unique". An "entity" which is a unitive structure is also parsed into "three phases". The term "epochal" is important which qualifies "process" as though process in general implies gradation or tiers, epochal identifies the procession of time-events. 'The entity arises out of "objectifications" in the actual world relative to it' is an interesting technical lexicon which partakes of the discourse of embedded causality and systemic arising. Personality is a "character 'given' by the past" conditioning. The "final cause" or "lure" smacks of entelechy. The "superjective" as no longer subject as it has fulfilled its purpose or charge God-given, qualifies the transcendent with the "datum" of lived-experience. I find it very interesting that the system from which the "Superject" arises may be both "enriched" and "impoverished" through the onflow effects the fruit, actions or qualities of "satisfaction"-''qua''-entelechy-fulfilled. Now, this may be all heresay and interpretation of Whitehead and of no value, but then maybe not. As we do not have the direct words of Whitehead: unless the words in double quotation marks within Wilmot are inferred as those of Whitehead. Though extrapolated out of context and without definitive lexical key it is all representation approaching misrepresentation. But then it is still at best analysis and most pristine, representation. But it is definitely an exercise in process thought whatever that is. Now, I intuit Process Thought holds stock in the synthesis of divergent and dissonant narratives and discourses. The value of this exploration as partaking of such discourse synthesizing divergence is what is denoted by the English term "nondual" and its inflections: deconstrucing as it is being constructed, embodying as it is disembodied: historicity ahistorical: there is value in Process Thought in the global field of play where the different geographies of texts conjoin in fertile muddy terrain and orient -- mutually inform and iterate -- and disorient -- confuse and confound. But there is the [[w:Will-o'-the-wisp|will-o'-the-wisp]] of [[w:wayfinding|wayfinding]] of Systems Theology by implication of the systemic arising of text within context. In furtherance of that, I proffer that all texts are embedded in context. Text becomes context: context becomes text. Intertextuality as deixes, indices and interstices. Context is always open as text: open in the denotation of no closure, no certainty, no definitive understanding. The ambiguity of openness is the boon/bane of signification. Context contextualizes the closure of text but never succeeds. The closure of every text is extrapolation. Nay, conception as definitive closure is interpolation. The narrative discourse and [http://everything2.com/title/Wheels+within+wheels wheels within wheels] of the ''Arabian Nights''. There is value in confusion for the Human as this is a fertile opportunity in the process of prolem-solving for the establishment of new neural connectivity. *http://books.google.com.au/books?id=dGpi0doazb0C&pg=PA242&lpg=PA242&dq=stein+whitehead&source=bl&ots=S4AGPi2JoP&sig=B21WtEjFLkp850vQgpVcDz337uk&hl=en&ei=J8LzS8pRj-jsA-6YzY8M&sa=X&oi=book_result&ct=result&resnum=5&ved=0CCgQ6AEwBA#v=onepage&q=stein%20whitehead&f=false ==Nonduality as psychoactively induced experience== The following was extracted from Wikipedia and is just as a springboard and will be rewritten: In 1968, [[w:R. Gordon Wasson|Wasson]] proposed that ''A. muscaria'' was the ''[[w:Soma|Soma]]'' talked about in the [[w:Rig Veda|Rig Veda]],<ref>Wasson, ''Soma:Divine Mushroom of Immortality'', p 10.</ref> which received widespread publicity and popular support at the time.<ref name=Letcher145>Letcher, p 145.</ref> He noted that descriptions of ''Soma'' omitted description of roots, stems or seeds, which suggested a mushroom,<ref>Wasson, ''Soma:Divine Mushroom of Immortality'', p 18.</ref> and used the adjective ''hári'' "dazzling" or "flaming" which the author interprets as red.<ref>Wasson, ''Soma:Divine Mushroom of Immortality'', p 36–37.</ref> One line described men urinating ''Soma''; this recalled the practice of recycling urine in Siberia. Soma is mentioned as coming "from the mountains", which Wasson interpreted as being brought with the Aryan invaders from the north.<ref>Wasson, ''Soma:Divine Mushroom of Immortality'', p 22–24.</ref> However, Indian scholars Santosh Kumar Dash and Sachinanda Padhy noted that both the eating of mushrooms and drinking of urine were proscribed, using as a source the ''[[w:Manusmṛti|Manusmṛti]]''.<ref name=Letcher146>Letcher, p 146.</ref> In 1971, Vedic scholar John Brough from Cambridge University rejected Wasson's theory; he noted the language was too vague to determine a description of Soma.<ref>{{cite journal|author=Brough, J.|title=Soma and ''Amanita muscaria''|journal=Bulletin of the School of Oriental and African Studies (BSOAS)|volume=34|year=1971|pages=331–62}}</ref>. In his 1976 survey, ''Hallucinogens and Culture'', anthropologist Peter T. Furst evaluated the evidence for and against the identification of the Fly Agaric mushroom as Vedic Soma, concluding cautiously in its favor. <ref>{{cite book |title=Hallucinogens and Culture |last=Furst |first=Peter T.|year=1976|publisher=Chandler & Sharp |location= |isbn={{ISBN|0-88316-517-1}} |pages=96–108}}</ref> [[File:Venus von Willendorf 01.jpg|thumb|Venus von Willendorf]] I feel it important to have this section as important to enter the discourse. The relationship of [[w:Amanita muscaria|Amanita muscaria]] and the ''[[w:Venus of Willendorf|Venus of Willendorf]]'' that iconically resembles the fungi has recently been published and well-received. Stephen R. Berlant has suggested a possible connection with a mushroom cult, based on visual similarities between the figurine and typical young ''Amanita muscaria'' mushrooms, a natural [[w:psychotrope|psychotrope]].<ref>Stephen R. Berlant, "The Prehistoric Practice of Personifying Psychotropic Mushrooms". The Journal of Prehistoric Religion, Vol. XIII, 1999, pp. 22-29.</ref> The ''The Sacred Mushroom and the Cross'' (1970) was first printed in the the heyday of the hippy and flowerpower [[w:counterculture|counterculture]] and its reception by the majority of his peers destroyed [[w:John_Marco_Allegro|Allegro]]'s academic career. In November 2009 it was reprinted in a 40th anniversary edition with among other inclusions, a 30 page addendum by [[w:Carl A. P. Ruck|Ruck]] of Boston University with new linguistic evidence that supports Allegro's theories. Ruck states in that addendum that: <blockquote>"The concerted and biased attempts to destroy Allegro's discoveries have failed. The confirmatory evidence is mounting in his favor. The critics can now raise their voices again. Let us hope that they do, since the matter is not settled, but they should be advised to do so with more careful consideration. This book that many have prized in secret is now available again. It demands the serious consideration of theologians, mythologists, and students of religion. No account of the history of the Church, both West and East, can afford to leave the poor despicable fungus unconsidered, nor the role that entheogens in general have played in the evolution of European civilization."</blockquote> *[http://books.google.com.au/books?id=fXHf8qdq9kYC&printsec=frontcover&dq=Judith+Anne+Brown&source=bl&ots=SYykiEAd8j&sig=Qai4HTrTHhKshENWXuIIURg5R5o&hl=en&ei=NwDsS6LSJ47U7AOM--SLBg&sa=X&oi=book_result&ct=result&resnum=8&ved=0CDYQ6AEwBw#v=onepage&q&f=false] *http://books.google.com.au/books?id=J3LN5mBxYj4C&pg=RA3-PA179&dq=The+Sacred+Mushroom+and+the+Cross&hl=en&ei=tv7rS87NApDi7AOvu7ihBg&sa=X&oi=book_result&ct=result&resnum=6&ved=0CEEQ6AEwBQ#v=onepage&q=The%20Sacred%20Mushroom%20and%20the%20Cross&f=false The [[w:Beat Generation|Beat Generation]], [[w:Allen Ginsberg|Ginsberg]] (1926 – 1997) was an initiate into the Ganachakra and though I have not found a source to demonstrate this, it is untenable that he was not. He was a disciple of [[w:Chögyam Trungpa|Trungpa]] (1939 – 1987) (who was an initiating adept in the Kagyupa and Nyingmapa traditions, in both these traditions the Ganachakra is a central rite): <blockquote>"This submission to Trungpa Guru, this Surrender is it correct, a transfer of God failure debasement to a living Being so at least the Adoration Devotion is to a Real Entity not an Image--And Merwin's War is it mine, 'gainst the vulgar drunken Guru Sangha?--Am I a fool? (Notebooks and journals)"<ref>Source: [http://books.google.com.au/books?id=0ovGKWrrePAC&pg=PA157&dq=Ginsberg+vajrayana&hl=en&ei=BQ7sS9b5MYro7AOPr-WaBg&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDcQ6AEwAw#v=onepage&q=Ginsberg%20vajrayana&f=false] (accessed: Friday May 14, 2010)</ref></blockquote> The Ganachakra rite with its sacrament of alcohol amongst other offerings (particularly sexual union which was later transmuted from an outer rite to an inner visualized one so those with monastic vows did not break samaya, but the Ganachakra was the fruit of [[w:Anuyoga|Anuyoga]] and the [[w:Mahasiddha|Mahasiddha]] traditions), with its iconoclastic antinomian ethos founded in the doha-poetic Mahasiddha tradition would have resonated with him. After an accident Trungpa gave back his monastic vows which was quite common in the indigenous Nyingma tradition which was for the most part householder and lay, or non-monastic and non-authoritarian though there were monastic traditions post rise of the Sarma. Trungpa drank, smoked and fucked as and whom he pleased like a true Mahasiddha. Conner, Sparks, Sparks, Mariya (1997:p.160) hold that: <blockquote>Ginsberg, Allen (1926-) Gay poet of Russian Jewish heritage, luminary of the Beat Movement and the Hippie Movement, now one of the most honoured poets of the US. Best known for his 1956 epic poem Howl, he has been deeply inspired by the writings of the English mystical poet William Blake (1757-1827) and by Tibetan Buddhism, as well as by periods of vagabondage and visionary experiences triggered by peyote. Actively involved in the anti-war movement during the US's conflict with Vietnam, he composed, in 1966, "Wichita Vortex Sutra," as an incantation meant to aid in ending the war. In a 1972 interview with gay activist and writer Allen Young, Ginsberg described a ceremony of UNION which he had celebrated privately with poet Peter Orlovsky in 1954 in a cafeteria in downtown San Francisco. "We made a vow to each other...to do everything we wanted to, sexually or intellectually, and in a sense explore each other until we reached the mystical 'X' together, emerging two merged souls. We had the understanding that when our...erotic desire was ultimately satisfied by being satiated (rather than denied), there would be a lessening of desire, grasp...craving and attachment; and that ultimately we would both be delivered free in heaven together. And so the vow was that neither of us would go into heaven unless we could get the other one in - like a mutual Bodhisattva's vow... So we held hands, took a vow: I do, I do, you promise? yes, I do. At that instant we looked in each other's eyes and there was a kind of celestial fire that crept over us and blazed up and illuminated the entire cafeteria and made it an eternal place." In this interview, Ginsberg also spoke of his concern that the emerging gay liberation movement was not addressing certain issues, among them aging, death, sexual and/or relationship addiction: "An element in the gay lib struggle and metaphysics that I don't think has been taken up is that of disillusionment with the body. I'm not trying to be provocative in that - just the age-old realization of...[the] grinning skeleton, with the spiritual lesson behind it, of detachment from neurotic desire. I think there's a genuine eros between men that isn't dependent on neurotic attachment and obsession, that's free and light and holy and lambent...If there's too much of a neurotic grasping to gaiety, to gayness, even to gay lib, then it makes everything too tense, and the lightness of love is lost. So the gay lib movement will have to come to terms sooner or later with the limitations of sex." Typical of Ginsberg's (proto-) Queer-Spiritual poetry is his beautifully composed "Elegy for Neal Cassidy" (1968), in which he chants: "Tender Spirit, thank you for touching me with tender hands / Sir spirit, give me your blessing again, Sir Spirit forgive my phantom body's demands, / Sir Spirit thanks for your kindness past."<ref>Conner, Randy P.; Sparks, David Hatfield; Sparks, Mariya (editors) & Anzaldua, Gloria E. (wrote Forward) (1997). ''Cassell's Encyclopedia of Queer Myth, Symbol, and Spirit: Gay, Lesbian, Bisexual, and Transgender Lore''. London, U.K.: Cassell. {{ISBN|0304337609}}., p.160.</ref> </blockquote> Nepal, is one of the few remaining cultures where both shamanic and tantric techniques are still alive and in full practice today as a native tradition. The result of eighteen years of field research, [[w:Claudia Müller-Ebeling|Müller-Ebeling]], [[w:Christian Rätsch|Rätsch]] & Shahi (2002)<ref>[[w:Claudia Müller-Ebeling|Müller-Ebeling, Claudia]] and [[w:Christian Rätsch|Christian Rätsch]] and Surendra Bahadur Shahi (2002). ''Shamanism and Tantra in the Himalayas''. Transl. by [[w:Annabel Lee (artist)|Annabel Lee]]. Rochester, Vt.: Inner Traditions International.</ref> presents for the first time a comprehensive overview of Himalayan shamanism that is based on the knowledge and experience of the different tribes from that region. Included are original statements from the various ethnic groups and reproductions of 135 color [[w:thangkas|thangkas]], which function as visual guides to the specific practices of the tantric tradition. In addition to the thangkas, the work is generously illustrated with numerous photographs of different shamanic healing ceremonies, ritual objects, and culturally significant plants that have never been published before. The book also contains a wealth of original recipes, smoking mixtures, scientific tables, charts, and descriptions of more than 20 plants whose psychoactive properties and uses by shamans have never before been researched or documented. Dowman & Downs (1985) in their English translation and annotation to a work attributed to medieval work attributed to Abhayadatta clearly identify the relationship of Ganachakra and medicine: <blockquote>"This is to exclude the homeopathic, alchemical ''bdud rtsi chos sman'' and similar herbal panaceas distributed by the Lamas during Long-life (T. ''tshe-grub'') and Ganacakra (''tshogs 'khor'') rites, because their efficacy depends as much upon sympathetic magic as upon the potency of the constituent herbal and other organic and non-organic substances."<ref>Abhayadatta (author), Keith Dowman (translator, annotator) & Hugh R. Downs (editor, annotator) (1985). ''Masters of Mahāmudrā: songs and histories of the eighty-four Buddhist siddhas''. SUNY series in Buddhist studies: SUNY Press. {{ISBN|0887061583}}, 9780887061585. Source: [http://books.google.com.au/books?id=okPNx4B1eoEC&pg=PR10&dq=mahasiddha&hl=en&ei=fBXsS_n5OYzm7AP0jeSPBg&sa=X&oi=book_result&ct=result&resnum=6&ved=0CDwQ6AEwBQ#v=onepage&q=ganacakra&f=false] (accessed: Friday May 14, 2010)</ref></blockquote> <blockquote>"The broad definition of entheogen used here is: "plants or substances capable of producing visionary experiences which are used for magico-religious or psychospiritual purposes." The use of entheogens in the Vajrayana tradition has been documented by such scholars as Ronald M Davidson, William George Stablein, Bulcsu Siklos, David B. Gray, Benoytosh Bhattacharyya, Shashibhusan Das Gupta, Francesca Fremantle, Shinichi Tsuda, David Gordon White, Rene de Nebesky-Wojkowitz, James Francis Hartzell, Edward Todd Fenner, Ian Baker, Dr. Pasang Yonten Arya and numerous others. The research of these scholars has established that these plants were definitely used in Vajrayana (within limited contexts) and that they were used in a manner largely consistent with their use in Saivite and shamanic traditions."<ref>Source: [http://people.tribe.net/sahajananda/blog/db107f43-2404-4e61-87c2-1e5b29b5d9da] (accessed: Saturday May 15, 2010)</ref></blockquote> ==Alchemy== *George Herbert (1593-1633) 'The Elixir' https://tspace.library.utoronto.ca/html/1807/4350/poem974.html ==Nonduality: householder/lay, monastic/cloistered, hermetic/solitary retreat == <blockquote>“Everyone strives after the law,” says the man, “so how is that in these many years no one except me has requested entry?” The gatekeeper sees that the man is already dying and, in order to reach his diminishing sense of hearing, he shouts at him, “Here no one else can gain entry, since this entrance was assigned only to you..."</blockquote> ::[http://records.viu.ca/~johnstoi/kafka/beforethelaw.htm Franz Kafka, 'Before the Law' rendered into English by Ian Johnston] This section is interested how the discourse of nonduality has informed and qualified the social experience of the Human. Historically culturally specific examples in different times and places. Gender, sexuality, sex. Vows: taking vows and giving them back. ===Sacred/Secular=== [[w:Sacred-profane_dichotomy|Sacred-profane_dichotomy]] [[w:Émile Durkheim|Durkheim]] (1858 – 1917) Eliade [http://books.google.com.au/books?id=zBzzv977CLgC&dq=eliade+sacred+and+the+profane&printsec=frontcover&source=bn&hl=en&ei=0t_zS5frBpPm7AOo4p2YDA&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDAQ6AEwAw#v=onepage&q&f=false The Sacred and the Profane] Is what is secular to some sacred? Is what is sacred to others secular? Is this dichotomy real? Who determines what is sacred? How is what is sacred determined? Are there varieties and gradation of the sacred and holiness? Is there a field of related elements that is profane in all cultures? profane as human cultural universal? *S. S. Acquaviva, Patricia Lipscomb The Decline of the Sacred in Industrial Society. *http://books.google.com.au/books?id=domdhtsnQt0C&dq=sacred+secular&printsec=frontcover&source=in&hl=en&ei=tcDzS522BoHY7APOnoCHDA&sa=X&oi=book_result&ct=result&resnum=12&ved=0CFkQ6AEwCw#v=onepage&q&f=false *Morality, immoral, amoral *left handed / right handed *human cultural universals *situation-specific morality as different to rules... *rules are for those who lack awareness like young children... do adults need the imposition of a colonizing morality? whose morality is it then? is a morality contextual? morality as approprateness and inappropriateness... *time, place and circumstance... *morality in nondual traditions is often the subject of contentation with dualist traditions... puritanical/tyrannical *morality as function of Jungian Shadow *shamefaced as virtue (this is counter-intuitive in English)... shame as positive experience ===Human aspects=== People have different personalities and propensities and through living form unique though in large-part often shared worldviews. People change and their needs and aspirations and relationship to their world and their spirituality changes throughout their life. This section is an ongoing meditation on nonduality in different evocations in different ways of being Human. Essentially this is in regards to the social and societal aspects of the Human intersect with "nondual" (nondual as experience, teaching, nondual as traditions). A person, being Human participates in a particular family, genetic heritage, kinship system, friendship networks, employ and trade, information and knowledge exchange, etc. Certain people choose to congregate, whilst others choose otherwise. Some people do not choose but there are advantages or influences which determine which choice is appropriate. Some people may not have a choice, a compulsion. Why are these choices made in context? What are nondual spiritual disciplines or practices done in private or solitary? What are nondual spiritual disciplines enacted in community or in public? 'Fuga mundi' (Latin): "Flight from the world", an aspect of desert and/or monastic spirituality that stresses the monastic life as a separation from the cares and concerns of the secular life. see also Extra mundum nulla salus. [http://books.google.com.au/books?id=jduLeQnA2doC&pg=PA53&dq=fuga+mundi+latin&hl=en&ei=bbDzS-CwI4Ge6gPMv42ODA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDEQ6AEwAQ#v=onepage&q&f=false] What is secular life? Is secular in contradistinction to sacred? The sacred/secular distinction is informed and challenged by the discourse of nonduality. Many "Westerners" who buy packaged nondual spirtuality, customized and priced for their consumption. Now the language choice was not dispassionate but provocative with purpose. Marketing exploits duality. Here is something that is valuable or desirable or worthwhile and you do not have it but we may furnish it. So there is a fundamental tension between the marketing of nonduality. Marketing is a study in duality and the manipulation of perceived lack. There is an exploitation whether intentional, overt or covert of people's need or desire for exoticism. This exploits a duality in Westerners who through the modern demands of contemporary society feel very "secular" as viery different and other to that which is "sacred". A prestige is given by the "secular" personage to the "sacred" personage? Is this duplicitous? I rarely use the term "exploit" and "dominant" but they need to be dissected I feel. Devil's Advocate. ===Householder life=== The Mahasiddha and Ngakpa provide historical examples of nonduality in both families and in trade, "in the world" and otherwise. Nyingma had strong householder traditions indeed, the strength of the Nyingma due to their political decentralization and non-monastic institutionalization in certain times of their history was a demonstrable strength. Many transmission lineages held by monastic institutions would not be unbroken if it was not for householder lineages. Nondual texts in the Mahayana were very popular with householders, they may be the product of householders or may have been propagated by householder lineages and communities hence their transmission through time. Nondual traditions and teachings hold salience for the contemporary world and its challenges and innovative responses to the Human Condition. Most people in the world live in what we call householder situations and householder is just as valid just as holy as any other situation. That said, most householders would consider a monastic as de facto holier than a non-monastic which just isn't defensible. ===Institutional life=== What is an institution? Institutionalized Zen & Gelukpa. Monasteries. Tradition as institution. Institution as a place of shelter that has specific power structures and specializations. Institution as community. Institution as closed community. Institution as open community. Who control access to institutions? What bonds people in an institution? *http://en.wikipedia.org/wiki/SWOT_analysis I imagine that cloisted life would be rich, a hothouse opportunity and nexus of efflorescence of culturally specific cultural tokens, etc. People being people. Some people would be close and intimate, others distanced and estranged. There is a social-security and reciprocity in Institution. Distribution of work and specialization of engagement. Institution becomes fixed with traditions.Tradition facilitates transmission through time of knowledge. Knowledge in an isolated environment becomes may become insular. Not enuff new challenges to diversify and respond and grow. Develop dependency on culurally specific symbols that bind the institution and they become reified and normalized and preferred. Tradition has a sense of timelessness. Tradition also has a developmental aspect. Tradition is different in different places and times. Is a nunnery/monastery always an institution? ===Retreat=== Short term, fixed period, long-term, permanent retreat. Can a person be on retreat whilst being in the world that is in the so called "secular" world? Is what is secular to some sacred? Is what is sacred to others secular? Does the "secular" world drive the "sacred" away or does the "sacred" person become positioned as different or other than their society? ==Bhagavad Gita: as English discourse, discourse as pervasion== [[File:Raja_Ravi_Varma_-_Dattatreya.jpg|thumb|left|Dattatreya]] Though the Gita on the face of it may be viewed other than as a nondual text, through the divine person being identified as '''the''' Divine Personality Krishna as different to the personality of Arjuna and all the other personages on the battlefield of Kurukshetra as lesser-deities (deva and ashura) or indeed at the last analysis as jiva, there are many views of their entwined relationships and status within the Greater Tradition. Nondualism and Bhakti are not mutually exclusive as some Bhakti traditions outwardly and vehemently maintain. Some Bhakti traditions place an emphasis on bhakri rasa-prema '''relationship''' with the Divine Person (however that is envisioned) and some hold that such a relationship precludes the unity of advaita understood or misunderstood as "monism". That said, the Gita has many indigenous commentaries and a famous nondual commentary by Shankara that was very influential in the indigenous tradition. The importance of this point and not only the possibility but value of nondual bhakti will be examined in due course. In delaration this is my personal view and realized experience where Personalism and Impersonalism are nondual and really only academic and philosophical constructions and partial descriptions of a whole that is unknowable except by grace or revelation though may be intuited though intellectual inquiry as championed by the Adi-Avadhuta [[w:Dattatreya|Dattatreya]]. It has often been misunderstood or underemphasized by English and Western scholarship that Shankara was actually part of the Bhakti movement. And as the Bhagavad Gita is principally a Vaishnava and/or Krishnaite text, it should be noted that there are Vaishanva traditions of nonduality and qualified nonduality. Banerji (1971, 1989: p.600) gives a brief overview of the English discourse of the Gita <blockquote>"The Bhagavadgita appears to have exercised deep influence on European literature. Coleridge (1772-1832) read this work in translation. In some of his poetical compositions, there are trances of the influence of the Gita. Wordsworth (1770-1850), in some poems, e.g. ''Tintern Abbey'' and Immortality Ode, reveals ideas very similar to those of the Gita. Matthew Arnold's 'disinterested endeavour' is a literal translation of Niskama Karma of the Gita. In recent times, Yeats and Elliot appear to have been influenced by this work. The Spanish poet, Himeneth, was inspired by it. Huxley repeatedly stressed the universal value of this great work. His Perennial Philosophy should be particularly mentioned in this connection. He has pointed out close similarity between certain basic ideas of the Gita and those of Christianity and Islamic doctrines. William Blake's (1757-1827) writings testify to the study of the Gita. Eliot, in his Dry Salvages and Burnt Norton, shows familiarity with the doctrines laid down in the Gita. M. Taylor, in his Tara, refer to the above Indian scripture. Keats (1795-1821) in his Endymion, reveals his acquaintance with the Gita."<ref> Banerji, Sures Chandra (1971; 1989 2nd Ed.). ''A companion to Sanskrit literature: spanning a period of over three thousand years, containing brief accounts of authors, works, characters, technical terms, geographical names, myths, legends and several appendices''. Motilal Banarsidass Publishers. {{ISBN|812080063X}}, 9788120800632. Source: [http://books.google.com.au/books?id=JkOAEdIsdUsC&pg=PA600&lpg=PA600&dq=coleridge+sanskrit&source=bl&ots=SLaiwWOWBp&sig=M99aD8zi27nM2qg18X_ebCgBsA8&hl=en&ei=OdfrS66MNovU7AOvsemZBg&sa=X&oi=book_result&ct=result&resnum=7&ved=0CC8Q6AEwBg#v=onepage&q=coleridge%20sanskrit&f=false] (accessed: Thursday May 13, 2010)</ref></blockquote> Bhagavad Gita 14.22-25: <poem> śrī-bhagavān uvāca prakāśaṁ ca pravṛttiṁ ca moham eva ca pāṇḍava na dveṣṭi sampravṛttāni na nivṛttāni kāṅkṣati udāsīna-vad āsīno guṇair yo na vicālyate guṇā vartanta ity evaṁ yo 'vatiṣṭhati neṅgate sama-duḥkha-sukhaḥ sva-sthaḥ sama-loṣṭāśma-kāñcanaḥ tulya-priyāpriyo dhīras tulya-nindātma-saṁstutiḥ mānāpamānayos tulyas tulyo mitrāri-pakṣayoḥ sarvārambha-parityāgī guṇātītaḥ sa ucyate </poem> [[w:A. C. Bhaktivedanta Swami Prabhupada|Prabhupada]] renders Bhagavad Gita 14.22-25 thus: <blockquote>"The Blessed Lord said: He who does not hate illumination, attachment and delusion when they are present, nor longs for them when they disappear; who is seated like one unconcerned, being situated beyond these material reactions of the modes of nature, who remains firm, knowing that the modes alone are active; who regards alike pleasure and pain, and looks on a clod, a stone and a piece of gold with an equal eye; who is wise and holds praise and blame to be the same; who is unchanged in honor and dishonor, who treats friend and foe alike, who has abandoned all fruitive undertakings-such a man is said to have transcended the modes of nature."<blockquote></blockquote></blockquote> ===Essence, Nature, Spirit: a unitary, unitive, unifying Numen=== <poem> "Or come we of an Automaton Unconscious of our pains? . . . Or are we live remains Of Godhead dying downwards, brain and eye now gone? :[http://en.wikisource.org/wiki/Nature's_Questioning Thomas Hardy, ''Nature's_Questioning'', extract] </poem> [[File:JuanSimónGutiérrez.jpg|thumb|A representation of the Holy Ghost as a dove]] In many nondual traditions there is a related though somewhat individuated concept but when understood in the broadest denotation and connotation of their usages without sectarian bias cover a comparable semantic field. To my knowledge no comparative study unifying these concepts has been undertaken but I am going to mention them here as they all share a commonality in my experience: [[w:Holy Ghost|Holy Ghost]] of my Grandmother, [[w:Paramatman|Paramatman]] of the Upanishads and particularly my experience of the Bhagavata Purana, Buddha Nature (and its many synonyms) of the various Buddhadharma traditions such as Zen, Mahayana Buddhism, Emerson's Over-soul for example. <blockquote>[http://en.wikisource.org/wiki/Essays:_First_Series/The_Over-Soul "Man is a stream whose source is hidden. Our being is descending into us from we know not whence."]</blockquote> <poem> "He who sees the In- -finite in all things sees God." Blake, ''[[s:There_is_no_natural_religion|There is no natural religion]]'' extract </poem> <blockquote>16Don't you know that you are God's shrine and the Spirit of God lives among you? 17If anyone ruins the shrine of God, God will ruin him. For the shrine of God is holy, which is what you yourselves are. 18Let no one deceive himself; if anyone considers himself to be wise among you in this age, let him become foolish, so that he might become wise. For the wisdom of this world is foolishness alongside God; for it has been written, He catches the wise in their trickery. 20And again, The Lord knows the thoughts of the wise, that they are empty. [[s:1_Corinthians#Chapter 3|1 Corinthians 3:16-20]]</blockquote> <poem> vadanti tat tattva-vidas tattvaḿ yaj jñānam advayam brahmeti paramātmeti bhagavān iti śabdyate Śrīmad Bhāgavatam 1.2.11 </poem> ====Dhatu==== [[File:Karmapa3.jpg|thumb|Third Karmapa]] Dhatu "realm" or "element" is the "Buddha-nature". This verse is from the no longer extant ''Mahayana Abhidharma Sutra'' and it is found within a number of texts which attest to its veracity such as Asaṅga's ''Mahāyānasaṃgraha'', the ''Nyingpo Tenpa'' (de bzin gsegs pa'i sin po bstan pa'i bstan bcos) of the [[w:Third Karmapa|Third Karmapa]] and in the ''[[w:Ratnagotravibhāga_(text)|Ratnagotravibhāga]]''. Why it is important is that though the Buddha-nature may be understood as being in all phenomena (dharmas) and all sentient-beings it may also be understood as being the common-ground of all dharmas and all sentient-beings. Stated differently, as the the Buddha-nature is the realm without beginning and it envelops and permeates all, the Buddha-nature is One: a Monad, it is the Common-Ground (samāśrayaḥ)<ref>This Common Ground is the [[w:Ground_of_Being_(Dzogchen)|Ground-of-Being]] and I would like to affirm that I chose 'Common Ground' as the English rendering as it is an informed variant of [[w:The_commons|The Commons]] or the [[w:Village_green|Village Green]] which have ancient associations as collective Sacred Ground in the English language. Refer William Blake.</ref> of all-dharmas (sarvadharma):<br > <blockquote>'''anādikāliko dhātuḥ sarvadharmasamāśrayaḥ/'''<br > '''tasmin sati gatiḥ sarvā nirvāṇādhigamo 'pi ca//'''<br > The realm without beginning is the base of all the dharmas: it being so,<br > there is every destiny, as well as the attainment of Nirvana.<ref> Wayman, Alex & Tajima, Ryujun (1992). ''The Enlightenment of Vairocana''. Volume 18 of Buddhist traditions. Motilal Banarsidass Publishers. {{ISBN|8120806409}}, 9788120806405. Source: [http://books.google.com.au/books?id=rlhQwTAYR-oC&pg=PA60&lpg=PA60&dq=The+realm+without+beginning+is+the+base+of+all+the+dharmas:+it+being+so,+there+is+every+destiny,+as+well+as+the+attainment+of+Nirvana.&source=bl&ots=KqG3CiZL76&sig=F1O7QY8D8KLudTxAkeiZO6sA7js&hl=en&ei=j3X1S6PMDor46QOe2o2HBg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBoQ6AEwAQ#v=onepage&q=The%20realm%20without%20beginning%20is%20the%20base%20of%20all%20the%20dharmas%3A%20it%20being%20so%2C%20there%20is%20every%20destiny%2C%20as%20well%20as%20the%20attainment%20of%20Nirvana.&f=false] (accessed: Friday May 21, 2010)</ref><br> </blockquote> ==Abrahamic traditions== ===Jewish traditions and Hasidism=== Michaelson (2009: p.130) identifies that nonduality was unambigously evident in the medieval Jewish textual tradition which peaked in [[w:Hasidism|Hasidism]]: <blockquote>"As a Jewish religious notion, nonduality begins to appear unambigously in Jewish texts during the medieval period, increasing in frequency in the centuries thereafter and peaking at the turn of the nineteenth century, with the advent of Hasidism. It is certainly possible that earlier Jewish texts may suggest nonduality -- as, of course, they have been interpreted by traditional nondualists -- but...this may or may not be the most useful way to approach them."<ref>Michaelson, Jay (2009). ''Everything Is God: The Radical Path of Nondual Judaism''. Shambhala Publications. {{ISBN|1590306716}}, 9781590306710. Source: [http://books.google.com.au/books?id=jmrgoeaQvlwC&pg=PA130&dq=i+and+thou+martin+buber+nondual&hl=en&ei=0I3iS_jIFdGgkQX50dHtDw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDYQ6AEwAA#v=onepage&q=i%20and%20thou%20martin%20buber%20nondual&f=false] (accessed: Thursday May 6, 2010), p.130</ref></blockquote> ===William Blake=== <poem> "Therefore God becomes as we are, that we may be as he is" Blake, ''[[s:There_is_no_natural_religion|There is no natural religion]]'' extract </poem> I had an intuition I would find "nonduality" associated with Blake (1757-1827). I was correct. When we get to nonduality being discussed in relation to Blake I feel it needs to be said is this a colonial pervasion of advaita upon other discourses, other possibilities, or is nonduality as a term approaching an experiential awareness that approaches a human cultural universal? It is only natural in the Human Condition that if we have a new cultural paradigm, a new conceptual or experiential tool and in this example that is "nonduality", that the paradigm will permeate other disciplines and discourses, as this embedded narrative demonstrates. That it will become a paradigm that will become fashionable to perceive and filter, that our new "term" or "technology" recasts our understanding and appreciation of history as we actively reflect. It gives me pause to consider that a Sanskrit philosophical term in literal English translation through revisionist discourse is now associated with an English Pre-Romantic or Romantic period unorthodox Christian-influenced artist-mystic. I am surprised that nonduality and Blake is not a new association. George Frederick Wingfield Digby in his work ''Symbol and image in William Blake'' (1957: p.116) mentions: <blockquote>"The Tantra shows the way back to purity and unity by means of the use of images. For duality is the nature of consciousness, and yet reality is non-dual. Consciousness must therefore find the means to transform itself. The seer and the seen, the experiencer and the experience, must again achieve unity..."</blockquote> I can't figure out how to sort out this reticulum of sections... but I know it will take form in time as themes emerge rather than arbitrary designation according to physical locality, chronology or outer tradition and/or social affiliation <blockquote>"In Blake's earliest engraved work, "All Religions are One," the nondual state appears as the Poetic Genius..." (p.2)</blockquote> Freeman (1997) in her dedicated work on Blake gives "nondualism" a headline mention. Freeman (1997: p.32) identifies that although evidence is currently scant whether Blake read [[w:Charles Wilkins|Wilkins]]' (1749 – 1836) first English translation of the ''[[w:Bhagavad Gita|Bhagavad Gita]]'', Blake was significantly inspired enough by its translation and Wilkins' endeavour to use this as the subject of one of his representational artworks, unfortunately no longer extant: <blockquote>"Although there is scant evident that Blake had direct exposure to such works, it is probable, though as yet unconfirmed, that Blake read Sir Charles Wilkins's translation of the ''Bhagavad Gita'', the most influential text of Eastern nondual philosophy. In his descriptive catalog Blake lists a now-lost portrait of Wilkins translating the ''Gita'': "This subject is, Mr. Wilkin, translating the Geeta; an ideal design, suggesting for the first publication of that part of the Hindoo Scriptures, translated by Mr. Wilkin" (E. 548). Blake links the Poetic Genius to the east, as Ezekiel tells the narrator of ''The Marriage of Heaven and Hell'': "The philosophy of the east taught the first principles of human perception....[W]e of Israel taught that the Poetic Genius (as you now call it) was first principle and all the others merely derivative" (E. 39, pl.12). From Blake's association of the Poetic Genius with Eastern thought, one can infer that Blake's attraction to Eastern philosophy was its nondualism.<br > Although these references suggest a direct influence of Eastern thought on Blake, it is not presented here as the source of Blake's nondual vision."<ref>Source: [http://books.google.com.au/books?id=1L2ELlWaVrYC&pg=PA31&dq=bhagavad+gita+nonduality&hl=en&ei=t9jrS9agMo7o7APXstygBg&sa=X&oi=book_result&ct=result&resnum=5&ved=0CEEQ6AEwBA#v=onepage&q=bhagavad%20gita%20nonduality&f=false] (accessed: Thursday May 13, 2010)</ref></blockquote> Although Blake's use of English is non-standard and famously so, it is worth noting that there was considerable diversity in the English language historically and especially so due to the florescence of different speech communities prior to standardization through the ascendance of particular 'prestige' speech varieties and their codification, availability and dissemination through the medium of the "dictionary". It is curious that if Blake had actually read Wilkins' literary work of the ''Gita'' prior to his crystallization of the translation-process as art, he would have spelt Wilkins' name correctly instead of incorrectly as "Wilkin"; but then, maybe not. Entwine Frye and ''[[w:Fearful_Symmetry_(Frye)|Fearful Symmetry]]'' and Avatamsaka and interpenetration. *http://en.wikisource.org/wiki/All_religions_are_one *http://books.google.com.au/books?id=1L2ELlWaVrYC&printsec=frontcover&dq=nondualism&hl=en&ei=buPqS_LqNpHg7APe3YWXCA&sa=X&oi=book_result&ct=result&resnum=4&ved=0CEAQ6AEwAw#v=onepage&q&f=false *http://books.google.com.au/books?id=HOxpOMQ_Pa8C&pg=PA330&dq=blake+%22poetic+genius%22&hl=en&ei=a-TqS4uUJYvg7APtkKWhCA&sa=X&oi=book_result&ct=result&resnum=5&ved=0CEQQ6AEwBA#v=onepage&q=blake%20%22poetic%20genius%22&f=false [[File:William Blake by Thomas Phillips.jpg|thumb|left|'William Blake' painted by Thomas Phillips]] *http://ebooks.adelaide.edu.au/b/blake/william/job/index.html *http://books.google.com/books?id=XMyxBQOSy5kC&printsec=frontcover&dq=isbn:0691012911&ei=_YztS4DFNpOOkASf-92fDA&cd=1#v=onepage&q&f=false ===Meister Eckhart=== My first exposure to Meister Eckhart was through the non-critical translations of [[w:Matthew Fox_(priest)|Fox]] (1983)<ref>Matthew Fox (1983). ''Meditations with Meister Eckhart''. Bear & Company, {{ISBN|0-939680-04-1}}</ref> and [[w:D.T._Suzuki|Suzuki]] (1957)<ref>[http://www.sacred-texts.com/bud/mcb/index.htm Suzuki, Daisetz Teitaro (1957). ''Mysticism, Christian and Buddhist''.]</ref>. I don't know which edition of these texts either as it was a long time ago. Much later, but still four years from the current now, I was also aware of him through Otto. I now understand that most English renderings of Meister Eckhart are little more than loose paraphrases verging on misattributions that are more the ideology and partisan view of the 'translator' than that held by the historical Eckhart. Fox resides in this category, as does Suzuki. I am not denigrating their work, but highlighting that it is interpretative translation as different to critical. That distinction also becomes problematic as each translation is a product of its time written for a particular audience and for a particular reason with overt or covert agenda. I would like to reserve judgment on Otto as he was a very literate man, meaning literate in many languages, and may have read the originals and performed his own translations into English. Fox performed his own translations as well into English from the German, but I don't know whether he went back to the Middle High German source texts. It all becomes nebulous. That said, I still would like to discuss Eckhart but the 'Discourse of Eckhart' and what the historical Eckhart actually thought and felt in respect to his relationship with the divine are to be understood as differing and therefore, problematic for any understanding of a "fact" or "truth". But then truth and the factual are problematic anyway. [http://www.academici.com/blog.aspx?bid=6171 I have been unable to source reputable source material for Eckhart and it appears that this is the state of play.] Primary source material attributed to Eckhart are in Latin and [[w:Middle High German|Middle High German]]. Eckhart and the linkage with nonduality may be tentatively pinned upon Suzuki in his association of Eckhart with Mahayana Buddhism with the variety being a 'peasant tradition' of Japanese Zen. I should probably deal with Suzuki as the main study and view Eckhart through him but I will rest upon that. I will necessarily have to discuss Suzuki elsewhere particularly in Zen which I haven't even really begun. I have mentioned him in parsing in relation to the Wikipedia article of the Dzogchen ground. The Dzogchen triune appears to have its grounding in the Awakening of Mahayana Faith which general Buddhist tradition ascribes an an original Sanskrit composition of Ashvagosha, though now the original is long lost. A number of scholars but importantly Buswell in his work on the Chinese Buddhist Apocrypha consider the Awakening of Mahayana Faith a Chinese production. This will be all the more interesting when these two Absolute triunes that of the Awakening of Mahayana Faith and that of Dzogchen are treated by scholars on common ground, which as yet they have not been. :From Wikipedia: Eckhart expressed himself both in learned Latin for the clergy in his tractates, and more famously in the German vernacular (at that time Middle High German) in his sermons. Because, as he said in the defense he gave at his trial, his sermons were meant to inspire in listeners (the non clergy) the desire above all to do some good; he frequently used unusual language or seemed to stray from the path of orthodoxy. His unorthodox teachings made him suspect to the Catholic Church during the tension filled years of the Avignon Papacy, and he was tried for heresy in the final years of his life. It is fascinating that this tradition of misattribution starts from [[w:Franz Pfeiffer|Pfeiffer]] (1815 - 1868) whose gathering of Eckhart held many unattributed writings in a similar manner and style. Many of these have been contested by subsequent scholarship but have permeated through popular Eckhartian lore in English. The early translators of Eckhart into English, Evans () and Blakney (), draw heavily upon Pfeiffer for their source material. Field (c1909) renders Eckhart thus: <blockquote>[http://en.wikisource.org/wiki/Sermons_(Meister_Eckhart)/True_Hearing "The eye with which I see God is the same with which God sees me. My eye and God's eye is one eye, and one sight, and one knowledge, and one love."]</blockquote> A rendering of Eckhart with similar import in another quotation was famously promulgated by Hegel in a series of lectures from 1824 (''Lectures on the Philosophy of Religion'' Vol.1: pp.347-48) in similar wording: <blockquote>"The eye with which God sees me is the same eye by which I see Him, my eye and His eye are one and the same. In righteousness I am weighed in God and He in me. If God did not exist nor would I; if I did not exist nor would He."</blockquote> This has been identified as a "quilt quotation" and "Hegelian interpolation" by Magee (2001: p.25), that is a fabricated patchwork of threaded elements sewn together that had been sourced from different Eckhart sermons.<ref>Source: [http://books.google.com.au/books?id=NFIOpySKxw0C&dq=Hegel+and+the+Hermetic+Tradition&printsec=frontcover&source=bn&hl=en&ei=ccf-S9qFGsGGkAXEt7CoDQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0CCIQ6AEwAw#v=onepage&q&f=false] (accessed: Friday May 28, 2010)</ref> This hints at something that I want to say but don't know how... but it is something like this: Religious people and religious texts are not insular and fixed they are institutions and a dynamic. They are perpetuated and promulgated by a series of factors. The success of this marketing as institution determines how the branding of them enter and maintain their place within the general discourses of various communities to whom what is communicated holds worth. That which is deemed to hold worth is also transformed in the experience of a particular individual that invests it with worth and this is what is attributed as a tradition that is traded through time. <blockquote>"...the doctrine that Eckhart, following Aquinas, expounds is fundamentally the perennial "doctrine of nonduality." (p.148)</blockquote> http://books.google.com.au/books?id=69fazWQXP9IC&pg=PA147&dq=meister+eckhart+nonduality&hl=en&ei=TebqS5OxJJDs7AORkdi_Cg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDkQ6AEwAA#v=onepage&q=meister%20eckhart%20nonduality&f=false ===Christianity=== [[s:Bible_%28King_James%29/Acts_of_the_Apostles#Chapter_4|Acts 4:32, ''King James'']]<br /> And the multitude of them that believed were of one heart and of one soul: neither said any of them that ought of the things which he possessed was his own; but they had all things common. <blockquote> "The conclusion I have reached is that in each religion there is a movement which culminates in advaita. In Hinduism and Buddhism it is obvious, but it is equally true in Taoism and Sikhism. Islam and Judaism are the main problem. They are both deeply dualistic--God separate from the world and humanity, good and evil, heaven and hell. But in both religions the mystical tradition--in Sufism and the Kabbala--transcends the dualism and reaches pure advaita. In Christianity I see a gradual movement from the dualism of its Jewish origins to the non-duality of the Fourth Gospel, where Jesus prays "that they all may be one, as thou, Father, in me and I in thee--that they may be perfectly one"--in other words a pure advaita sharing in the inner life of God."<ref>Bede Griffiths correspondence to Judson B. Trapnell dated June 21, 1991. Source: [http://books.google.com.au/books?id=D7qLzpQDPgoC&pg=PA255&dq=sikhism+nonduality&hl=en&ei=AiZcTOTcLZCmvQPbuZj5AQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CEQQ6AEwBA#v=onepage&q=sikhism%20nonduality&f=false] (accessed: Saturday August 7, 2010), p.255</ref> </blockquote> [[w:Bede Griffiths|Griffiths]]' (1906 – 1993) form of [[w:Vedanta|Vedanta]]-inspired or nondual Christianity has been given the nomenclature '[[w:Wisdom Christianity|Wisdom Christianity]]' or 'Sapiential Christianity'.<ref>Barnhart, Bruno (2007). ''The future of wisdom: toward a rebirth of sapiential Christianity''. Continuum. {{ISBN|0826427677}}, 9780826427670</ref><ref>Barnhart, Bruno (2009). "Christianity in the Light of Asian Nonduality". Summary of a paper presented by Bruno Barnhart at the Monastic Symposium on Purity of Heart/Contemplation at New Camaldoli in June 2000. Source: [http://www.bedegriffiths.com/wisdom-christianity/christianity-in-the-light-of-asian-nonduality.html] (accessed: Saturday May 1, 2010)</ref> Barnhart (1999: p.238) explores Christian nondual experience in a dedicated volume and states that he gives it the gloss of "unitive" experience and "[[w:perennial philosophy|perennial philosophy]]".<ref>Barnhart, Bruno (1999). ''Second simplicity: the inner shape of Christianity''. Paulist Press. {{ISBN|0809138328}}, 9780809138326. Source: [http://books.google.com.au/books?id=5agB2wJvZqUC&printsec=frontcover&dq=Bruno+Barnhart&source=bl&ots=4E0znu6UYS&sig=hnche8WKEgRLKHN32Z-goJVKLC8&hl=en&ei=qB_cS-OSNMyLkAWj4tyfBw&sa=X&oi=book_result&ct=result&resnum=7&ved=0CCgQ6AEwBg#v=onepage&q=nondual&f=false], p.238</ref> Further, Barnhart (2009) holds that: <blockquote>"It is quite possible that nonduality will emerge as the theological principle of a rebirth of sapiential Christianity ('wisdom Christianity') in our time."<ref>Barnhart, Bruno (2009). "Christianity in the Light of Asian Nonduality". Summary of a paper presented by Bruno Barnhart at the Monastic Symposium on Purity of Heart/Contemplation at New Camaldoli in June 2000. Source: [http://www.bedegriffiths.com/wisdom-christianity/christianity-in-the-light-of-asian-nonduality.html] (accessed: Saturday May 1, 2010)</ref></blockquote> The [[w:Catholic Church|Catholic Church]], for example, teach in their [[w:Catechism of the Catholic Church|Catechism]] that evil - and any manifestations of it - exists in the sense that "only the ''whole'' of the Christian faith can constitute a response"<ref>{{cite web|url=http://www.vatican.va/archive/compendium_ccc/documents/archive_2005_compendium-ccc_en.html|title=The Catechism of The Catholic Church|accessdate=2009-09-30}}</ref>. That is, the search or struggle for ones truth is in a sense a search for non-dualism. Sin is a manifestation of erring from that search, Jesus represents hope to find/follow that search and the unity of the Holy Spirit represents divine faith (regardless of brand) in all of us. Perceiving the devil is a corruption of faith and a descent into dualistic thinking. The [[w:Congregation for the Doctrine of the Faith|Holy Office]] have historically struggled to maintain a non-dualistic message through the times of the [[w:Galileo affair]] and [[w:Protestant Reformation|Protestant Reformation]]. The politics surrounding the papacy of [[w:Pope Pius XII|Pope Pius XII]] and his 1943 [[w:papal encyclical|papal encyclical]] [[w:Mystici Corporis|Mystici Corporis]] published during [[w:World War II|World War II]] are also characteristic of this struggle. See also [[w:Hitler's Pope|Hitler's Pope]] and [[w:The Myth of Hitler's Pope|The Myth of Hitler's Pope]]. [[w:Saint Francis of Assisi|Saint Francis of Assisi]] (c.1181 - 1226), often called the "most Christ-like" of the Catholic Saints with his [[w:gift of spirit|gift of spirit]] of the divine [[w:stigmata|stigmata]], embodies a nonduality with his beloved [[w:Scapegoating|scapegoat]], sacrificial or [[w:eucharistic|eucharistic]] [[w:Christ|Christ]], the [[w:Lamb of God|Lamb-of-God]], as Egan (1991: p.217) making reference to ''[[w:Canticle of Brother Sun|Canticle of Brother Sun]]'' which may be viewed as a nondual ecological testament<ref>Esbjörn-Hargens, Sean & Zimmerman, Michael E. (2009). ''Integral ecology: uniting multiple perspectives on the natural world''. Illustrated edition. Shambhala Publications. {{ISBN|1590304667}}, 9781590304662. Source: [http://books.google.com.au/books?id=Ver-iHURIjMC&pg=PA237&dq=canticle+to+the+sun+nondual&hl=en&ei=PS_cS6HhH5GgkQW67Z2sBw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDkQ6AEwAA#v=onepage&q=canticle%20to%20the%20sun%20nondual&f=false] (accessed: Saturday May 1, 2010), p.237</ref>, states: <blockquote>In September 1224 on Mount Alvernia, he [ that is Saint Francis] received that "final seal" (Dante), the first documented stigmata in Christian history. He bore now the wounds of the crucified Christ not only in spirit, but also on his body. During two more years of increasingly painful illness, Francis composed his classic hymn, ''Canticle of Brother Sun'', which expressed profound Christian love for God and creation. Francis's participation in the hierarchical, sacramental Church never wavered. He had embraced fully the crucified and eucharistic Christ whose wounds he wore on his person. Having renounced the world, he wanted only to have, to know, and to be totally like Christ crucified. In this way, he found all things in God and God in all things.<ref>Egan, Harvey D. (1991). ''An Anthology of Christian mysticism''. Second Edition. Liturgical Press. {{ISBN|0814660126}}, 9780814660126. Source: [http://books.google.com.au/books?id=LHdS5hNH7zsC&printsec=frontcover&dq=stigmata+nondual&hl=en&ei=ISXcS7fHHtKHkAXzipy9Bw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDgQ6AEwADgK#v=snippet&q=stigmata&f=false] (accessed: Saturday May 1, 2010), p.217</ref></blockquote> [[w:A Course in Miracles|A Course in Miracles]] or ''ACIM'' is a modern day Christian non-dualistic teaching that is not inclusive of physical reality. Physical reality is denied valid existence all together accept as a wrong (or evil) mis-thought. This tradition states, "''Nothing real can be threatened. Nothing unreal exists. Herein lies the peace of God.''"<ref>Schucman (1992) pp ??</ref> [[w:Christian Science|Christian Science]] is very similar to "ACIM" above. In a glossary of terms written by the founder, Mary Baker Eddy, matter is defined as illusion and when defining individual identity she writes "There is but one I, or Us, but one divine Principle, or Mind, governing all existence".<ref>{{cite book |last=Eddy|first=Mary Baker|authorlink=Mary Baker Eddy |title= [[Science and Health with Key to the Scriptures]] |isbn= {{ISBN|0-87952-259-3}} }}</ref> ===Gnosticism=== [[File:Eg-NagHamadi-map.png|thumb|Nag Hammadi]] Robinson, Smith & the Coptic Gnostic Library Project (1996: p.536) in regards to the [[w:Nag Hammadi library|Nag Hammadi codices]] affirm: <blockquote>"The present collection of texts from Nag Hammadi shows us that what we call Gnosticism can range between a hierarchical monism to strict dualism."<ref>Robinson, James McConkey; Smith, Richard & Coptic Gnostic Library Project (1996). ''The Nag Hammadi library in English''. Fourth edition, revised. BRILL. {{ISBN|9004088563}}, 9789004088566. Source: [http://books.google.com.au/books?id=UiSFUJ6al1IC&pg=PA25&dq=Nag+Hammadi+Scriptures,+The:+The+International+Edition&hl=en&ei=PMTxS8jkFpDc7APd5ciHDA&sa=X&oi=book_result&ct=result&resnum=5&ved=0CD4Q6AEwBA#v=onepage&q=monism&f=false] (accessed: Tuesday May 18, 2010), p.536</ref></blockquote> Since its beginning, [[w:Gnosticism|Gnosticism]] has been characterized by many dualisms and dualities, including the doctrine of a separate God and [[w:Manichaean|Manichaean]] (good/evil) dualism. The discovery in 1945 of the ''[[w:Gospel of Thomas|Gospel of Thomas]]'', however, has led some scholars to believe that Jesus' original teaching may have been one accurately characterized as nondualism.<ref>{{cite book|last=Miller|first=Ronald|authorlink=Ronald H. Miller|title=The Gospel of Thomas: A Guidebook for Spiritual Practice}}</ref> An English rendering from The Gospel of Thomas that showcases a nondual vision of reconciling opposites which are also preserved, that is "make the two one": <blockquote>When you make the two one, and when you make the inside like the outside and the outside like the inside, and the above like the below, and when you make the male and the female one and the same...then you will enter [the Kingdom].<ref>Source: [http://books.google.com.au/books?id=i_KOOn0BaCoC&pg=PA145&dq=Gospel+of+Thomas+nondual&ei=r4_QS_L8C6ailASGnszQCA&cd=10#v=onepage&q=Gospel%20of%20Thomas%20nondual&f=false] (accessed: Friday April 23, 2010), p.145</ref></blockquote> The ''[[w:Gospel of Philip|Gospel of Philip]]'' also conveys nondualism: :''"Light and Darkness, life and death, right and left, are brothers of one another. They are inseparable. Because of this neither are the good good, nor evil evil, nor is life life, nor death death. For this reason each one will dissolve into its earliest origin. But those who are exalted above the world are indissoluble, eternal."'' <ref>{{cite web|url=http://www.gnosis.org/naghamm/gop.html|title=The Gospel of Philip|accessdate=2007-09-06}}</ref> *http://books.google.com.au/books?id=UiSFUJ6al1IC&pg=PA25&dq=Nag+Hammadi+Scriptures,+The:+The+International+Edition&hl=en&ei=PMTxS8jkFpDc7APd5ciHDA&sa=X&oi=book_result&ct=result&resnum=5&ved=0CD4Q6AEwBA#v=onepage&q&f=false ====Codex XI: 3==== I find the "triadic monism" of [[w:Allogenes|Allogenes]] of interest given the Gankyil and as an offset for [[w:Valentinus (Gnostic)|Valentinus]]' (c. 100 - c. 160) trinity before it was appropriated and transformed by the discourse of prestige ''qua'' institutional power (?) of the [[w:Holy Roman Church|Holy Roman Church]] (?): <blockquote>The main issue at stake in interpreting Allogenes is the origin of its unusual combination of gnostic motifs and philosophical triadic monism. Did this philosophy develop within a gnostic community as greater philosophical sophistication forced it toward new affirmations, possibly resulting in some influence on Plotinus and Porphyry? Or is the Neoplatonism a conceptual veneer, adopted without roots in gnostic mythology? The answer must lie somewhere between these alternatives. The new sophistication must have been triggered by some kind of active philosophical interchange, although the dominant motivation continues to be religious and the forms of speech remain those of initiation and revelation.<ref>Robinson, James McConkey (2000). ''The Coptic Gnostic Library: A Complete Edition of the Nag Hammadi Codices, Volume 2.'' BRILL: Institute for Antiquity and Christianity. {{ISBN|9004117024}}, 9789004117020. Source: [http://books.google.com.au/books?id=qibkoKs-hDMC&dq=The+Coptic+Gnostic+Library&printsec=frontcover&source=bn&hl=en&ei=GMnxS6uGLY7U7AOY78H-Cw&sa=X&oi=book_result&ct=result&resnum=5&ved=0CDkQ6AEwBA#v=onepage&q=monism&f=false] (accessed: Tuesday May 8, 2010)</ref></blockquote> Valentinus professed to have derived his ideas from Theodas or [[w:Theudas (teacher of Valentinius)|Theudas]], a disciple of St. Paul. Valentinus drew freely on some books of the New Testament. Unlike a great number of other gnostic systems, which are expressly [[w:dualism|dualist]], Valentinus developed a system that was more monistic, albeit expressed in dualistic terms.<ref>'Valentinian gnosticism [...] differs essentially from dualism' ([[Elaine Pagels]], ''The Gnostic Gospels'', 1978); 'a standard element in the interpretation of Valentinianism and similar forms of Gnosticism is the recognition that they are fundamentally monistic' (William Schoedel, 'Gnostic Monism and the Gospel of Truth' in ''The Rediscovery of Gnosticism, Vol.1: The School of Valentinus'', edited by Bentley Layton, E.J.Brill, Leiden, 1980).</ref> ===A Course in Miracles=== ''[[w:A Course in Miracles|A Course in Miracles]]'' is a an expression of nondualism that represents itself as independent of any religious denomination. Really? How can any book not partake of the tradition, that is, how can it be other than of Christian discourse as it employs the popular mythic language and symbols of Christianity? That said, ''Miracles'' as a [[w:New Age|New Age]] reparation of popular Christian dualism, fulfilled a function in time. Upton understands the Course as Neo-Gnosticism (2005: p.221).<ref>Upton, Charles (2005). ''The System of Antichrist: Truth and Falsehood in Postmodernism and the New Age''. Sophia Perennis. {{ISBN|0900588381}}, 9780900588389. Source: [http://books.google.com.au/books?id=CwUo8zcsinoC&pg=PA221&dq=course+in+miracles+critique&hl=en&ei=aLfxS975LYre7AO4xoiUBg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDEQ6AEwAA#v=onepage&q=course%20in%20miracles%20critique&f=false] (accessed: Tuesday May 18, 2010)</ref> [[w:Charles Upton (poet)|Upton]] (2005: p.221): <blockquote> "There is a great deal of profound truth in ''A Course in Miracles'': the uncompromising sense of God as Absolute Truth and Love; deep insight into the convoluted games the ego plays to escape this Truth and Love; an understanding that the subject/object mode of consciousness cannot directly witness Absolute Truth; the doctrine of one and only choice which is completely free, that between Truth and illusion; the primacy granted to forgiveness in the process of '[[w:metanoia|metanoia]]', that total change of mind by which Truth is chosen and illusion dismissed; the doctrine--entirely true in ''one'' sense--that humanity never really fell into sin, never entered into the illusion of separation from God.<ref>Upton, Charles (2005). ''The System of Antichrist: Truth and Falsehood in Postmodernism and the New Age''. Sophia Perennis. {{ISBN|0900588381}}, 9780900588389. Source: [http://books.google.com.au/books?id=CwUo8zcsinoC&pg=PA221&dq=course+in+miracles+critique&hl=en&ei=aLfxS975LYre7AO4xoiUBg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDEQ6AEwAA#v=onepage&q=course%20in%20miracles%20critique&f=false] (accessed: Tuesday May 18, 2010)</ref></blockquote> Upton's praise of Miracles positions it as appropriate in this discourse and his reservations will be mentioned in due course as well. I will endeavour to find other voices to provide texture. A Course in Miracles presents an interpretation of nondualism that recognises only "God" (i.e. absolute reality) as existing in any way, and nothing else existing at all. In a book entitled ''[[w:The Disappearance of the Universe|The Disappearance of the Universe]]'', which explains and elaborates on A Course in Miracles, it says in its second chapter that we: <blockquote>"don't even ''exist'' in an individual way - not on ''any'' level. There is no separated or individual soul. There is no Atman, as the Hindus call it, except as a mis-thought in the mind. There is only God."<ref>''[[w:The Disappearance of the Universe|The Disappearance of the Universe]]'', [[w:Gary Renard|Renard]], page 33</ref></blockquote> This amazes me, Renard has built their reinterpretation of their understanding of the Christian tradition informed by the Atman of the Puranas and the Buddhadhara discourse of Pratityasamutpa and Shunyata and the flux of Heraclitus and then not only do they not give due attribution, but misrepresent the tradition that they have appropriated. Class act really. This citation was already in the Nonduality article prior to my editing it so its inclusion here is welcome, but I would not have necessarily gone looking for it and I am not going to dismiss it out of turn. As I have a story about The Course in Miracles and I bought the very expensive book when I was very young circa 14 and lent it to my Aunt on the day of purchase who subsequently lent it to her friend without my express permission. Her girlfriend never returned it. My Aunt didn't remember lending it when I enquired a year later why she hadn't returned my book. She didn't apologize as she didn't remember borrowing it nor lending it. My Aunt was not generally absent minded in my experience and her girlfriend was a very good friend of many years. So, this is curious. I am not sure why but I feel it is important that this section stays and I am going to explore Miracles further. I have learned to trust my intuition and it has been repeatedly vindicated and If I chose the book as a child it was appropriate. I just didn't need to read it. A verse from the course itself that displays its interpretation of nondualism is found in Chapter 14: <blockquote>"The first in time means nothing, but the First in eternity is God the Father, Who is both First and One. Beyond the First there is no other, for there is no order, no second or third, and nothing but the First."<ref>''[[w:A Course in Miracles|A Course in Miracles]]'', 3rd edition, page 279 or Chapter 14, Section IV, Paragraph 1, verses 7-8</ref></blockquote> === Judaism === Michaelson (2009) explores nonduality in the tradition of Judaism.<ref>Michaelson, Jay (2009). ''Everything Is God: The Radical Path of Nondual Judaism.'' Shambhala Publications. {{ISBN|1590306716}}, 9781590306710. Source: [http://books.google.com.au/books?id=jmrgoeaQvlwC&dq=i+and+thou+martin+buber+nondual&source=gbs_navlinks_s] (accessed: Saturday may 8, 2010)</ref> ''[[w:Judaism|Judaism]]'' has within it a strong and very ancient mystical tradition that is deeply nondualistic. "[[w:Ein Sof|Ein Sof]]" or infinite nothingness is considered the ground face of all that is. God is considered beyond all proposition or preconception. The physical world is seen as emanating from the nothingness as the many faces "partsufim" of god that are all a part of the sacred nothingness. Sometimes the faces are referred to as colored spheres "sphirot" that are the same as chakras in eastern traditions. sphirot are seen as eminations or fruit of the tree of life in the sacred garden of paradise. The tree exists and emanates through many, sometimes infinite, stages or levels of reality. All is considered one nondualistic whole. nothingness and somethingness are considered one united and inseparable thing. Duality is seen as an illusion of brokenness or contraction and enlightenment is the act of inner restoration or repair "tikkun" of god's unity. === Islam === :"Whithersoever ye turn there is the Face of God." (Quran 2:115; rendered in English majestically by [[w:Hossein Nasr|Seyyed Hossein Nasr]]) Islam and Sufism is not an area of my current knowledge so I am going to enjoy this meditation. My first contact with Sufi mysticism was in dancing, not whirling dervishes but circle-dancing in a round as well as I am somewhat familiar with the poetry of Rumi in English translation. I find it interesting that Islam, Advaita Vedanta and Vaishanvism had significant interaction in Medieval Bengal. Vaishnavism also has 'advaita' or nondual traditons and texts and these have not really been foregrounded in Western scholarship to my knowledge. I will repair this in my humble way within this discourse. This is only from my personal knowledge stated here without current investigation. I felt I would just add this as a declaration. Also, I know that the Bhakti movement of which Shankara was a part was incited into its devotion through the impetus forded through the fervor it encountered in Islam. No culture is an island in the river of time. ====Al-Fatiha and the transignification of resurrection==== :"I shall tell you a great secret my friend. Do not wait for the last judgement, it takes place every day." :- Albert Camus, from [[w:The Fall (Albert Camus novel)|'La Chute (The Fall)']] (1956) (NB: unsure if Camus rendered this in English from the French or if this is a gloss from an unnamed, unspecified translator, need to investigate and credit as appropriate.) The ''[[w:Al-Fatiha|Al-Fatiha]]'' the "Opening" of the Qur'an is a prayer constituted by seven pith verses. The verse which concerns us is the fourth and the central verse of the seven: :1:4 مَـالِكِ يَوْمِ الدِّي :[[w:Malik|Māliki]] [[w:Islamic view of the Last Judgment|yawmi d-dīn]] :Master of the Day of Judgment The following large extract is from the masterwork of Nasr (2007: p.18) and particular concerns the abovecited verse: :The next verse, "Master of the Day of Judgment," concerns the flow of time at the end of which there is death and a meeting with God. To be aware of our human condition is to realize that we are on a journey in this life, which ends with death followed by resurrection, and that we are destined for the unavoidable meeting with God, which means that although we die, we are also immortal. The profound reality of our consciousness cannot be eradicated by the accident of bodily death. The verse speaks not only of the Day beyond all days, but also of Judgment. This eschatological assertion is of the utmost significance for our life here on earth. It reveals the grandeur of the human state and the fact that actions in this life on earth have consequences beyond the life of this world. :Now, these are matters widely accepted by people of faith everywhere. The Sufis take a further step, however, and seek to die and be resurrected here and now and to experience the encounter with God while still here in this world through spiritual practices and by climbing the ladder of perfection. In the deepest sense those who have already achieved the goal have already died, been resurrected, met the Master of the Day of Judgment, been judged by the Supreme Judge, and rest in the Paradise of Divine Proximity. The Prophet of Islam was once asked about death and resurrection. The Prophet answered, "Look at me; I have died and been resurrected many times."<ref>Nasr, Seyyed Hossein (2007). ''The Garden of Truth: The Vision and Promise of Sufism, Islam's Mystical Tradition''. HarperCollins. {{ISBN|006162599X}}, 9780061625992</ref> Now, I cite this for its salience for an English Christocentric audience for its interest in reframing the centrality of Christ and the Christian resurrection and body theology and making of every pure and accomplished Sufi a master on a comparable platform as that of Christ. Thematically, I wish to evoke Plath's poem ''[[w:Lady Lazarus|Lady Lazarus]]'' and [[w:E._E._Cummings|Cummings']] poem ''[http://www.poetry-chaikhana.com/C/cummingsee/ithankYouGod.htm i thank You God for most this amazing]'' with it's: <poem> ...(i who have died am alive again today, and this is the sun's birthday; this is the birth day of life and of love and wings... </poem> [[w:Sufism|Sufism]] (Arabic تصوف taṣawwuf, meaning "Mysticism") is often considered a mystical tradition of [[w:Islam|Islam]]. There are a number of different Sufi orders that follow the teachings of particular spiritual masters, but the bond that unites all Sufis is the concept of ego annihilation (removal of the subject/object dichotomy between humankind and the divine) through various spiritual exercises and a persistent, ever-increasing longing for union with the divine. "The goal," as [[w:Reza Aslan|Aslan]] writes, "is to create an inseparable union between the individual and the Divine." '''Wahdat al-Wajud''' (Arabic: وحدة الوجود) the "Unity of Being" is a [[w:Sufi metaphysics|Sufi metaphysical]] philosophy emphasizing that 'there is no true existence except the Ultimate Truth (God)'. Or in other phrasing that the only truth within the Universe is God, and that all things exist within God only. All of His creations emerge from ''`adim'' (عدم non-existence) to ''wujood'' (existence) out of His thought only. Hence the existence of God is the only truth (''[[w:Haqq|Haqq]]''), and the concept of a separate created Universe is falsehood, Arabic: (''[[w:Batil|Batil]]''). [[w:Ibn Arabi|Ibn Arabi]] is most often characterized in Islamic texts as the originator of the doctrine of wahdat al-wujud, however, this expression is not found in his works and the first who employed this term was perhaps, in fact, the Andalusian mystical thinker [[w:Ibn Sabin|Ibn Sabin]].<ref>[[w:Seyyed Hossein Nasr|S.H. Nasr]] (2006), ''Islamic Philosophy from Its Origin to the Present: Philosophy in the Land of Prophecy'', State University of New York Press, p. 156</ref> Although he frequently makes statements that approximate it, it cannot be claimed that "Oneness of Being" is a sufficient description of his ontology, since he affirms the "manyness of reality" with equal vigor. <ref name=autogenerated2>Imaginal worlds, William Chiittick(1994), pg.15</ref> The central doctrine of Sufism, sometimes called [[w:Wahdat-ul-Wujood|Wahdat-ul-Wujood]] or Wahdat al-Wujud or Unity of Being, is the Sufi understanding of Tawhid (the oneness of God; absolute monotheism). Put very simply, for Sufis, Tawhid implies that all phenomena are manifestations of a single reality, or Wujud (being), which is indeed al-Haq (Truth, God). The essence of Being/Truth/God is devoid of every form and quality, and hence unmanifest, yet it is inseparable from every form and phenomenon, either material or spiritual. It is often understood to imply that every phenomenon is an aspect of Truth and at the same time attribution of existence to it is false. The chief aim of all Sufis then is to let go of all notions of [[w:duality|duality]] (and therefore of the individual self also), and realize the divine unity which is considered to be the truth. <blockquote>The expression wahdat al-wujud is built from two words - wahda and wujud - both of which were important for Islamic throught from early times. Islamic theory and practice is grounded in the shahada or the giving witness that "There is no god but God," an expression often called kalimat al-tawhid, the "statement through which God's Unity is declared." The basic sense of tawhid or the declaration of God's Unity is that everything in creation derives from God, who is One Reality. The word tawhid comes from the same root as wahda, as do other related and often discussed terms such as ahad and wahid ("one") and ahadiyya and wahdaniyya ("oneness" or "unity"). Alrady int he saying of 'Ali we come across areference to four different meanings for the apparently simple statement, "God is One." [note, diacritics need to be added and italics as well as proofing]<ref>Source: [http://books.google.com.au/books?id=Ko8F4EJRFv4C&pg=PA70&dq=Wahdat+al-Wujud&hl=en&ei=h8XrS9iDI4vY7AOP88iLBg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwAA#v=onepage&q=Wahdat%20al-Wujud&f=false] (accessed: Thursday May 13, 2010), p.70</ref></blockquote> [[w:Al-Ghazali|Al-Ghazali]] (1058 - 1111) important for the dialogue and acceptance between 'orthodox' Inslamic Sunni praxis and soteriology and that of the 'mystical' Sufi... need to cite and ensure correct. NB: Beauford, refer book downloaded from Scribd regarding the impact Ghazali has had on Western thought and Western Philosophical thought as a point of entry. [[w:Jalal ad-Din Muhammad Rumi|Rumi]], (1207–1273), one of the most famous Sufi masters and poets, has written that what humans perceive as duality is in fact a veil, masking the reality of the Oneness of existence. "All desires, preferences, affections, and loves people have for all sorts of things," he writes, are veils. He continues: "When one passes beyond this world and sees that Sovereign (God) without these 'veils,' then one will realize that all those things were 'veils' and 'coverings' and that what they were seeking was in reality that One." The veils, or rather, duality, exists for a purpose, however, Rumi contends. If God as the divine, singular essence of all existence were to be made fully manifest to us, he counsels, we would not be able to bear it and would immediately cease to exist as individuals. [[w:Ali-Shir Nava'i|Ali-Shir Nava'i]] (9 February 1441 – 3 January 1501) wrote Lisan-ol-tayr (لسان الطیر or "Language of Birds", following Attar's Manteq-ol-tayr منطق الطیر or Speeches of Birds), in which he expressed his philosophical views and Sufi ideas. Lison ut-Tayr ( The Language of the Birds ) - An epic poem that is an allegory for our need to seek God, whatever our excuses may be. The story begins with the birds of the world realizing that they are far from their king and need to seek him. They begin the long and hard journey with many complaints, but a wise bird encourages them through admonishment and exemplary stories. It was written in 1498-99, and consists of 3598 couplets. In the introduction Navoi noted he wrote this epic poem as a response to "Mantiq-ut Tayr" by Fariduddin Attar and used the pseudonym "Foniy". ====Asma Path==== The [http://www.bahaistudies.net/asma/ Asma Path] (Arabic: asmāʾ, names), an online contemplative community (jamāʿa) that hold to the 'Oneness of God' (Tawḥīd). God has no partners (shirk). Anyone drawn to the Ṣūfī background or context of the Baháʾí Faith may enter this Ṣūfī path (ṭarīqah). It is: informal, service-oriented (“engaged”), postmodern (“emerging”), without any clergy, and NOT an official Baháʾí activity. ==Poetry and nonduality== [[File:Flores.gif|thumb]] <blockquote>"What a lovely thing a rose is!….There is nothing in which deduction is so necessary as in religion…It can be built up as an exact science by the reasoner. Our highest assurance of the goodness of Providence seems to me to rest in the flowers. All other things, our powers, our desires, our food, are all really necessary for our existence in the first instance. But this rose is an extra. Its smell and its colour are an embellishment of life, not a condition of it. It is only goodness which gives extras, and so I say again that we have much to hope from the flowers".</blockquote> :-- Sherlock Holmes extracted from: Doyle, Sir Arthur Conan, ''Memoirs of Sherlock Holmes'', [[s:The Adventure of the Naval Treaty|<nowiki>[</nowiki>The Adventure of<nowiki>]</nowiki> The Naval Treaty]]; London: 1894. I have been reading Orphic Hymns today amongst other works through the inspiration of McEvilley and I feel the importance of moving away from the mechanization of sections established through chronology or locality. This Intranet and metameditation is going to get significantly more chaotic before order manifests. I don't want to force a structure and therefore, my creative flow until themes and an organic way of ordering the material traces of nonduality emerge. I was following an intelligible order based upon distinctions according to Western scholarship such as Ancient, Classical, etc. But that was a vestige from the Wikipedia Nonduality article. The changing fortunes of the wheel are given significance through an understanding of the incarnation and re-embodiment lineages. So much Dharma discourse that has entered English lacks the poetry of the original. This is a significant flaw. The Vedas are hymns with rhyme, rhythm, meter and assonance. The Upanishads and Puranas are poetry as much as they are story and thereby philosophy as the narratives and stories are embedded, interpenetrate, delimit and offset oneanother. Somewhere on my learning Tibetan weblog here at Wikiversity when I noted elements of Sanskrit for my excursion into Tibskrit and apprenticeship with the Jigme Lingpa's Longchen Nyingtik Ngondro, that poetry is one of the traditional disciplines for a person to be finished, in the classical Victorian sense as well as the soteriological of the Dharma. [[File:Saraha.jpg|thumb|Saraha|right]] ===Mahasiddhas=== *http://www.scribd.com/doc/45973468/The-Alchemical-Body-Siddha-Traditions-in-Medieval-India ====Saraha==== Gunther (1973: p.5) relates the following awakening-story of [[w:Saraha|Saraha]] the [[w:Brahmin|Brahmin]] (also known as Rahula) who outwardly contravenes Vedic injuction apportioned to his '[[w:Varna (Hinduism)|class]]' in consuming intoxicants and cavorting with youthful forward buxom-forefold nay fourfold pretty-upfront wenches bearing intoxicants, so primed he has a visionary trance or pure-vision wherein a quest is entrusted from a Natha which he fulfills and thereby wins wisdom hard-won from a craftsmaiden-arrowsmith of the marketplace: <blockquote>Once when this Brahmin Rahula was roaming in his district and came to a garden, the four Brahmin girls approached him with cups of beer and begged him to drink them. Although he protested he succumbed to their entreaties and drank the four cups of beer in large gulps. He had four particularly pleasant sensations and, as had been prophesied about him, he met the Bodhisattva Sukhanatha face to face. Blessed by him he was exhorted: "In this city there lives a mysterious arrowsmith woman who is making a four-piece arrow. Go to her and many beings will profit by it." With these words the vision disappeared.<br > Through the sustaining power of his vision the mystic awareness of the coemergence of both transcendence and immanence was born in him. Thinking that he would have to act after this instantaneous realization of spiritual freedom, he went to the big market place and there he saw a young woman cutting an arrow-shaft, looking neither to the right nor to the left, wholly concentrated on making an arrow. Coming closer he saw her carefully straightening a reed with three joints, cutting it both at the bottom and at the top, inserting a pointed arrowhead where she had cut the bottom into four sections and tying it with a tendon, putting four feathers where she had split the top into two pieces and then, closing one eye and opening the other, assuming the posture of aiming at a target. When he asked her whether she was a professional arrowsmith she said: "My dear young man, the Buddha's meaning can be known through symbols and actions, not through words and books." Then and there the spiritual significance of what she was doing dawned upon him.<ref>[http://books.google.com.au/books?id=FTU-AAAAIAAJ&printsec=frontcover&dq=The+Royal+Song+of+Saraha:+A+Study+in+the+History+of+Buddhist+Thought&source=bl&ots=dFSH7UyYpz&sig=uSnDCV4yn_Yfphv88V8MAYopqOw&hl=en&ei=OV3tS6zlN43Y7AO9_IioBg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBQQ6AEwAA#v=onepage&q&f=false] </ref></blockquote> The arrowsmith was in 'one-pointed focus' (Sanskrit: ekagraha-drishti), a dynamic flow meditation with support, wherein specifically she is crafting an arrow and sporting the Tao of the craft-master wherein through drishti which extends through samyama into nondual pratyaksha wherein subject, object and action are nondual. It was in this nondual darshan that she gave her Mahamudra transmission to Rahula prophesied by the 'Master-of-Bliss' (Sukhanatha). There is further import not conveyed by Gunther who is translating into English from the Tibetan. The arrow metaphor and motif in recurrent in Saraha's teachings and is visually foregrounded in his Tibetan iconography, where as pictured herewith he is often depicted with an 'arrow' or '[[w:Dadar (ritual tool)|dadar]]' (Tibetan: mda' dar). Further to this, the comment of [[w:Judith Simmer-Brown|Simmer-Brown]] (2001: p.&nbsp;359) as follows is sage: <blockquote>The word for ''arrow'' is ''mda<nowiki>'</nowiki>'', which is identical in pronunciation to the word for ''symbol'', ''brda<nowiki>'</nowiki>''.<ref>Simmer-Brown, Judith (2001). ''Dakini's Warm Breath: the Feminine Principle in Tibetan Buddhism''. Boston, USA: Shambhala. {{ISBN|1-57062-720-7}} (alk. paper). p.359</ref> </blockquote> Homonyms are often exploited in humour and in adverting and are pervasive throughout manifold spiritual traditions as teaching tools. Hononyms are as a general rule speech-community specific and may not even be intelligible in different dialects. Homonyms are generally lost in the translation as are other [[w:Category:Poetic_devices|poetic]] and [[w:literary device|literary device]] so important for an accomplished work. Saraha was a famed Marathi poet and singer of songs, a Mahasiddha. The enumerations of 'three' and 'four' repeated are twilight language for the wheel-of-joy in its three-fold and four-fold emanations. Importantly, the Four Joys (S: caturananda; T: dga'ba bzhi) are an experience of Mahamudra.<ref>Source: [http://books.google.com.au/books?id=-3804Ud9-4IC&pg=PA252&lpg=PA252&dq=%22four+joys%22+beer&source=bl&ots=FPHCYYKgMv&sig=cSMiHQNXbXkvBxB672l_IxGgN0g&hl=en&ei=HXrtS57YEIuC7QOQ6NmmBg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBgQ6AEwAQ#v=onepage&q&f=false] (accessed: Saturday May 15, 2010), p.252</ref> ===Northrop Frye=== *[[w:Northrop Frye|Fye]] (1912 – 1991) and the Avatamsaka... (NB: excellent source on Korean Tradition of Avatamsaka) *http://books.google.com.au/books?id=riySOk90TJMC&pg=PA291&dq=The+Flower+Ornament+Scripture+:+A+Translation+of+the+Avatamsaka+S%C5%ABtra&hl=en&ei=FortS7zRFZGgkQWmuZjoBg&sa=X&oi=book_result&ct=result&resnum=7&ved=0CEYQ6AEwBg#v=onepage&q&f=false *http://books.google.com.au/books?id=EC11dO5u4akC&pg=PA6&lpg=PA6&dq=The+World+Text+avatamsaka+metaphor&source=bl&ots=twiCwaCWe7&sig=owcLtKG4JsqLEopMPPqRbmV-oUw&hl=en&ei=L4_tS7OoBoHi7APMiIGdBg&sa=X&oi=book_result&ct=result&resnum=8&ved=0CDQQ6AEwBzgK#v=onepage&q=The%20World%20Text%20avatamsaka%20metaphor&f=false ==Circle dances== This is another area of cultural practice that I intuit has some import for nonduality and a resolution of plurality into unity or establish a sense of the mystical relationship of the many to the One. *http://books.google.com.au/books?id=pIHbKCpj9HEC&pg=PA228&lpg=PA228&dq=dga'+ba+bzhi&source=bl&ots=u66Lyb_hhj&sig=sbro0uDJQrpYEJZTZO4ZyWVCyL0&hl=en&ei=cIDtS7OAEI_g7AOigOWYBg&sa=X&oi=book_result&ct=result&resnum=4&ved=0CCUQ6AEwAw#v=onepage&q=dga'%20ba%20bzhi&f=false ==The Mystery of that which sustains== <blockquote>"Let it here be noted that the Greek fables originated in spiritual mystery and real vision, which are lost and clouded in fable and allegory..."</blockquote> Blake, ''[[s:Life_of_William_Blake_(1880),_Volume_2/Prose_writings/A_Vision_of_the_Last_Judgment Life of William Blake (1880), Volume 2/Prose writings/A Vision of the Last Judgment|A Vision of the Last Judgment]]'', extract To fully embrace this discourse an awareness of Humanity as embedded in the Web of Life a reticulum of dynamic evolving systems organic and inorganic is required. One factor that is not so visible in such analyses of nonduality that I have so far encountered is a grounding in the spiritual disciplines and praxis that institute such an experience. Well, as yet, this has not been published and presented in text. This is just a meditation and file note the salience of which is drawn from my own personal experience, research and realizations. Hopefully, it is clear why I open with Christian traditions for English speaking peoples as to ignore them or to treat them tangentially is distorting historicity. It is only a recent phenomenon that other cultures are no longer presented to English speaking peoples by way of Christan mores, recent in this context is a play of a hundred years. There were portents and precursors far earlier than this, but they were not commonplace. [[w:Grant_Allen|Allen]] (1848 – 1899) in his work of 1897, identified Christ as partaking of the [[w:mythos|mythos]] (such would be known later by Jung as "archetype") of the Corn God (1897, 2000: pp.136-137) states: <blockquote>"The earliest known rite of the Christian Church was the sacramental eating and drinking of bread and wine together; which rite was said to commemorate the death of the Lord and his last supper, when he eat and drank bread and wine with his disciples. The language put into his mouth on this occasion in the Gospels, especially the Fourth, is distinctly that of the corn and wine god. "I am the true vine; ye are the branches." "I am the bread of life." "Take, eat, this my body." "This is my blood of the new testament." Numberless other touches of like kind are scattered through the speeches. In early Christian art, as exhibited in the catacombs at Rome, the true vine is most frequently figured; as are also baskets of loaves, with the corresponding miracle of the loaves and fishes. Multiplication of bread and wine are the natural credentials of the corn and wine god."<ref>Source: [http://books.google.com.au/books?id=N2CeS1MGOIQC&pg=PA137&lpg=PA137&dq=christ+as+corn+god&source=bl&ots=VvnRHqHVL9&sig=35dXWSVlu-y2jNW5S_n1137ZBdI&hl=en&ei=ZQfpS-uUMtGIkAWItf2CCw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBkQ6AEwAA#v=onepage&q=christ%20as%20corn%20god&f=false] (accessed: Tuesday May 11, 2010), pp.136-137</ref></blockquote> <blockquote>"And Jesus answered them, saying, The hour is come, that the Son of man should be glorified. Verily, verily, I say unto you, Except a corn of wheat fall into the ground and die, it abideth alone: but if it die, it bringeth forth much fruit. He that loveth his life shall lose it; and he that hateth his life in this world shall keep it unto life eternal."—John 12:23-25.</blockquote> I feel it important to remind myself and my reader that it is the [[w:Holy Grail|Holy Grail]] which is the vessel of Christ at the Last Supper, and that this partakes of the mythos of the Drinking Horn and the Horn of Plenty, as well as the Sacred Quest and Sacred Feast. Also, in the Bible the word "corn" is used to refer to cereal crops such as wheat and does not solely denote "corn" proper. In this passage corn should be understood as being grain: <blockquote>"Israel then shall dwell in safety alone; the fountain of Jacob shall be upon a land of corn and wine; also his heavens shall drop down dew" (Deuteronomy, 33.28 This divine substance was something special, refined.</blockquote> If we tentatively put aside the very involved interiority of John and his mystical language and its manifold representations into English, let us just be aware of the association of Jesus with the corn that yields fruit upon its death. Death yields life. Food is key in human society for without it we do not have the energy to maintain our bodily processes and the constituents of our bodies disband. Subsistence is in essence a mystery. Imbibing is nonduality as praxis. Mindful eating was a teaching of the Shakyamuni Buddha. The [[w:cornucopia|cornucopia]] (Latin: Cornu Copiae) is a symbol of food and abundance dating back to the 5th century BC, also referred to as the food of worship and holiness, Horn of Amalthea, harvest cone, and horn of plenty. Animals horns were often used anciently by many peoples as storage devices for food and water as well as ritual instruments, eg. [[w:Drinking horn|Drinking horn]]. It is poignant that the horn is secured from sacrifice and that it yields plenty. There is an interesting myth as to its origin which has the reciprocity between species as key. Food in the form of sheaf of corn is one of the few grounded elements of the [[w:Eleusinian Mysteries|Eleusinian Mysteries]] available to modern scholarship. The Eleusinian Mysteries are the most famous of what is known as the [[w:Greco-Roman mysteries|Greco-Roman mysteries]]. Another mystery cycle is known as the [[w:Mithraic Mysteries|Mithraic Mysteries]] and it this rite [[w:animal sacrifice|animal sacrifice]] and feast were central. And it is within this context that the rite of Christ though not set as he was Semitic, but it is through the Romans that the textual tradition as they interpreted it was transmitted. Hence, Christos the savior partakes of the Corn God, Eucharist as "blood on the corn" the archetype of the dying god. Also Christ as the Lamb-of-God partakes in the ancient practice of scapegoating. The rite of the [[w:Blót|Blót]] for the Norse wherein "blood" as "blessed" terms which entered the English from the Norse and are etymologically related terms, as are similarly related terms from the Greek "sacred" and "sacrifice". Christ and the Last Supper. The [[w:Agape feast|Agape Feasts]]. The [[w:Ganachakra|Ganachakra]] Rite with its liturgy and group-practice incorporates aspects of the [[w:Generation Stage|Generation Stage]] and [[w:Completion Stage|Completion Stage]] as part of the rite. <blockquote> "In the Shaivite tradition, the God's companions are described as a troupe of freakish, adventurous delinquent and wild young people, who prowl in the night, shouting in the storm, singling, dancing and ceaselessly playing outrageous tricks on sages and gods. They are called Ganas, the 'vagabonds', corresponding to the Cretan Korybantes and the Celtic Korrigans (fairies' sons). Like the Sileni and Satyrs, some of them have goat's or bird's feet. The Ganas mock the rules of ethics and social order. The personify the joy of living, courage and imagination, which are all youthful values. They live in harmony with nature and oppose the destructive ambition of the city and the deceitful moralism which both hides and expresses it. These delinquents of heaven are always there to restore true values and to assist the 'God-mad' who are persecuted and mocked by the powerful. They personify everything which is feared by and displeases bourgeois society and which is contrary to the good morale of a well-policed city and its palliative concepts."</blockquote> *[http://books.google.com.au/books?id=QDQK7l13WIIC&lpg=PP1&ots=S8I1fsO1RA&dq=Gods%20of%20Love%20and%20Ecstasy&pg=PA99#v=onepage&q&f=false From Shiva and Dionysus - by Shri Alain Daniélou (reprinted as Gods of Love and Ecstasy) page 99] The rite of the meal and its symbology and liturgy of the [[w:Freemasons|Freemasons]]. The Freemason's were and arguably are still, a Mystery tradition. <blockquote>"The early Mason was accustomed to elaborate and extensive Feasts that might encompass several days. To give you an idea of the magnitude of a Feast, a partial bill of fare for a banquet (50 people) in 1506 included the following: 36 chickens, 1 swan, 4 geese, 9 rabbits, 2 rumps of beef tails, 6 quails, 50 eggs, 4 breasts of veal. The meal would be enjoyed in a formal gathering where the master would preside over the ceremonies attending the meal and direct a series of toasts."<ref>Source: [http://www.themasonictrowel.com/Articles/General/craft_files/masonic_feasts_gltx.htm] (accessed: Tuesday May 11, 2010)</ref></blockquote> The harvest has so many rich metaphorical associations in the Biblical tradition but let us just be clear in understanding how primary, how fundamental, the harvest of food is and embodying it as a sacred act. I tender that the Mysteries have always been part of the Human Condition as part of the rite of sacred play and that they just change name and form. One of the problems of being where we are presently is that we reify history as actuality and I find it mindful to meditate on all the unknown contacts between cultures that have been made since time immemorial which have influenced each other. Just as each person, if they meditate on how chance meetings and individuals they don't even know have shaped their life and worldview. *NB: B9HH this book is new at State Library of Victoria: Bowden, Hugh (2010). ''Mystery Cults of the Ancient World''. London: Thames & Hudson Ltd. {{ISBN|9780691146386}} ==Deep Ecology== [[File:The Earth seen from Apollo 17.jpg|thumb|Earth ''qua'' Gaia]] {{center top}} ''But ask the animals, and''<br /> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;''they will teach you;''<br /> ''the birds of the air, and''<br /> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;''they will tell you;''<br /> ''ask the plants of the earth,''<br /> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;''and they will teach you;''<br /> ''and the fish of the sea will''<br /> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;''declare to you.''<br /> '''[[w:Book of Job|Job]] 12:7-8''' {{center bottom}} I have always wanted to write on [[w:Deep Ecology|Deep Ecology]] and now it is presenting itself. ===Economy and ecology as nondual: the textured and checkered discourse of Capitalism and the System=== Capitalism isn't a monolithic entity. Capitalism isn't an evil that is being done to us. We collectively assent to the convention. Humanity instituted this system and it has been valuable in providing advanced technology and significant material comfort. This technology and comfort is disproportionately distributed. This inequity is always and evermoreso receding and being addressed and redressed, sometimes pure tokenism sometimes deep egalitarianism. Inequity to the environment is being addressed and is informed by the discourse of embedded narratives, embedded systems. Capitalism like money is a cultural convention, a cultural construction. Capitalism and our economy may both be modelled on organic systems which is curious and very important to note. They are dynamic evolving adaptive systems. Feedback models, [[w:Gregory Bateson|Bateson]]. The interdisciplinary nous of [[w:Permaculture|Permaculture]] with its [[w:ecological principles|ecological principles]] interpenetrated with the education model, the unity of the artist-craftsperson and the Industrial pragmatism of the bauhaus where '[[w:Form follows function|form (ever) follows function]]' and the subsequent qualifications as per design discourse and designing for integration would significantly address all areas of environmental concern. This must be qualified with the need of Humanity to integrate with as well as preserve wild nature: wild nature of both the natural world and our own primordial nature: integrate nature and culture, informed by [[w:Transpersonal Psychology|Transpersonal Psychology]] to redress spiritual crisis through the discourse of science. I would like to say that in my experience Capitalism and Democracy breed specializations, compartmentalized knowledges. In my experience both Capitalism and Democracy create a wealthy and intellectual echelon that is an overlapping but not a discrete sets. Capitalism still thrives on industrialization where workers, the powerhouse of the system do menial tasks, unengaging work that perpetuate dissatisfaction. Dissatisfaction that is never quenched by acquisition. The workers who are in general lower-paid as well as many of those who have the higher-paying jobs and managerial power have very little time in truth as they sell their time, the stuff of their life for what they perceive as their gain. Their specializations in their work function tends to ensure that they are not informed as to the richness of the world, a world that is dynamic and ever-diversifying and they are unaware for the most part of the manifold rich discourses that are available. Popular culture which is fundamentally disposable for the most part and necessarily so at it must always and evermoreso churn to keep the wheels of acquisition and desire in motion, and keen the feelings of envy and lack which fuels the drive for acquisition. Democracy of all kinds to function as a form of legitimate governance requires the constituency to be informed on matters of concern. Time and quality information are mandatory to provide the humus forming valid cognition in the ecology of mind based upon sound discourse. [[w:Democracy|Democracy]] is in general [[w:Representative Democracy|Representative Democracy]] and it must be outlined that we now have the means for [[w:Direct Democracy|Pure Democracy]]. We may even design and implement more workable and equitable systems if we truly set our endeavour to do so. I perceive the key to redress as an intention issue a matter of human heartfulness and one of integrated interdisciplinary design. [[w:Sustainable_city|Sustainable city]] *http://books.google.com.au/books?id=nGPBwDgTvAYC&pg=PA362&dq=permaculture+bauhaus&hl=en&ei=dPnzS7DPBIuK6gPY48iEDA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDQQ6AEwAA#v=onepage&q=permaculture%20bauhaus&f=false Using the body metaphor and the body as an organic system, a living system that is a dynamic evolving adaptive system, Loy () demonizes unresponsive Capitalism as cancer [[w:metastasized|metastasized]] upon the body of Gaia: <blockquote>Capitalism made more sense a couple [of] centuries ago when the Earth seemed infinite and capital was relatively scarce. Today the obvious metaphor is cancer on a planetary scale. Cells become cancerous when they mutate into uncontrolled growth and spread throughout the body to disrupt its healthy functioning. Unfortunately, that is not a bad description of our collective situation now.</blockquote> "Greed is not a virus that has infected the economic hard drive; it has become the software that runs our economy."<br > "Greed also takes two forms, according to our means. “In a consumer society there are two kinds of slaves: the prisoners of addiction and the prisoners of envy” (Ivan Illich)." <blockquote>The ultimate irony is that money in itself is literally worthless: whether pieces of paper or numbers in bank accounts, money has value only because it is our socially-agreed medium of exchange. A $100 bill is just a piece of paper. We can’t eat it, drink it, ride on it, etc. We forget that money is a social construct — a kind of group fantasy. The anthropologist Weston LaBarre called it a psychosis that has become normal, “an institutionalized dream that everyone is having at once.”</blockquote> *http://www.shambhalasun.com/sunspace/?p=11692 ===Deep Ecology discourse=== [[File:WuXianGuan-Immortals-footprint-0519.jpg|thumb|The supposed "footprint of a ''xian''", a little pond in Guangzhou's Temple of the Five Immortals]] <poem> 'Tis refreshing to old-fashioned people like me To meet such a primitive Pagan as he, In whose mind all creation is duly respected As parts of himself--just a little projected; And who's willing to worship the stars and the sun, A convert to--nothing but Emerson. </poem> Robert Lowell, ''[[s:A_Fable_for_Critics|A Fable for Critics]]'' extract *World as lover, World as self... extract handwritten notes on J Macy from oldskool notebook *[[w:Uddhava Gita|Uddhava Gita]] from the Bhagavat Purana *Dogen and Zen metaphorical extension *subject as context, subject and context become self-reflexive, Zen [http://ccbs.ntu.edu.tw/FULLTEXT/JR-MISC/60780.htm Loy, David (1997). ''Loving the World as Our Own Body: The Nondualist Ethics of Taoism, Buddhism and Deep Ecology.''] <poem> '''[[w:īśopaniṣad|Īśopaniṣad]], verse 6''' yas tu sarvāṇi bhūtāny ātmany evānupaśyati sarva-bhūteṣu cātmānaḿ tato na vijugupsate </poem> [[w:Jizang|Jizang]] ([[w:Chinese_language|Chinese]]: 吉蔵; [[w:Pinyin|Pinyin]]: Jízàng; [[w:Wade–Giles|Wade–Giles]] Chi-tsang) (549–623) was a Chinese Buddhist monk and scholar who is often regarded as the founder of the [[w:Three Treatise School|Three Treatise School]]. He is also known as Jiaxiang or Master Jiaxiang (嘉祥, Chia-hsiang), because he acquired fame at the Jiaxiang Temple. Jizang was a Chinese author of Persian origin who wrote "Plants Become Buddhas" (草木成佛Chinese: caomu chengfo; Japanese: somoku jobutsu) and this was very innovative and controversial in the Tang Dynasty. NB: Beauford the Chinese characters of the title in sequence denote: grass [[:wikt:草|草]] , tree/wood [[:wikt:木|木]] , become [[:wikt:成|成]] , buddha [[:wikt:佛|佛]]. (This may be the primary resource: http://zh.wikisource.org/zh/%E4%BD%9B%E5%AD%B8%E5%A4%A7%E8%BE%AD%E5%85%B8/%E8%8D%89%E6%9C%A8%E6%88%90%E4%BD%9B ) *http://www.scribd.com/doc/47341101/Plants-in-Early-Buddhism-the-Far-Eastern-Idea-of-the-Buddha-Nature-of-Grasses-and-Trees *Rambelli, Fabio (2001). ''Vegetal Buddhas: Ideological Effects of Japanese Buddhist Doctrines on the Salvation of Inanimate Beings''. Italian School of East Asian Studies Occasional papers 9. Kyoto: Scuola Italiana di Studi sull' Asia Orientale. Source: http://www.scribd.com/doc/45813044/Vegetal-Buddhas ==The Middle Way== The ''[[w:Doctrine of the Mean|Doctrine of the Mean]]'' (Chinese: 中庸; zhōng yōng), is at once a teaching tool and way of being as well as one of the books of Neo-Confucianism. The composition of the text is attributed to Zisi (or Kong Ji) the only grandson of Confucius. This is a text on the teaching of "everything in moderation", including on the rare occasion moderation as well *heheheheeh*. [http://books.google.com.au/books?id=JiALtt67xicC&pg=PA194&dq=%22Constant+Mean%22+(James,+Legge)&ei=riDoS5SqC5XOlQTR6uXICQ&cd=1#v=onepage&q&f=false Chung-nî said, "The superior man embodies the course of the Mean; the mean man acts contrary to the course of the Mean."] http://chinese.dsturgeon.net/text.pl?node=10262&if=en == Taoism == [[Image:Eight Immortals Crossing the Sea - Project Gutenberg eText 15250.jpg|thumbnail|left|250px|"The [[w:Eight Immortals|Eight Immortals]] Crossing the Sea", from ''Myths and Legends of China'', 1922, E.T.C. Werner.]] <blockquote>"You can also read about Taoist immortals in the Taoist Canon, the collection of books that form the scriptures of Taoism. Personally, I've never been thrilled by the way the immortals are portrayed in these biographies. The entries read like articles from an encyclopedia and the characters appear dull and remote. After reading one, I always felt that I learned ''about'' the immortals rather than ''from'' them. On the other hand, in the operas, radio plays, and stories told by my grandmother and the Banyan Tree Park storytellers, the characters came alive. At the end of each story, I felt that I had not only met the immortals but had learned from them."<ref>Wong, Eva (2001). ''Tales of the Taoist immortals.'' Edition: illustrated. Shambhala Publications. {{ISBN|1570628092}}, 9781570628092. Source: [http://books.google.com.au/books?id=SisxRpSwdrcC&printsec=frontcover&dq=taoist+immortals&ei=5U3_S6ahFY3slQS_oIzdDA&cd=1#v=onepage&q&f=false] (accessed: Friday May 28, 2010), p.3</ref></blockquote> In ancient times, the 'feather men' ([[:wikt:羽|羽]][[:wikt:人|人]]; yu ren) were flying 'immortals' ([[:wikt:仙|仙]]; [[w:Xian (Taoism)|Xian]]), whose bodies were covered with a coat of feathers. Yu literally means "feather". Yuren it is an alternative designation for a Taoist priest. http://www.pantheon.org/articles/y/yu-ren.html The belief in sprites (jing) is founded on the conception that as living beings age, they accumulate spirit (jing). http://findarticles.com/p/articles/mi_go2081/is_n2_v114/ai_n28648194/pg_5/ Mountain gods were less purposefully evil than amoral, self-centered and egocentric. <blockquote>"All mountains, whether large or small, have gods and spirits."<ref>Baopuzi (Basic Sinological Series ed.; rpt., Taibei: Shijie shuju, 1969): 17.76. Cf. James Ware, Alchemy, Medicine, and Religion in the China of A.D. 320 (Cambridge, Mass.: M.I.T. Press, 1966), 279-80.</ref></blockquote> [[w:Ge Hong|Ge Hong]] (283-343), rendered in English by unknown translator [[File:Yin yang.svg|thumb|[[w:Taijitu|Taijitu]]]] The following is an English rendering from an extract drawn from the Tonggyong Taejon: <blockquote>"I have a divine spell; its name is a mysterious medicine: its shape is a circle [two bows put together] or the Great Ultimate."<ref>Source: [http://books.google.com.au/books?id=MOb9thbs-IcC&pg=PA378&lpg=PA378&dq=Wei+Mo+Chieh+So+Shuo+Ching&source=bl&ots=nLeZ3J1DPO&sig=3_xt-DtLoP-WVwZC1SEvKUcuVos&hl=en&ei=kRLoS4rxLZWekQWjwYW8Bg&sa=X&oi=book_result&ct=result&resnum=7&ved=0CCwQ6AEwBg#v=onepage&q=Wei%20Mo%20Chieh%20So%20Shuo%20Ching&f=false] (accessed: Tuesday May 11, 2010), p.38</ref></blockquote> Dechar (2005: p.5-6) identifies that the terms "Tao" and "[D]harma" are etymologically rooted by identifying the [[:wikt:etymon|etymon]] "da": <blockquote>"The word Tao has no exact English translation, but it relates most closely to the Western idea of wholeness, to the unknowable unity of the divine. When used by the Taoist philosophers, Tao became the Way, the path or cosmic law that directs the unfolding of every aspect of the [U]niverse. So Tao is the wisdom of the divine made manifest in nature and in my individual life. The Chinese word Tao has an etymological relationship to the Sanskrit root sound "da", which means "to divine something whole into parts". The ancient Sanskrit word dharma is also related to this root. In the Buddhist tradition, dharma means "that which is to be held fast, kept, an ordinance or law...the absolute, the real." So, both dharma and Tao refer to the way that the One, the unfathomable unity of the divine, divides into parts and manifests in the world of form."<ref>Dechar, Lorie Eve (2005). ''Five spirits: alchemical acupuncture for psychological and spiritual healing''. Illustrated edition. Lantern Books, {{ISBN|1590560922}}, 9781590560921. Source: [http://books.google.com.au/books?id=xupf6B1OpZsC&pg=PA5&dq=tao+dharma&hl=en&ei=EaXiS5XIMc6GkAWp6LAZ&sa=X&oi=book_result&ct=result&resnum=7&ved=0CEoQ6AEwBg#v=onepage&q=tao%20dharma&f=false] (accessed: Thursday May 6, 2010), p.5-6</ref></blockquote> [[w:Taoism|Taoism]]'s ''[[w:wu wei|wu wei]]'' (Chinese ''wu'', not; ''wei'', doing) is a term with various translations (e.g. inaction, non-action, nothing doing, without ado) and interpretations designed to distinguish it from passivity. From a nondual perspective, it refers to activity that does not imply an ''"I"''. The concept of [[w:Yin and Yang|Yin and Yang]], often mistakenly conceived of as a symbol of dualism, is actually meant to convey the notion that all apparent opposites are complementary parts of a non-dual whole. The [[w:Tao Te Ching|Tao Te Ching]] has been seen as a nondualist text; from that perspective, the term "[[w:Tao|Tao]]" could be interpreted as a name for the Ultimate Reality (which, as the Tao Te Ching itself notes, is not the reality itself). *http://www.wikilivres.info/wiki/The_Tao_Te_Ching *http://books.google.com.au/books?id=vGFs3yIMCFoC&pg=PA342&dq=flower+ornament+sutra+cleary&hl=en&ei=gpTtS_ayEcyGkAWDhcHoBg&sa=X&oi=book_result&ct=result&resnum=6&ved=0CEkQ6AEwBQ#v=onepage&q&f=false ===Bagua=== [[File:Bagua-name-earlier.svg|thumb]] The '''ba gua''' (Chinese: 八卦; Pinyin: bā guà; Wade-Giles: pa kua; literally 'eight symbols') are eight diagrams used in [[w:Taoist|Taoist]] [[w:cosmology|cosmology]] to represent the fundamental principles of reality, seen as a range of eight interrelated concepts. Each consists of three lines, each line either "broken" or "unbroken," representing [[w:Yin and yang|yin]] or yang, respectively. Due to their tripartite structure, they are often referred to as "trigrams" in English. ===Lao Zi=== *[[w:Laozi|Lao Zi]] ===Ziran=== '''[[w:Ziran|Ziran]]''' (Chinese: [[:wikt:自然|自然]]; [[w:Pinyin|Pinyin]]: zìrán; [[w:Wade-Giles|Wade-Giles]]: tzu-jen; Sanskrit: [[w:Sahaja|Sahaja]]) is a key concept in [[w:Daoism|Daoism]] that literally means "self so; so of its own; so of itself" and thus "naturally; natural; spontaneously; freely; in the course of events; of course; doubtlessly" (Slingerland 2003, p.&nbsp;97; Lai, p.&nbsp;96). This Chinese word is a two-character [[w:compound (linguistics)|compound]] of ''zi'' (自) "nose; self; oneself; from; since" and ''ran'' (然) "right; correct; so; yes", which is used as a ''-ran'' suffix marking adjectives or adverbs (roughly corresponding to English ''-ly''). It is worth mentioning that in Chinese culture, the nose (or zi) is a common metaphor for a person's point of view (Callahan, 1989). The word 'ziran' first occurs in the [[w:Daodejing|Daodejing]] (17, 23, 25, 51) and refers to the structure of [[w:Tao|Dao]], which cannot be referred back to anything else. It is generally accepted that the philosopher Laozi, author of the Daodejing, coined the term. Ziran is a central concept of Daoism, closely tied to the practice of [[w:Wu wei|wuwei]], or effortless action. Ziran can be seen as the positive side of the Dao, with wuwei opposing it as the negative. *http://www.confuchina.com/05%20zongjiao/Lao%20Zi%27s%20Concept%20of%20Zi%20Ran.htm *http://books.google.com/books?id=Q20vRqOhwbIC&pg=PA103&lpg=PA103&dq=ziran+tzu-jen&source=bl&ots=6GkWh3_6A2&sig=POQKL9kMYfWBlNqqh-ZKUvYmxcc&hl=en&ei=KrA5TZmlHoO4vQOT74TCCg&sa=X&oi=book_result&ct=result&resnum=4&ved=0CCoQ6AEwAw#v=onepage&q&f=false *http://books.google.com/books?id=sO4FoJtiDQgC&printsec=frontcover&dq=Lai,+Karyn.+Learning+from+Chinese+Philosophies&source=bl&ots=mDGdFf1rpl&sig=R1gKC712oRmDLhs9T8EJboYae9I&hl=en&ei=WLY5TfC8DoeAvgO5gKmJCg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCEQ6AEwAg#v=onepage&q&f=false ==Confucianism== ===Mencius=== tzu-jen (refer above) is a key concept in [[w:Mencius|Mencius]], he may be the earliest! *http://books.google.com/books?id=hDUykaZSZHMC&pg=PA12&dq=tzu-jen&hl=en&ei=-bA5Te38JorIvQOSmcXoCg&sa=X&oi=book_result&ct=result&resnum=6&ved=0CD4Q6AEwBQ#v=onepage&q=tzu-jen&f=false ==Charnel Ground== [[Image:Gana Dancing Deogarh India.jpg|thumb|A Dancing gana, Dashavatara temple, [[Deogarh]]]] 'Charnel ground' (Devanagari: श्माशान; Romanized Sanskrit: śmāśāna; Tibetan pronunciation: durtrö; {{bo|t=དུར་ཁྲོད|w=dur khrod}})<ref>Rigpa Shedra (July 2009). 'Charnel ground'. Source: [http://www.rigpawiki.org/index.php?title=Charnel_ground] (accessed: Saturday December 19, 2009)</ref> is a very important location for [[w:sadhana|sadhana]] and ritual activity for Indo-Tibetan traditions of [[w:Dharma|Dharma]] particularly those traditions iterated by the [[w:Tantra|Tantric]] view such as [[w:Kashmiri Shaivism|Kashmiri Shaivism]], [[w:Kaula|Kaula]] tradition, [[w:Esoteric Buddhism|Esoteric Buddhism]], [[w:Vajrayana|Vajrayana]], [[w:Mantrayana|Mantrayana]], [[w:Dzogchen|Dzogchen]], and the sadhana of [[w:Chöd|Chöd]], [[w:Phowa|Phowa]] and [[w:Zhitro|Zhitro]], etc. The charnel ground is also an archetypal liminality that figures prominently in the literature and liturgy and as an artistic motif in Dharmic Traditions and cultures iterated by the more [[w:antinomian|antinomian]] and esoteric aspects of traditional Indian culture. My review of the literature and culture of Dharmic nondual traditions has time and time again pointed to the charnel ground, cremation ground, site of sky burial, cemetery, bodily relics, stupa and crypt. Though a charnel ground may have demarcated locations within it functionally identified as burial grounds, [[w:cemetery|cemeteries]] and [[w:crematorium|crematoria]] it is distinct from these as well as from [[w:crypt|crypt]]s or [[w:Burial vault (tomb)|burial vault]]s. Specifically, a charnel ground is an aboveground site for the putrefaction of bodies, generally human, where formerly living tissue is left to decompose uncovered. This unsanitary practice is now known to foster disease and generally runs counter to an orderly and well-governed nation-state. Throughout [[w:History of India|Ancient India]] and [[w:Medieval India|Medieval India]], charnel grounds in the form of open air [[w:crematorium|crematorium]]s were historically often located along rivers and many ancient famous charnel sites are now 'sanitized' pilgrimage sites (Sanskrit: [[w:tirtha|tirtha]]) and areas of significant domestic income through cultural tourism. In the [[w:Himalaya|Himalaya]] where tillable topsoil for burial and fuel for cremation is scarce and a valuable commodity, the location of a '[[w:Sky burial|Sky burial]]' is identified with a charnel ground.<ref>Gold, Peter (1994). ''Navajo and Tibetan Sacred Wisdom: The Circle of the Spirit'' (Paperback). Inner Traditions. {{ISBN|089281411X}}, {{ISBN|978-0892814114}}, p.141</ref> The [[w:Sutrayana|Sutrayana]] tradition of the 'Nine Cemetery Contemplations' (Pali: nava sīvathikā-manasikāra) of the ''[[w:Satipatthana Sutta|Satipatthana Sutta]]'' demonstrate that charnel ground meditations were part of [[w:Early Buddhism|Early Buddhism]]. 'Cemetery contemplations', as described in ''[[w:Mahasatipatthana Sutta|Mahasatipatthana Sutta]]'' ([[w:Digha Nikaya|DN]]: 22) and the ''[[w:Satipaṭṭhāna Sutta|Satipaṭṭhāna Sutta]]'' ([[w:Majjhima Nikaya|MN]]: 10): <blockquote>"...have as their objects a corpse one or two or three days old, swollen up, blue-black in colour, full of corruption; a corpse eaten by crows, etc.; a framework of bones; flesh hanging from it, bespattered with blood, held together by the sinews; without flesh and blood, but still held together by the sinews; bones scattered in all direction; bleached and resembling shells; heaped together after the lapse of years; weathered and crumbled to dust. At the end of each of these contemplations there follows the conclusion: "This body of mine also has this nature, has this destiny, cannot escape it. Similar are the 10 objects of loathsomeness (asubha q.v.)."<ref>Nyanatiloka (1980). ''Buddhist Dictionary: Manual of Buddhist Terms and Doctrines.'' Fourth Revised Edition edited by Nyanaponika. Kandy, Sri Lanka: Buddhist Publication Society. Source: [http://www.palikanon.com/english/wtb/s_t/siivathikaa.htm] (accessed: Tuesday December 24, 2009)</ref></blockquote> On the face of it or alternatively the cosmetic level, the charnel ground is simply a locality often [[w:chthonic|chthonic]] where bodies are disposed of, either by [[w:cremation|cremation]] or [[w:burial|burial]].<ref>Rigpa Shedra (July 2009). 'Charnel ground'. Source: [http://www.rigpawiki.org/index.php?title=Charnel_ground] (accessed: Saturday December 19, 2009)</ref> Though the charnel ground is to be understood as a polysemy and metaphor it must be emphasized that holy people as part of their sadhana and natural spiritual evolution grappling with death, impermanence and transition did historically in both India, China and Tibet as well as in other localities, frequent charnel grounds, crematoriums and cemeteries and were often feared and despised by people who did not understand their 'proclivities' (Sanskrit: anusaya). From a deeper structural significance and getting to the substantive bones of the Vajrayana spiritual point of view however, the charnel ground is full of profound [[w:Transpersonal psychology|transpersonal]] significance. It represents the 'death of [[w:ego|ego]]' (Sanskrit: atmayajna), and the end of: *'''attachment''' (Sanskrit: ''[[w:Upādāna|Upādāna]]''; Tibetan: ''[http://rywiki.tsadra.org/index.php/len_pa len pa]'') to this body and life *'''craving''' (Sanskrit: ''[[w:tṛṣṇā|Tṛṣṇā]]''; Tibetan: ''[http://rywiki.tsadra.org/index.php/sred_pa sred pa]'') for a body and life in the future *'''fear of death''' (Sanskrit: abhiniveśa) *'''aversion''' (Sanskrit: dveṣa; Wylie: zhe sdang) to the decay of 'impermanence' (Sanskrit: [[w:anitya|anitya]]).<ref>Rigpa Shedra (July 2009). 'Charnel ground'. Source: [http://www.rigpawiki.org/index.php?title=Charnel_ground] (accessed: Saturday December 19, 2009)</ref> It is worth noting that 'attachment', 'craving', 'fear' and 'aversion' abovecited in bold font are somewhat standardized and hence less-rich lexical choices for the [[w:semantic field|semantic field]] represented by the four of the 'Five Poisons' (Sanskrit: pancha [[w:klesha|klesha]]) they denote. Prior to spiritual realization, charnel grounds are to be understood as terrifying places, full of 'roaming spirits' (Sanskrit: [[w:gana|gana]]) and '[[w:hungry ghosts|hungry ghosts]]' (Sanskrit: [[w:pretas|pretas]]) indeed localities that incite consuming fear. In a charnel ground there are bodies everywhere in different states of decomposition: freshly dead bodies, decaying bodies, skeletons and disembodied bones.<ref>Rigpa Shedra (July 2009). 'Charnel ground'. Source: [http://www.rigpawiki.org/index.php?title=Charnel_ground] (accessed: Saturday December 19, 2009)</ref> [[w:Judith Simmer-Brown|Simmer-Brown]] (2001: p.127) conveys how the 'charnel ground' experience may present itself in the modern Western [[w:mindstream|mindstream]] situations of emotional intensity, protracted peak performance, marginalization and extreme desperation: <blockquote>"In contemporary Western society, the charnel ground might be a prison, a homeless shelter, the welfare roll, or a factory assembly line. The key to its successful support of practice is its desperate, hopeless, or terrifying quality. For that matter, there are environments that appear prosperous and privileged to others but are charnel grounds for their inhabitants--Hollywood, Madison Avenue, Wall Street, Washington, D.C. These are worlds in which extreme competitiveness, speed, and power rule, and the actors in their dramas experience intense emotion, ambition, and fear. The intensity of their dynamics makes all of these situations ripe for the Vajrayana practice of the charnel ground."<ref>[[w:Judith Simmer-Brown|Simmer-Brown, Judith]] (2001). ''Dakini's warm breath: the feminine principle in Tibetan Buddhism''. Boston, Massachusetts, USA: Shambhala Publications, Inc. {{ISBN|9781570629204}} (paperback), p.127</ref></blockquote> *[[w:Shiva|Shiva]] as [[w:Bhairava|Bhairava]] *'Outcastes' (Sanskrit: [[w:Chandala|Chandala]]) literally cast out from the castes, outside the [[w:Varnashrama Dharma|Varnashrama Dharma]] of the ''[[w:status quo|status quo]]''. [[File:GanpatiTibetan.jpg|thumb|left|Ganapati, Maha Rakta]] Tibetan representations of Ganesha show ambivalent or richly different depictions of him.<ref name="Nagar, p. 185">Nagar, p. 185.</ref> In one Tibetan form he is shown being trodden under foot by [[w:Mahakala|Mahākala]], a popular Tibetan deity.<ref name="Nagar, p. 185"/><ref>Getty, p. 42</ref> Other depictions show him as the Destroyer of Obstacles, sometimes dancing.<ref>Nagar, pp. 185-186.</ref> This play of Ganesha as both the "creator and remover of obstacles" as per his [[w:epithet|epithet]] as well in the two Vajrayana iconographic depictions of him as that which consumes (Maha Rakta) and that which is consumed (danced upon by Vignantaka) is key to the reciprocity rites of the charnel ground. ''Ganapati, Maha Rakta'' (Tibetan: tsog gi dag po, mar chen. English: The Great Red Lord of Hosts or [[w:Gana|Ganas]]) is a Tantric Buddhist form of Ganapati ([[w:Ganesha|Ganesha]]) related to the [[w:Chakrasamvara|Chakrasamvara]] Cycle of Tantras. The Sanskrit term 'rakta' holds the [[w:semantic field|semantic field]] of "blood" and "red". This form of Ganapati is regarded as an emanation of [[w:Avalokiteshvara|Avalokiteshvara]]. "...beside a lapis lazuli rock mountain is a red lotus with eight petals, in the middle a blue rat expelling various jewels, [mounted on his rat 'steed' (Sanskrit: [[w:vahana|vahana]])] Shri Ganapati with a body red in colour, having an elephant face with sharp white tusks and possessing three eyes, black hair tied in a [[w:topknot|topknot]] with a [[w:cintamani|wishing-gem]] and a red silk ribbon [all] in a bundle on the crown of the head. With twelve hands, the six right hold an axe, arrow, [[w:ankus|hook]], [[w:vajra|vajra]], sword and spear. The six left [hold] a [[w:pestle|pestle]], bow, [[w:khatvanga|khatvanga]], [[w:skullcup|skullcup]] filled with blood, skullcup filled with human flesh and a shield together with a spear and [[w:dhvaja|banner]]. The peaceful right and left hands are signified by the vajra and skullcup filled with blood held to the heart. The remaining hands are displayed in a threatening manner. Wearing various silks as a lower garment and adorned with a variety of jewel ornaments, the left foot is extended in a dancing manner, standing in the middle of the bright rays of red flickering light." ([[w:Ngorchen Konchog Lhundrup|Ngorchen Konchog Lhundrup]], 1497-1557)<ref>A Gift of Dharma to Kublai Khan By Chogyal Phagpa, Seventh Patriarch of Sakya. Ngorchen Konchog Lhundup, Ngor chos 'byung, folia 323?328. Translated by Jared Rhoton, 1976)</ref>. [[File:Vignataka.jpg|thumb|right|Vignantaka trampling or dancing upon Vinayaka]] This form of Ganapati belongs to a set of three powerful deities known as the 'mar chen kor sum' or the Three Great Red Deities included in a larger set called 'The Thirteen Golden Dharmas' of [[w:Sakya|Sakya]]. The other two deities are Kurukulle and Takkiraja. In depictions of the six-armed protector [[w:Mahakala|Mahakala]] (Skt: Shad-bhuja Mahakala, Wylie: mGon po phyag drug pa), an elephant-headed figure usually addressed as Vinayaka is seen being trampled by the Dharma Protector, but he does not appear distressed. In [[w:Vajrayana|Vajrayana]] and cognate Buddhist art, He is depicted as a subdued god trampled by Buddhist deities like [[w:Aparajita|Aparajita]], [[w:Parnasabari|Parnasabari]] and [[w:Vignataka|Vignataka]]. The Tibetan Ganesha appears, besides bronzes, in the resplendent Thangka paintings alongside the Buddha. In "''Ganesh, studies of an Asian God''," edited by Robert L. BROWN, State University of New York Press, 1992, page 241-242, he wrote that in the Tibetan Ka'gyur tradition, it is said that the Buddha had taught the "Ganapati Hridaya Mantra" (or "Aryaganapatimantra") to disciple [[w:Ananda|Ananda]]. The sutra in which the Buddha teaches this mantra can be found here[http://ganesha.bravehost.com/index.html]. The 'pastime' and 'play' (Sanskrit: [[w:lila|lila]]) of dancing and its representation in charnel ground literature and visual representations is endemic: [[w:Ganapati|Ganapati]] as son of Shiva, Ganapati as Lord of Gana, the demonic host of Gana, dancing ganesha, dancing gana, the dance of life and death, what is dance but a continuum of forms, dancing is energetic, dancing is symbolic of spiritual energy in iconography, particularly chthonic imagery of Dharmic Traditions. Energy moves (and cycles) between forms as does dance. The 'wheel' (Sanskrit: [[w:chakra|chakra]], [[w:mandala|mandala]]) in all its permutations and efflorescence is a profound Dharmic cultural artifact enshrining the energetic dance of the Universe. This is particularly applicable in the iconography of [[w:Nataraja|Nataraja]] and the 'wrathful deities' (Sanskrit: [[w:Heruka|Heruka]]) of Vajrayana which are depicted with a flaming [[w:aureole|aureole]], a flaming wheel. This resonates with the deep symbolism of the mystery rite and folklore and folk custom and high culture of [[w:circle dance|circle dance]]s which approaches a human [[w:cultural universal|cultural universal]]. [[w:Namkhai Norbu|Namkhai Norbu]], a famed [[w:Dzogchen|Dzogchen]] master in the [[w:Bonpo|Bonpo]] tradition and the [[w:Dharmic Tradition|Dharmic Tradition]]s of [[w:esoteric Buddhism|esoteric Buddhism]] has revealed a number of [[w:terma (Buddhism)|terma]] (Tibetan) of circle dances such as the 'Dance of the [[w:Six Lokas|Six Lokas]] of [[w:Samantabhadra|Samantabhadra]]'. The dance is a restricted initiatory rite and its process may not be disclosed as so doing would be a contravention of 'commitments ' (Sanskrit: [[w:samaya|samaya]]) but it may be affirmed that the rite is enacted on a colourful mandala of the [[w:Five Pure Lights|Five Pure Lights]] and the 'central point' (Sanskrit: [[w:bindu|bindu]]) of the dance mandala is illuminated with a sacred candle known as the '[[w:garbha|garbha]]' (Sanskrit) within the International Dzogchen Community. This terma dance is all clearly applicable to the charnel ground when taken as the 'wheel of becoming' (Sanskrit: [[w:bhavachakra|bhavachakra]]) which generally is demarcated by six distinct 'places' (Sanskrit: [[w:loka|loka]]). [[w:Dattatreya|Dattatreya]] the [[w:avadhuta|avadhuta]], to whom has been attributed the esteemed nondual medieval song, the ''[[w:Avadhuta Gita|Avadhuta Gita]]'', was a sometime denizen of the charnel ground and a founding deity of the [[w:Aghor tradition|Aghor tradition]] according to Barrett (2008: p.33): <blockquote>"...Lord Dattatreya, an antinomian form of Shiva closely associated with the cremation ground, who appeared to Baba Kina Ram atop Girnar Mountain in Gujarat. Considered to be the ''adi guru'' (ancient spiritual teacher) and founding deity of Aghor, Lord Dattatreya offered his own flesh to the young ascetic as ''prasād'' (a kind of blessing), conferring upon him the power of clairvoyance and establishing a guru-disciple relationship between them."<ref>Barrett, Ron (2008). Aghor medicine: pollution, death, and healing in northern India. Edition: illustrated. University of California Press. {{ISBN|0520252187}}, 9780520252189. Source: [http://books.google.com.au/books?id=BSeYR93tIzwC&pg=PA33&dq=dattatreya+aghor&ei=fhCBS-zuIYfulQSl8oCUCg&client=opera&cd=1#v=onepage&q=dattatreya%20aghor&f=false], p.33</ref></blockquote> Barrett (2008: p.161) discusses the sadhana of the '[[w:Aghora|Aghora]]' (Sanskrit; Devanagari: अघोर)<ref>'Aghora' in ''Monier Williams Sanskrit-English Dictionary'' [Online]. Source: [http://www.sanskrit-lexicon.uni-koeln.de/monier/indexcaller.php] (accessed: Tuesday February 9, 2010)</ref> in both its left and right-handed proclivites and identifies it as principally cutting through attachments and aversion and foregrounding primordiality, a view uncultured, undomesticated: <blockquote>"The gurus and disciples of Aghor believe their state to be primordial and universal. They believe that all human beings are natural-born Aghori. Hari Baba has said on several occasions that human babies of all societies are without discrimination, that they will play as much in their own filth as with the toys around them. Children become progressively discriminating as they grow older and learn the culturally specific attachments and aversions of their parents. Children become increasingly aware of their mortality as they bump their heads and fall to the ground. They come to fear their mortality and then palliate this fear by finding ways to deny it altogether. In this sense, Aghor ''sādhanā'' is a process of unlearning deeply internalized cultural models. When this ''sādhanā'' takes the form of ''shmashān sādhanā'', the Aghori faces death as a very young child, simultaneously meditating on the totality of life at its two extremes. This ideal example serves as a prototype for other Aghor practices, both left and right, in ritual and in daily life."<ref>Barrett, Ron (2008). ''Aghor medicine: pollution, death, and healing in northern India.'' Edition: illustrated. University of California Press. {{ISBN|0520252187}}, 9780520252189. Source: [http://books.google.com.au/books?client=opera&id=BSeYR93tIzwC&dq=dattatreya+aghor&q=dattatreya#v=snippet&q=dattatreya&f=false], (accessed: Monday February 22, 2010) p.161</ref></blockquote> *[[w:Vetala|Vetala]] (Sanskrit) *[[w:Shaivites|Shaivites]] *Practitioners of [[w:Anuyoga|Anuyoga]]. The Anuyoga class of tantras of the Nyingmapa is understood as the "Mother Tantras" of the Sarma Schools, this class of literature is also known as "Yogini Tantras" and there is a voluminous Shaivite or Shakta Tantra by the same name, Yoginitantra. *In the dance of reciprocity that is the beauty and cruelty of the Mystery of living and dying, the rite of [[w:Ganachakra|Ganachakra]] is celebrated, indeed all forms participate in the Ganachakra, there is nothing in the Three Worlds ([[w:Triloka|Triloka]]) that is not a charnel ground... Beer (2003: p.102) relates how the symbolism of the [[w:khatvanga|khatvanga]] that entered [[w:esoteric Buddhism|soteric Buddhism]] (particularly from [[w:Padmasambhava|Padmasambhava]]) was a direct borrowing from the [[w:Shaivite|Shaivite]] [[w:Kapalika|Kapalika]]s who frequented places of austerity such as charnel grounds and [[w:cross roads|cross roads]] etcetera as a form of 'left-handed path' (Sanskrit: [[w:vamamarga|vamamarga]]) 'spiritual practice' (Sanskrit: [[w:sadhana|sadhana]]): <blockquote>"The form of the Buddhist ''khatvanga'' derived from the emblematic staff of the early Indian Shaivite ''yogins'', known as ''kapalikas'' or 'skull-bearers'. The ''kapalikas'' were originally miscreants who had been sentenced to a twelve-year term of penance for the crime of inadvertently killing a Brahmin. The penitent was prescribed to dwell in a forest hut, at a desolate crossroads, in a charnel ground, or under a tree; to live by begging; to practice austerities; and to wear a loin-cloth of hemp, dog, or donkey-skin. They also had to carry the emblems of a human skull as an alms-bowl, and the skull of the Brahmin they had slain mounted upon a wooden staff as a banner.These Hindu ''kapalika'' ascetics soon evolved into an extreme outcaste sect of the 'left-hand' tantric path (Skt. vamamarg) of ''shakti'' or goddess worship. The early Buddhist tantric ''yogins'' and ''yoginis'' adopted the same goddess or ''dakini'' attributes of the ''kapalikas''. These attributes consisted of; bone ornaments, an animal skin loincloth, marks of human ash, a skull-cup, ''damaru'', flaying knife, thighbone trumpet, and the skull-topped tantric staff or ''khatvanga''."<ref>Beer, Robert (2003). ''The handbook of Tibetan Buddhist symbols''. Serindia Publications. {{ISBN|1932476032}}, 9781932476033 Source: [http://books.google.com.au/books?id=-3804Ud9-4IC&pg=PA102&lpg=PA102&dq=shaivite+charnel+ground&source=bl&ots=FPGERYPfHx&sig=CzaZR-I2RrnJubdI-QyZDZZm_N4&hl=en&ei=Fc1oS4C4CISssgPi5_3zBA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CAwQ6AEwAQ#v=onepage&q=shaivite%20charnel%20ground&f=false]</ref></blockquote> Sadhana in the charnel ground within the Dharmic Traditions may be traced to ancient depictions of the chthonic [[w:Shiva|Shiva]] and his [[w:chimera|chimeric]] son [[w:Ganapati|Ganapati]] (Ganesha) who was decapitated and returned to life with the head of an elephant. In certain narratives, Shiva made the Ganesha 'lord of the [[w:gana|gana]]' (Sanskrit: Ganapati). Such depictions of Shiva, Ganesha and the ganas are evident in literature, architectural ornamentation and iconography, etc. In the Indian traditions of Tantra the charnel ground is very important. In must be remembered that the seat of Shiva and his locality of sadhana is the sacred Mount [[w:Kailasha|Kailasha]] in the Himalaya. In some non-Buddhist traditions of [[w:Ganachakra|Ganachakra]] such as the Kaula the leader of the rite is known as 'ganapati', which is a title of respect. The Eight Great Charnel Grounds are important in the life of [[w:Padmasambhava|Padmasambhava]]. This is one definite way the importance of the charnel ground in sadhana entered and became replicated in the Himalayan Dharmic Tradition. The charnel ground is a particular place that holds powerful teachings on impermanence and is important for slaying the ego. In this, the charnel ground shares with the tradition of [[w:dark retreat|dark retreat]] that was foregrounded in some Himalayan practice lineages. [[w:Judith Simmer-Brown|Simmer-Brown]] (2001: p.127) conveys how great [[w:Mahasiddha|Mahasiddha]]'s in the [[w:Nath|Nath]] and [[w:Mantrayana|Mantrayana]] [[w:Buddhadharma|Buddhadharma]] traditions such as [[w:Tilopa|Tilopa]] (988–1069) and [[w:Goraksa|Gorakṣa]] ([[w:floruit|fl.]] 11th - 12th century) yoked adversity to till the soil of the path and accomplish the fruit, the '[[w:Ground of Being (Dzogchen)|ground]]' (Sanskrit: āśraya; Wylie: gzhi)<ref>Dudjom Rinpoche and Jikdrel Yeshe Dorje (1991). ''The Nyingma School of Tibetan Buddhism: its Fundamentals and History''. Two Volumes. Translated and edited by Gyurme Dorje with Matthew Kapstein. Wisdom Publications, Boston. {{ISBN|0-86171-087-8}}, p.535 Index of Technical Terms</ref> of realization - worthy case-studies for those with spiritual proclivity: <blockquote>"The charnel ground is not merely the hermitage; it can also be discovered or revealed in completely terrifying mundane environments where practitioners find themselves desperate and depressed, where conventional worldly aspirations have become devastated by grim reality. This is demonstrated in the sacred biographies of the great siddhas of the Vajrayāna tradition. Tilopa attained realization as a grinder of sesame seeds and a procurer for a prominent prostitute. Sarvabhakṣa was an extremely obese glutton, Gorakṣa was a cowherd in remote climes, Taṅtepa was addicted to gambling, and Kumbharipa was a destitute potter. These circumstances were charnel grounds because they were despised in Indian society and the siddhas were viewed as failures, marginal and defiled."<ref>[[Judith Simmer-Brown|Simmer-Brown, Judith]] (2001). ''Dakini's warm breath: the feminine principle in Tibetan Buddhism''. Boston, Massachusetts, USA: Shambhala Publications, Inc. {{ISBN|9781570629204}} (paperback), p.127</ref></blockquote> Dyczkowski (1988: p.26) holds that [[w:Hāla|Hāla]]'s [[w:Prakrit literature|Prakrit literature]] poem the ''[[w:Gāthāsaptaśati|Gāthāsaptaśati]]'' (third to fifth century CE) is one of the first extant literary references to a [[w:Kapalika|Kapalika]] where in the poem the Kapalika, who is a female, anoints and besmears her body with the crematory ash from the funeral pyre of her recently deceased lover.<ref>Dyczkowski, Mark S. G. (1988). ''The canon of the Śaivāgama and The Kubjikā Tantras of the western Kaula tradition''. SUNY series in Kashmir Śaivism. SUNY Press. {{ISBN|0887064949}}, 9780887064944 Source: [http://books.google.com.au/books?id=qoZIJMGo4W0C&pg=PA39&lpg=PA39&dq=V%C4%81m%C4%81c%C4%81ra&source=bl&ots=MfmKHrqqkl&sig=gJGdk8G_oFGes6-jiv8ijNq1lk0&hl=en&ei=aDFqS_rHCdGTkAWewry2DQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CBYQ6AEwBDgK#v=onepage&q=&f=false] (accessed: Thursday February 4, 2010)</ref> This is a literary allusion to the anointing of 'sacred ash' (Sanskrit: [[w:bhasma|bhasma]]; [[w:vibhuti|vibhuti]]) especially associated with [[w:Shiva|Shiva]] who applies it all over His body. Dyczkowski (1988: p.26) relates how [[w:Krishna Mishra|Kṛṣṇa Miśra]] (c 1050-1100) casts the character of a [[w:Kāpālika|Kāpālika]] in his play, the ''[[w:Prabodhacandrodaya|Prabodhacandrodaya]]'' and then quotes verbatim a source that renders the creed of this character into English thus: <blockquote>"My charming ornaments are made from garlands of human skulls." says the Kāpālika, "I dwell in the cremation ground and eat my food from a human skull. I view the world alternately as separate from God (Īśvara) and one with Him, through the eyes that are made clear with the ointment of yoga... We (Kāpālikas) offer oblations of human flesh mixed with brains, entrails and marrow. We break our fast by drinking liquor (surā) from the skull of a Brahmin. At that time the god Mahābhairava should be worshipped with offerings of awe-inspiring human sacrifices from whose severed throats blood flows in currents.<ref>Dyczkowski, Mark S. G. (1988). ''The canon of the Śaivāgama and The Kubjikā Tantras of the western Kaula tradition''. SUNY series in Kashmir Śaivism. SUNY Press. {{ISBN|0887064949}}, 9780887064944 Source: [http://books.google.com.au/books?id=qoZIJMGo4W0C&pg=PA39&lpg=PA39&dq=V%C4%81m%C4%81c%C4%81ra&source=bl&ots=MfmKHrqqkl&sig=gJGdk8G_oFGes6-jiv8ijNq1lk0&hl=en&ei=aDFqS_rHCdGTkAWewry2DQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CBYQ6AEwBDgK#v=onepage&q=&f=false] (accessed: Thursday February 4, 2010)</ref></blockquote> In Vajrayana poetry, literature and song paraticularly that of the '[[w:songs of realization|songs of realization]]', charnel grounds are often described as containing "rivers of blood", "poisonous waterfalls", and depicted as localities containing dangerous [[w:wild animal|wild beast]]s. The [[w:two truths|two truths]] doctrine though iterates this view and when perceived differently, charnel grounds are peaceful places of [[w:beatitude|beatific]] [[w:solitude|solitude]] and this [[w:chthonic|chthonic]] symbolism and [[w:Twilight Language|Twilight Language]] and iconography accrues a rich [[w:polysemy|polysemy]]. When perceived differently, the charnel grounds are places of 'peace' (Sanskrit: [[w:shanti|shanti]]), pleasant groves, populated by [[w:wildflowers|wildflowers]] and [[w:fruit|fruit]]. [[w:Songbird|Songbirds]], tame lions and tigers, and the vast open vault of the sky, fruit and flowers are often used in Vajrayana iconography and poetry and Beer (1999)<ref>[[w:Robert Beer|Beer, Robert]] (1999). ''The Encyclopedia of Tibetan Symbols and Motifs'', (Hardcover). Shambhala Publications. {{ISBN|157062416X}}, {{ISBN|978-1570624162}}</ref> explains their symbolism and how they are understood in the tradition in fine detail. They are all included in depictions of the charnel ground. The [[w:Upa-yoga|Upa-yoga]] scriptures first appeared in 'Mount Jakang Chen' {{bo|t=རི་བྱ་རྐང་ཅན|w=ri bya rkang can}} (alternate names: Riwo Jakang, [[w:Mount Jizu|Mount Jizu]]) and the [[w:charnel ground|charnel ground]] of Cool Grove {{bo|t=བསིལ་བའི་ཚལ|w=bsil ba'i tshal}}.<ref>Tsogyal, Yeshe (composed); Nyang Ral Nyima Oser (revealed); Erik Pema Kunsang (translated); Marcia Binder Schmidt (edited) (1999). ''The Lotus-Born: The Life Story of Padmasambhava'' (Paperback). With forward by HH Dilgo Khyentse Rinpoche. Shambhala: South Asia Editions, p.292</ref> Cool Grove is also known as 'Śītavana' (Sanskrit).<ref>Rigpa Shedra (July 2009). 'Shitavana'. Source: [http://www.rigpawiki.org/index.php?title=Shitavana] (accessed: October 3, 2009)</ref> Gray (undated: c2009) provides an excellent survey of chthonic charnel ground accoutrement motif such as skull imagery in the textual tradition of the [[w:Yogini tantras|Yogini tantras]].<ref>Gray, David B. (n.d.). 'Skull Imagery and Skull Magic in the Yogini Tantras'. Santa Clara University. Source: [http://www.scribd.com/doc/26244245/Gray-David-B-n-d-Skull-Imagery-and-Skull-Magic-in-the-Yogini-Tantras-Santa-Clara-University] (accessed: Tuesday February 2, 2010)</ref> [[w:Baital Pachisi|Baital Pachisi]] [[w:Somadeva|Somadeva]] In the charnel grounds of Vajrayana, there are no [[social convention]]s to conform to and no distractions to be seduced by for the [[w:siddha|siddha]].<ref>Rigpa Shedra (July 2009). 'Charnel ground'. Source: [http://www.rigpawiki.org/index.php?title=Charnel_ground] (accessed: Saturday December 19, 2009)</ref> Dakas and [[w:dakini|dakini]]s gather there to celebrate ceremonial tsok 'feasts' (Sanskrit: [[w:ganachakra|ganachakra]]). The lion's roar of [[w:Dharma|Dharma]] discourse resound as do the [[w:liturgy|liturgy]] and the specific 'hourglass drums' (Sanskrit: [[w:damaru|damaru]]) of the [[w:chöd|chödpa]] and the light of the inner 'joy of bliss' (Sanskrit: [[w:ananda|ananda]]) radiates and this dynamic movement is represented iconographically by the 'bliss-whirling' (Sanskrit: [[w:gankyil|ananda-chakra]]). In his ''Manual'' on the practice of the ''[[w:Longchen Nyingtik|Longchen Nyingtik]]'', [[w:Dzongsar Khyentse Rinpoche|Dzongsar Khyentse Rinpoche]] holds that: <blockquote>"Right now, our minds are very fickle. Sometimes you like a certain place, and it inspires, and yet with that same place, if you stay too long, it bores you. […] As you practice more and more, one day this kind of habit, this fickle mind will just go. Then you will search for the bindu interpretation of the right place, and according to the classic tantric texts, that is usually what they call the “eight great charnel grounds”. So then, you have to go to a cemetery, especially to one of the eight cemeteries. There, under a tree, in the charnel ground, wearing a tiger skin skirt, holding a [[w:kapāla|kapāla]] and having this indifference between relatives and enemies, indifference between food and shit, you will practise. Then your [[w:bindu|bindu]] will flow. At that time, you will know how to have intercourse between emptiness and appearance."<ref>Dzongsar Khyentse Rinpoche, from ''Longchen Nyingtik Practice Manual'' (page 66).</ref><ref>Rigpa Shedra (July 2009). 'Charnel ground'. Source: [http://www.rigpawiki.org/index.php?title=Charnel_ground] (accessed: Saturday December 19, 2009)</ref> </blockquote> In the life in which a [[w:pratyekabuddha|pratyekabuddha]] attains the fruit of their 'path' (Wylie: lam), they are naturally drawn to charnel grounds. "When reflecting on the bones found there, the pratyekabuddha inquire "Where do these bones come from?"<ref>Rigpa Shedra (July 2009). 'Charnel ground'. Source: [http://www.rigpawiki.org/index.php?title=Charnel_ground] (accessed: Saturday December 19, 2009)</ref> This [[w:samyama|samyama]] (Sanskrit) on the bones awakens knowledge of their many lifetimes of investigation into the [[w:Twelve Nidanas|'Twelve Links of Dependent Origination']]. These twelve links then unfold in their [[w:mindstream|mindstream]] as a 'blessing' (Sanskrit: [[w:adhishthana|adhishthana]]), in both forward and reverse sequence and on that foundation they yield 'realisation' (Sanskrit: [[w:siddhi|siddhi]]).<ref>Rigpa Shedra (July 2009). 'Charnel ground'. Source: [http://www.rigpawiki.org/index.php?title=Charnel_ground] (accessed: Saturday December 19, 2009)</ref> The charnel ground, cremation ground and cemetery is evident as a specific region within wrathful Indo-Tibetan [[w:sand mandala|sand mandala]] iconography. As the anthropologist Gold (1994: p.141) relates in his comparative study drawn from his professional fieldwork into the symbolic universals of the sacred circles and [[w:sand-painting|sand-painting]]s of the [[w:Navajo people|Navajo]] and [[w:Tibetan people|Tibetan people]]s, parses the sacred precinct and motif of the charnel ground as a locality in the symbolic grammar of the Indo-Tibetan 'fierce yidam' or '[[w:wrathful deities|wrathful deity]]' (Sanskrit: [[w:heruka|heruka]]) sand mandala: <blockquote>The third concentric ring [from the circumference] is optional, in that it is only used in mandalas representing the reality of deities of fierce power. It represents the charnel grounds wherein bodies are cut up and offered to birds of prey as a "sky burial." This ring signifies the cutting away of the bones and flesh of illusion on the way to the primordial ground at the mandala's center. In some mandalas, it is positioned outside of the Mountain of Fire ring.<ref>Gold, Peter (1994). ''Navajo and Tibetan Sacred Wisdom: The Circle of the Spirit'' (Paperback). Inner Traditions. {{ISBN|089281411X}}, {{ISBN|978-0892814114}}, p.141</ref></blockquote> The region of the charnel grounds in many wrathful mandala often hold eight specific charnel grounds where certain key events take place in the life of [[w:Padmasambhava|Padmasambhava]].<ref>[[w:Judith Simmer-Brown|Simmer-Brown, Judith]] (2001). ''Dakini's warm breath: the feminine principle in Tibetan Buddhism''. Boston, Massachusetts, USA: Shambhala Publications, Inc. {{ISBN|9781570629204}} (paperback), p.124</ref> Blood is thematic in Charnel Ground iconography where it may be understood as [[w:lifeblood|lifeblood]], a symbol of viscous 'compassion' (Sanskrit: [[w:karuna|karuna]]) of 'sacrifice' (Sanskrit: [[w:yajna|yajna]]), and attendant with the symbolism of blood, bones ground our shared humanity and solidarity in the wider [[w:Mandala|Mandala]] of life and the ancient lineage of '[[w:ancestors|ancestors]]' from which all sentient beings are of lineal descent. The tradition and custom of the 'sky burial' (Tibetan: ''jhator'') afforded [[w:Traditional Tibetan medicine|Traditional Tibetan medicine]] and ''[[w:thangka|thangka]]'' [[w:iconography|iconography]] such as the '[[w:Tree of physiology|Tree of physiology]]' with a particular insight into the interior workings of the human body. Pieces of the [[w:human skeleton|human skeleton]] were employed in ritual tools such as the [[w:Kapala|skullcup]], [[w:Kangling|thigh-bone trumpet]], etc. The 'symbolic bone ornaments' (Skt: {{IAST|aṣṭhiamudrā}}; Tib: rus pa'i rgyanl phyag rgya) are also known as "[[w:mudra|human skeleton]]" or 'seals' are also known as 'charnel ground ornaments'. The ''[[w:Hevajra Tantra|Hevajra Tantra]]'' identifies the Symbolic Bone Ornaments with the [[w:Five Wisdoms|Five Wisdoms]]] and [[w:Jamgon Kongtrul|Jamgon Kongtrul]] in his commentary to the ''Hevajra Tantra'' explains this further.<ref name="Kongtrul_2005_493">{{harvnb|Kongtrul|2005|p=493}}.</ref> The important [[w:Varnamala|Varnamala]] (or 'garland of [[w:bija|bija]] phonemes' in [[w:Twilight Language|Twilight Language]] is iconographically represented by a 'garland of severed heads or skulls' (Sanskrit: Mundamala). Beer (1999: pp. 277-278) relates how [[w:Padmasambhava|Padmasambhava]] received the siddhi of the [[w:Kīla_(Buddhism)|kīla]] [[w:esoteric transmission|transmission]] from a gigantic [[w:scorpion|scorpion]] at the charnel ground of [[w:Rajgriha|Rajgriha]]: <blockquote> The sting of the scorpion's whip-like tail transfixes and poisons its prey, and in this respect it is identified with the wrathful activity of the ritual dagger or kīla. Padmasambhava's biography relates how he received the siddhi of the kīla transmission at the great charnel ground of Rajgriha from a gigantic scorpion with nine heads, eighteen pincers and twenty-seven eyes. This scorpion reveals the kīla texts from a triangular stone box hidden beneath a rock in the cemetery. As Padmasambhava reads this terma text spontaneous understanding arises, and the heads, pincers, and eyes of the scorpion are 'revealed' as different vehicles or ''yanas'' of spiritual attainment. Here, at Rajgriha, Padmasambhava is given the title of 'the scorpion guru', and in one of his eight forms as Guru Dragpo or Pema Drago ('wrathful lotus'), he is depicted with a scorpion in his left hand. As an emblem of the wrathful kīla transmission the image of the scorpion took on a strong symbolic meaning in the early development of the Nyingma or 'ancient school' of Tibetan Buddhism...".<ref>[[Robert Beer|Beer, Robert]] (1999). ''The Encyclopedia of Tibetan Symbols and Motifs'', (Hardcover). Shambhala Publications. {{ISBN|157062416X}}, {{ISBN|978-1570624162}}, pp. 277-278</ref> </blockquote> The 'Eight Great Charnel Grounds (Sanskrit: aṣṭamahāśmāśāna; {{bo|t=དུར་ཁྲོད་ཆེན་པོ་བརྒྱད|w=dur khrod chen po brgyad}})<ref>Rigpa Shedra Wiki (July 2009). 'Eight great charnel grounds'. Source: [http://www.rigpawiki.org/index.php?title=Eight_great_charnel_grounds] (accessed: Monday December 21, 2009) ི</ref> 'The Most Fierce' ({{bo|t=གཏུམ་དྲག|w=gtum drag}})<ref>Dudjom Rinpoche and Jikdrel Yeshe Dorje (1991). ''The Nyingma School of Tibetan Buddhism: its Fundamentals and History''. Two Volumes. Translated and edited by Gyurme Dorje with Matthew Kapstein. Wisdom Publications, Boston. {{ISBN|0-86171-087-8}}, p.157 Enumerations</ref> 'Dense Thicket' ({{bo|t=ཚང་ཚིང་འཁྲིགས་པ|w=tshang tshing 'khrigs pa}})<ref>Dudjom Rinpoche and Jikdrel Yeshe Dorje (1991). ''The Nyingma School of Tibetan Buddhism: its Fundamentals and History''. Two Volumes. Translated and edited by Gyurme Dorje with Matthew Kapstein. Wisdom Publications, Boston. {{ISBN|0-86171-087-8}}, p.157 Enumerations</ref> 'Dense Blaze' ({{bo|t=འབར་འཁྲིགས་པ|w='bar 'khrigs pa}})<ref>Dudjom Rinpoche and Jikdrel Yeshe Dorje (1991). ''The Nyingma School of Tibetan Buddhism: its Fundamentals and History''. Two Volumes. Translated and edited by Gyurme Dorje with Matthew Kapstein. Wisdom Publications, Boston. {{ISBN|0-86171-087-8}}, p.157 Enumerations</ref> 'Endowed with Skeletons' ({{bo|t=ཀེང་རུས་ཅན|w=keng rus can}})<ref>Dudjom Rinpoche and Jikdrel Yeshe Dorje (1991). ''The Nyingma School of Tibetan Buddhism: its Fundamentals and History''. Two Volumes. Translated and edited by Gyurme Dorje with Matthew Kapstein. Wisdom Publications, Boston. {{ISBN|0-86171-087-8}}, p.157 Enumerations</ref> 'Cool Forest' or 'Cool Grove' (Sanskrit: Śītavana; Devanagari: शीतवन; {{bo|t=བསིལ་བུ་ཚལ|w=bsil bu tshal}})<ref>Dudjom Rinpoche and Jikdrel Yeshe Dorje (1991). ''The Nyingma School of Tibetan Buddhism: its Fundamentals and History''. Two Volumes. Translated and edited by Gyurme Dorje with Matthew Kapstein. Wisdom Publications, Boston. {{ISBN|0-86171-087-8}}, p.157 Enumerations</ref> 'Black Darkness' ({{bo|t=མུན་པ་ནག་པོ|w=mun pa nag po}})<ref>Dudjom Rinpoche and Jikdrel Yeshe Dorje (1991). ''The Nyingma School of Tibetan Buddhism: its Fundamentals and History''. Two Volumes. Translated and edited by Gyurme Dorje with Matthew Kapstein. Wisdom Publications, Boston. {{ISBN|0-86171-087-8}}, p.157 Enumerations</ref> 'Resonant with "Kilikili"' ({{bo|t=ཀི་ལི་ཀི་ལིར་སྒྲ་སྒྲོག་པ|w=ki li ki lir sgra sgrog pa}})<ref>Dudjom Rinpoche and Jikdrel Yeshe Dorje (1991). ''The Nyingma School of Tibetan Buddhism: its Fundamentals and History''. Two Volumes. Translated and edited by Gyurme Dorje with Matthew Kapstein. Wisdom Publications, Boston. {{ISBN|0-86171-087-8}}, p.157 Enumerations</ref> 'Wild Cries of "Ha-ha"' ({{bo|t=ཧ་ཧ་རྒོད་པ|w=ha ha rgod pa}})<ref>Dudjom Rinpoche and Jikdrel Yeshe Dorje (1991). ''The Nyingma School of Tibetan Buddhism: its Fundamentals and History''. Two Volumes. Translated and edited by Gyurme Dorje with Matthew Kapstein. Wisdom Publications, Boston. {{ISBN|0-86171-087-8}}, p.157 Enumerations</ref> [[w:Dudjom Rinpoche|Dudjom]] ''et al.'' (1991: p.626 History) relates how "earth, stone, water, and wood" gathered from The Eight Great Charnel Grounds and auspicious objects such as the flesh of a seven-times-born [[w:Brahmana|Brahmana]] and 'relics' (Sanskrit: [[w:Śarīra|Śarīra]]) of the [[w:Tathagatha|Tathagatha]] amongst other items were used to sculpt a statue of 'Yangdak Heruka' (Wylie: yang dag heruka; Sanskrit: Viśuddhaheruka)<ref>Rigpa Shedra Wiki (September 2008). 'Yangdak Heruka'. Source: [http://www.rigpawiki.org/index.php?title=Yangdak_Heruka] (accessed: Tuesday December 22, 2009)</ref> modelled on [[w:Zurcungpa|Zurcungpa]] (1014CE - 1074CE; alt. Zurcung Sherap-tra)<ref>[http://books.google.com.au/books?id=vkAJtFpXDP0C&pg=PA134&lpg=PA134&dq=Zurcungpa&source=bl&ots=VDGM-ToocV&sig=QNYxs3rOeK_SZ7rF2L8FerlDosg&hl=en&ei=3CMxS7zbFYGOkQXznM3vAg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CBUQ6AEwAg#v=onepage&q=Zurcungpa&f=false ''The presence of light: divine radiance and religious experience'']</ref> holding the aspect of the [[w:yidam|yidam]] after a [[w:vase empowerment|vase empowerment]] was given to the sculptors by the "master" (Zurcungpa): <blockquote>"The sculptors displayed great devotion, so first of all the master gave them the vase empowerment of the glorious Yangdak Heruka. They prepared a mixture which combined relics from the Tathagata's remains; the flesh of one who had been born as a brahman seven times; earth, stone, water, and wood from the eight charnel grounds; a variety of precious gems; and sacramental medicine refined by the awareness-holders of India and Tibet."<ref>Dudjom Rinpoche and Jikdrel Yeshe Dorje (1991). ''The Nyingma School of Tibetan Buddhism: its Fundamentals and History''. Two Volumes. Translated and edited by Gyurme Dorje with Matthew Kapstein. Wisdom Publications, Boston. {{ISBN|0-86171-087-8}}, p.626 History</ref></blockquote> ==Bibliography== *Nagar, Shanti Lal. ''The Cult of Vinayaka''. (Intellectual Publishing House: New Delhi, 1992). {{ISBN|81-7076-043-9}}. Chapter 17: "The Travels Abroad". * {{citation|last=Kongtrul Lodrö Tayé|first=Jamgön|ref=CITEREFKongtrul2005|authorlink=Jamgon Kongtrul<!--|last2=Guarisco|first2=Eliot|last3=McLeod|first3=Ingrid-->|year=2005|title=Systems of Buddhist Tantra, The Indestructible Way of Secret Mantra|series=The Treasury of Knowledge<!--(shes bya kun la khyab pa’i mdzod)-->, book 6, part 4|location=Boulder|publisher=Snow Lion|isbn=1-55939-210-X}}. * Gold, Peter (1994). ''Navajo and Tibetan Sacred Wisdom: The Circle of the Spirit'' (Paperback). Inner Traditions. {{ISBN|089281411X}}, {{ISBN|978-0892814114}}, p.141 *Getty, Alice. ''Gaņeśa: A Monograph on the Elephant-Faced God''. (Clarendon Press: Oxford, 1936). 1992 reprint edition, {{ISBN|81-215-0377-X}}. Individual chapters are devoted to individual countries and regions of the world. *{{ cite book | last = McDermott | first = Rachel F. | coauthors = Jeffrey J. Kripal | title = Encountering Kālī: in the margins, at the center, in the West | publisher = University of California Press | date = 2003 | isbn = {{ISBN|9780520232396}} }} *Tsuda, Shinichi (1978). 'The cult of śmaśana, the realities of Tantra'. In Goudriaan, Teun (ed.)(1978). ''The Sanskrit Tradition and Tantrism''. Leiden: E.J. Brill, pp.96-108. ==Notes== {{reflist|3}} == References== * Byron Katie, Mitchell Stephen (2007) "A Thousand Names for Joy: Living in Harmony with the Way Things Are". New York: Harmony Books. {{ISBN|978-0-307-33923-2}} * Castaneda, Carlos (1987). ''The Power of Silence''. New York: Simon and Schuster. {{ISBN|0-671-50067-8}}. * Downing, Jerry N. (2000) ''Between Conviction and Uncertainty'' {{ISBN|0-79144-627-1}} * Godman, David (Ed.) (1985). ''Be As You Are: The Teachings of Sri Ramana Maharshi''. London: Arkana. {{ISBN|0-14-019062-7}}. * Hawkins, David R. (October 2006). ''Discovery of the Presence of God: Devotional Nonduality''. Sedona, Arizona: Veritas Publishing. {{ISBN|0-9715007-6-2}} (Softcover); {{ISBN|0-9715007-7-0}} (Hardcover) * Jeon, Arthur (2004) City Dharma: Keeping Your Cool in the Chaos {{ISBN|1-40004-908-3}} * Katz, Jerry (Ed.) (2007). ''One: Essential Writings on Nonduality''. Boulder, CO: Sentient Publications. {{ISBN|1591810531}}. * Kent, John (1990) ''Richard Rose's Psychology of the Observer: The Path to Reality Through the Self'' [http://www.searchwithin.org/johnkent/index.html PhD Thesis] * Klein, Anne Carolyn (1995). ''Meeting the Great Bliss Queen: Buddhists, Feminists, and the Art of the Self''. Boston, Beacon Press. {{ISBN|0-8070-7306-7}}. * Kongtrül, Jamgön (1992). ''Cloudless Sky: The Mahamudra Path of The Tibetan Buddhist Kagyü School''. Boston: Shambhala Publications. {{ISBN|0-87773-694-4}}. * Lama, Dalai (2000). ''Dzogchen: The Heart Essence of the Great Perfection''. Ithaca: Snow Lion Publications. {{ISBN|1-55939-157-X}}. * Norbu, Namkhai (1993). ''The Crystal and the Way of Light: Sutra, Tantra and Dzogchen''. London: Arkana. {{ISBN|0-14-019314-6}}. * Schucman, Helen (1992) "A Course In Miracles". Foundation for Inner Peace, pg. 1. {{ISBN|0-9606388-9-X}}. * Tolle, Eckhart (2004) "The Power of Now: A Guide to Spiritual Enlightenment" {{ISBN|978-1577314806}} * Trungpa, Chögyam (1987). ''Cutting Through Spiritual Materialism''. Boston: Shambhala Publications. {{ISBN|0-87773-050-4}}. * Watson, Burton (Trans.) (1968). ''The Complete Works of Chuang Tzu''. New York: Columbia University Press. {{ISBN|0-231-03147-5}}. * Lumiere, Lynn-Marie and Lumiere-Wins, John (2000). "The Awakening West: Conversations with Today's New Western Spiritual Leaders" (Paperback) * Waite, Dennis (2004) "The Book of One: The Spiritual Path of Advaita" (Paperback) ==External links== *[http://www.gurusfeet.com/blog/neo-advaita-or-pseudo-advaita-and-real-advaita-nonduality-timothy-conway "Neo-Advaita or Pseudo-Advaita and Real Advaita-Nonduality by Timothy Conway" a commentary by Ravi dated Wednesday 09/7/2008] fskrbmdnlv5nmjg2qn1zd3s6b58kfcm 0 97456 2410934 2397840 2022-08-02T07:12:20Z Jtneill 10242 Update for 2022 wikitext text/x-wiki <noinclude> {{title|Schedule}}</noinclude> {{:Motivation and emotion/About/Schedule/Embed}} <noinclude> {{cot|Overview}} This [[Motivation and emotion/About/Schedule|schedule]] shows how the [[Motivation and emotion/Modules|modules]], [[Motivation and emotion/Lectures|lectures]], [[Motivation and emotion/Readings|readings]], [[Motivation and emotion/Tutorials|tutorials]], and [[Motivation and emotion/Assessment|assessment]] work together to help [[emerging scholar]]s achieve the [[Motivation and emotion/About/Learning outcomes|learning outcomes]]. ;Key dates * W02 (Fri) – Recommended topic sign up * W04 (Mon 9am) - Topic selection due * W04 (Mon 9am) - Quiz 1 due * W04 (Fri) – Census day - Last day to withdraw without penalty * W05 (Mon 9am) – Topic development due * W06 (Mon 9am) - Quiz 2 due * W08 – No classes - Mid-semester break * W08 (Mon 9am) - Quiz 3 due * W08 (Fri) – Withdraw without incurring fail * W11 (Mon 9am) - Quiz 4 due * W12 (Mon 9am) – Book chapter due * W13 (Mon 9am) - Quiz 5 due * W14 (Mon 9am) – Multimedia due * W15 (Mon 9am) – Quiz 6 due * W16 (Mon 3pm) – Results released {{cob}} <!-- ==See also== ;Previous schedules * [[/2021/]] * [[/2020/]] * [[/2019/]] * [[/2018/]] * [[/2017/]] * [[/2016/]] * [[/2015/]] * [[/2014/]] * [[/2013/]] * [[/2011/]] * [[/2010/]] * [[/2009/]] --> [[Category:Motivation and emotion/About/Schedule]] </noinclude> 6dw5g0rkiytkm34oh1vp8hxsdnsmdfg 2410935 2410934 2022-08-02T07:13:03Z Jtneill 10242 wikitext text/x-wiki <noinclude> {{title|Schedule}}</noinclude><p align = "center"> {{:Motivation and emotion/About/Schedule/Embed}}</p> <noinclude> {{cot|Overview}} This [[Motivation and emotion/About/Schedule|schedule]] shows how the [[Motivation and emotion/Modules|modules]], [[Motivation and emotion/Lectures|lectures]], [[Motivation and emotion/Readings|readings]], [[Motivation and emotion/Tutorials|tutorials]], and [[Motivation and emotion/Assessment|assessment]] work together to help [[emerging scholar]]s achieve the [[Motivation and emotion/About/Learning outcomes|learning outcomes]]. ;Key dates * W02 (Fri) – Recommended topic sign up * W04 (Mon 9am) - Topic selection due * W04 (Mon 9am) - Quiz 1 due * W04 (Fri) – Census day - Last day to withdraw without penalty * W05 (Mon 9am) – Topic development due * W06 (Mon 9am) - Quiz 2 due * W08 – No classes - Mid-semester break * W08 (Mon 9am) - Quiz 3 due * W08 (Fri) – Withdraw without incurring fail * W11 (Mon 9am) - Quiz 4 due * W12 (Mon 9am) – Book chapter due * W13 (Mon 9am) - Quiz 5 due * W14 (Mon 9am) – Multimedia due * W15 (Mon 9am) – Quiz 6 due * W16 (Mon 3pm) – Results released {{cob}} <!-- ==See also== ;Previous schedules * [[/2021/]] * [[/2020/]] * [[/2019/]] * [[/2018/]] * [[/2017/]] * [[/2016/]] * [[/2015/]] * [[/2014/]] * [[/2013/]] * [[/2011/]] * [[/2010/]] * [[/2009/]] --> [[Category:Motivation and emotion/About/Schedule]] </noinclude> ep2jb1sjqxiyzwb0l81jjrz3uypp9xu 2410936 2410935 2022-08-02T07:15:46Z Jtneill 10242 wikitext text/x-wiki <noinclude> {{title|Schedule}}</noinclude><div class = "center"> {{:Motivation and emotion/About/Schedule/Embed}}</div> <noinclude> {{cot|Overview}} This [[Motivation and emotion/About/Schedule|schedule]] shows how the [[Motivation and emotion/Modules|modules]], [[Motivation and emotion/Lectures|lectures]], [[Motivation and emotion/Readings|readings]], [[Motivation and emotion/Tutorials|tutorials]], and [[Motivation and emotion/Assessment|assessment]] work together to help [[emerging scholar]]s achieve the [[Motivation and emotion/About/Learning outcomes|learning outcomes]]. ;Key dates * W02 (Fri) – Recommended topic sign up * W04 (Mon 9am) - Topic selection due * W04 (Mon 9am) - Quiz 1 due * W04 (Fri) – Census day - Last day to withdraw without penalty * W05 (Mon 9am) – Topic development due * W06 (Mon 9am) - Quiz 2 due * W08 – No classes - Mid-semester break * W08 (Mon 9am) - Quiz 3 due * W08 (Fri) – Withdraw without incurring fail * W11 (Mon 9am) - Quiz 4 due * W12 (Mon 9am) – Book chapter due * W13 (Mon 9am) - Quiz 5 due * W14 (Mon 9am) – Multimedia due * W15 (Mon 9am) – Quiz 6 due * W16 (Mon 3pm) – Results released {{cob}} <!-- ==See also== ;Previous schedules * [[/2021/]] * [[/2020/]] * [[/2019/]] * [[/2018/]] * [[/2017/]] * [[/2016/]] * [[/2015/]] * [[/2014/]] * [[/2013/]] * [[/2011/]] * [[/2010/]] * [[/2009/]] --> [[Category:Motivation and emotion/About/Schedule]] </noinclude> cd9951soedfj00y2z3tw11anh2otrpx 2410942 2410936 2022-08-02T08:28:37Z Jtneill 10242 wikitext text/x-wiki <noinclude> {{title|Schedule}}</noinclude><div class = "center"> {{:Motivation and emotion/About/Schedule/Embed}}</div> <noinclude> {{cot|Overview}} This [[Motivation and emotion/About/Schedule|schedule]] shows how the [[Motivation and emotion/Modules|modules]], [[Motivation and emotion/Lectures|lectures]], [[Motivation and emotion/Readings|readings]], [[Motivation and emotion/Tutorials|tutorials]], and [[Motivation and emotion/Assessment|assessment]] work together to help [[emerging scholar]]s achieve the [[Motivation and emotion/About/Learning outcomes|learning outcomes]]. ;Key dates * W02 (Fri) – Recommended topic selection * W04 ** Mon 9am - Quiz 1 due ** Mon 9am - Topic selection due (optional) ** Fri – Census day - Last day to withdraw without penalty * W05 (Mon 9am) – Topic development due * W06 (Mon 9am) - Quiz 2 due * W08 ** No classes - Mid-semester break ** Mon 9am) - Quiz 3 due ** Fri – Withdraw without incurring fail * W11 (Mon 9am) - Quiz 4 due * W12 (Mon 9am) – Book chapter due * W13 (Mon 9am) - Quiz 5 due * W14 (Mon 9am) – Multimedia due * W15 (Mon 9am) – Quiz 6 due * W17 (Mon) – Results released {{cob}} <!-- ==See also== ;Previous schedules * [[/2021/]] * [[/2020/]] * [[/2019/]] * [[/2018/]] * [[/2017/]] * [[/2016/]] * [[/2015/]] * [[/2014/]] * [[/2013/]] * [[/2011/]] * [[/2010/]] * [[/2009/]] --> [[Category:Motivation and emotion/About/Schedule]] </noinclude> soz0ic7qqhgligbfonioot7hx6xivs7 2410943 2410942 2022-08-02T10:09:37Z Jtneill 10242 wikitext text/x-wiki <noinclude> {{title|Schedule}}</noinclude><div class = "center"> {{:Motivation and emotion/About/Schedule/Embed}}</div> <noinclude> {{cot|Overview}} This [[Motivation and emotion/About/Schedule|schedule]] shows how the [[Motivation and emotion/Modules|modules]], [[Motivation and emotion/Lectures|lectures]], [[Motivation and emotion/Readings|readings]], [[Motivation and emotion/Tutorials|tutorials]], and [[Motivation and emotion/Assessment|assessment]] work together to help [[emerging scholar]]s achieve the [[Motivation and emotion/About/Learning outcomes|learning outcomes]]. ;Key dates * W02 (Fri) – Recommended topic selection * W04 ** Mon 9am - Quiz 1 due ** Mon 9am - Topic selection due (optional) ** Fri – Census day - Last day to withdraw without penalty * W05 (Mon 9am) – Topic development due * W06 (Mon 9am) - Quiz 2 due * W08 ** No classes - Mid-semester break ** Mon 9am) - Quiz 3 due ** Fri – Last day to withdraw without incurring fail * W11 (Mon 9am) - Quiz 4 due * W12 (Mon 9am) – Book chapter due * W13 (Mon 9am) - Quiz 5 due * W14 (Mon 9am) – Multimedia due * W15 (Mon 9am) – Quiz 6 due * W17 (Mon) – Results released {{cob}} <!-- ==See also== ;Previous schedules * [[/2021/]] * [[/2020/]] * [[/2019/]] * [[/2018/]] * [[/2017/]] * [[/2016/]] * [[/2015/]] * [[/2014/]] * [[/2013/]] * [[/2011/]] * [[/2010/]] * [[/2009/]] --> [[Category:Motivation and emotion/About/Schedule]] </noinclude> bru7vou0r651rnnyxo0l5fx6ps02xu7 2410944 2410943 2022-08-02T10:14:31Z Jtneill 10242 wikitext text/x-wiki <noinclude> {{title|Schedule}}</noinclude><div align = "center"> {{:Motivation and emotion/About/Schedule/Embed}}</div> <noinclude> {{cot|Overview}} This [[Motivation and emotion/About/Schedule|schedule]] shows how the [[Motivation and emotion/Modules|modules]], [[Motivation and emotion/Lectures|lectures]], [[Motivation and emotion/Readings|readings]], [[Motivation and emotion/Tutorials|tutorials]], and [[Motivation and emotion/Assessment|assessment]] work together to help [[emerging scholar]]s achieve the [[Motivation and emotion/About/Learning outcomes|learning outcomes]]. ;Key dates * W02 (Fri) – Recommended topic selection * W04 ** Mon 9am - Quiz 1 due ** Mon 9am - Topic selection due (optional) ** Fri – Census day - Last day to withdraw without penalty * W05 (Mon 9am) – Topic development due * W06 (Mon 9am) - Quiz 2 due * W08 ** No classes - Mid-semester break ** Mon 9am) - Quiz 3 due ** Fri – Last day to withdraw without incurring fail * W11 (Mon 9am) - Quiz 4 due * W12 (Mon 9am) – Book chapter due * W13 (Mon 9am) - Quiz 5 due * W14 (Mon 9am) – Multimedia due * W15 (Mon 9am) – Quiz 6 due * W17 (Mon) – Results released {{cob}} <!-- ==See also== ;Previous schedules * [[/2021/]] * [[/2020/]] * [[/2019/]] * [[/2018/]] * [[/2017/]] * [[/2016/]] * [[/2015/]] * [[/2014/]] * [[/2013/]] * [[/2011/]] * [[/2010/]] * [[/2009/]] --> [[Category:Motivation and emotion/About/Schedule]] </noinclude> p7wf3ie2rd5kt4vlfyjdmovxg3ejnyb 2410945 2410944 2022-08-02T10:17:00Z Jtneill 10242 wikitext text/x-wiki <noinclude> {{title|Schedule}}</noinclude><div align = "center"> {{:Motivation and emotion/About/Schedule/Embed}}</div> <noinclude> {{cot|Key dates}} This [[Motivation and emotion/About/Schedule|schedule]] shows how the [[Motivation and emotion/Modules|modules]], [[Motivation and emotion/Lectures|lectures]], [[Motivation and emotion/Readings|readings]], [[Motivation and emotion/Tutorials|tutorials]], and [[Motivation and emotion/Assessment|assessment]] work together to help [[emerging scholar]]s achieve the [[Motivation and emotion/About/Learning outcomes|learning outcomes]]. * W02 (Fri) – Recommended topic selection * W04 ** Mon 9am - Quiz 1 due ** Mon 9am - Topic selection due (optional) ** Fri – Census day - Last day to withdraw without penalty * W05 (Mon 9am) – Topic development due * W06 (Mon 9am) - Quiz 2 due * W08 ** No classes - Mid-semester break ** Mon 9am) - Quiz 3 due ** Fri – Last day to withdraw without incurring fail * W11 (Mon 9am) - Quiz 4 due * W12 (Mon 9am) – Book chapter due * W13 (Mon 9am) - Quiz 5 due * W14 (Mon 9am) – Multimedia due * W15 (Mon 9am) – Quiz 6 due * W17 (Mon) – Results released {{cob}} <!-- ==See also== ;Previous schedules * [[/2021/]] * [[/2020/]] * [[/2019/]] * [[/2018/]] * [[/2017/]] * [[/2016/]] * [[/2015/]] * [[/2014/]] * [[/2013/]] * [[/2011/]] * [[/2010/]] * [[/2009/]] --> [[Category:Motivation and emotion/About/Schedule]] </noinclude> okkimcyi5cwx5rk8m58zovtw8qsw7v2 2410946 2410945 2022-08-02T10:17:40Z Jtneill 10242 wikitext text/x-wiki <noinclude> {{title|Schedule}}</noinclude><div align = "center"> {{:Motivation and emotion/About/Schedule/Embed}}</div> <noinclude> {{cot|Key dates}} * W02 (Fri) – Recommended topic selection * W04 ** Mon 9am - Quiz 1 due ** Mon 9am - Topic selection due (optional) ** Fri – Census day - Last day to withdraw without penalty * W05 (Mon 9am) – Topic development due * W06 (Mon 9am) - Quiz 2 due * W08 ** No classes - Mid-semester break ** Mon 9am) - Quiz 3 due ** Fri – Last day to withdraw without incurring fail * W11 (Mon 9am) - Quiz 4 due * W12 (Mon 9am) – Book chapter due * W13 (Mon 9am) - Quiz 5 due * W14 (Mon 9am) – Multimedia due * W15 (Mon 9am) – Quiz 6 due * W17 (Mon) – Results released {{cob}} <!-- ==See also== ;Previous schedules * [[/2021/]] * [[/2020/]] * [[/2019/]] * [[/2018/]] * [[/2017/]] * [[/2016/]] * [[/2015/]] * [[/2014/]] * [[/2013/]] * [[/2011/]] * [[/2010/]] * [[/2009/]] --> [[Category:Motivation and emotion/About/Schedule]] </noinclude> t8pkvwmii85jcmw949xsxvn30rnx8kn 2410947 2410946 2022-08-02T10:18:16Z Jtneill 10242 wikitext text/x-wiki <noinclude> {{title|Schedule}}</noinclude><div align = "center"> <!-- This [[Motivation and emotion/About/Schedule|schedule]] shows how the [[Motivation and emotion/Modules|modules]], [[Motivation and emotion/Lectures|lectures]], [[Motivation and emotion/Readings|readings]], [[Motivation and emotion/Tutorials|tutorials]], and [[Motivation and emotion/Assessment|assessment]] work together to help [[emerging scholar]]s achieve the [[Motivation and emotion/About/Learning outcomes|learning outcomes]]. --> {{:Motivation and emotion/About/Schedule/Embed}}</div> <noinclude> {{cot|Key dates}} * W02 (Fri) – Recommended topic selection * W04 ** Mon 9am - Quiz 1 due ** Mon 9am - Topic selection due (optional) ** Fri – Census day - Last day to withdraw without penalty * W05 (Mon 9am) – Topic development due * W06 (Mon 9am) - Quiz 2 due * W08 ** No classes - Mid-semester break ** Mon 9am) - Quiz 3 due ** Fri – Last day to withdraw without incurring fail * W11 (Mon 9am) - Quiz 4 due * W12 (Mon 9am) – Book chapter due * W13 (Mon 9am) - Quiz 5 due * W14 (Mon 9am) – Multimedia due * W15 (Mon 9am) – Quiz 6 due * W17 (Mon) – Results released {{cob}} <!-- ==See also== ;Previous schedules * [[/2021/]] * [[/2020/]] * [[/2019/]] * [[/2018/]] * [[/2017/]] * [[/2016/]] * [[/2015/]] * [[/2014/]] * [[/2013/]] * [[/2011/]] * [[/2010/]] * [[/2009/]] --> [[Category:Motivation and emotion/About/Schedule]] </noinclude> 890cuyfrnodbias8dxgnpsigrbd112i 2410948 2410947 2022-08-02T10:19:17Z Jtneill 10242 wikitext text/x-wiki <noinclude> {{title|Schedule}}</noinclude><div align = "center"> <!-- This [[Motivation and emotion/About/Schedule|schedule]] shows how the [[Motivation and emotion/Modules|modules]], [[Motivation and emotion/Lectures|lectures]], [[Motivation and emotion/Readings|readings]], [[Motivation and emotion/Tutorials|tutorials]], and [[Motivation and emotion/Assessment|assessment]] work together to help [[emerging scholar]]s achieve the [[Motivation and emotion/About/Learning outcomes|learning outcomes]]. --> {{:Motivation and emotion/About/Schedule/Embed}}</div> <noinclude> {{cot|Key dates}} * W02 (Fri) – Topic selection recommended * W04 ** Mon 9am - Quiz 1 due ** Mon 9am - Topic selection due (optional) ** Fri – Census day - Last day to withdraw without penalty * W05 (Mon 9am) – Topic development due * W06 (Mon 9am) - Quiz 2 due * W08 ** No classes - Mid-semester break ** Mon 9am) - Quiz 3 due ** Fri – Last day to withdraw without incurring fail * W11 (Mon 9am) - Quiz 4 due * W12 (Mon 9am) – Book chapter due * W13 (Mon 9am) - Quiz 5 due * W14 (Mon 9am) – Multimedia due * W15 (Mon 9am) – Quiz 6 due * W17 (Mon) – Results released {{cob}} <!-- ==See also== ;Previous schedules * [[/2021/]] * [[/2020/]] * [[/2019/]] * [[/2018/]] * [[/2017/]] * [[/2016/]] * [[/2015/]] * [[/2014/]] * [[/2013/]] * [[/2011/]] * [[/2010/]] * [[/2009/]] --> [[Category:Motivation and emotion/About/Schedule]] </noinclude> iamvyd66yj39ok1te6vi3vp1rmbsjal 2410949 2410948 2022-08-02T10:23:41Z Jtneill 10242 wikitext text/x-wiki <noinclude> {{title|Schedule}}</noinclude><div align = "center"> <!-- This [[Motivation and emotion/About/Schedule|schedule]] shows how the [[Motivation and emotion/Modules|modules]], [[Motivation and emotion/Lectures|lectures]], [[Motivation and emotion/Readings|readings]], [[Motivation and emotion/Tutorials|tutorials]], and [[Motivation and emotion/Assessment|assessment]] work together to help [[emerging scholar]]s achieve the [[Motivation and emotion/About/Learning outcomes|learning outcomes]]. --> {{:Motivation and emotion/About/Schedule/Embed}}</div> <noinclude> {{cot|Key dates}} * W02 (Fri) – [[Motivation and emotion/Assessment/Selection|Topic selection]] recommended * W04 ** Mon 9am - [[Motivation and emotion/Assessment/Quizzes|Quiz 1]] due ** Mon 9am - [[Motivation and emotion/Assessment/Selection|Topic selection]] due (optional) - no extensions ** Fri - Last day to withdraw without penalty (Census day) * W05 (Mon 9am) – [[Motivation and emotion/Assessment/Topic|Topic development]] due - no extensions * W06 (Mon 9am) - [[Motivation and emotion/Assessment/Quizzes|Quiz 2]] due * W08 ** No classes - Mid-semester break ** Mon 9am) - [[Motivation and emotion/Assessment/Quizzes|Quiz 3]] due ** Fri - Last day to withdraw without incurring fail * W11 (Mon 9am) - [[Motivation and emotion/Assessment/Quizzes|Quiz 4]] due * W12 (Mon 9am) – [[Motivation and emotion/Assessment/Chapter|Book chapter]] due * W13 (Mon 9am) - [[Motivation and emotion/Assessment/Quizzes|Quiz 5]] due * W14 (Mon 9am) – [[Motivation and emotion/Assessment/Multimedia|Multimedia presentation]] due * W15 (Mon 9am) – [[Motivation and emotion/Assessment/Quizzes|Quiz 6]] due * W17 (Mon) – Results released {{cob}} <!-- ==See also== ;Previous schedules * [[/2021/]] * [[/2020/]] * [[/2019/]] * [[/2018/]] * [[/2017/]] * [[/2016/]] * [[/2015/]] * [[/2014/]] * [[/2013/]] * [[/2011/]] * [[/2010/]] * [[/2009/]] --> [[Category:Motivation and emotion/About/Schedule]] </noinclude> f33rn32egug6h5ojbxjg5ro9sa1vxhv 0 99074 2410914 1823254 2022-08-02T05:52:25Z Jtneill 10242 wikitext text/x-wiki {{METT|Motivational Toxicity}} {{MET}} {{100%done}} __TOC__ {{MECR|http://screenr.com/0Yk}} ==Overview== [[File:Efimero.jpg|260px|thumb|right|Cocaine is a highly addictive illicit substance]][[w:Pleasure|Pleasure]] is the subjective interpretation of the feelings of [[w:Happiness|happiness]] and sense of satisfaction that is derived from an enjoyable event or stimulus (Adinoff, 2004). Pleasurable feelings are the brain’s way of rewarding particular behaviours and are experienced as the feeling of contentment after a good meal, the refreshing sensation of cold water easing a parched throat or the orgasmic rush of a sexual encounter (Foddy & Savulescu, 2007). Pleasurable experiences are beneficial in a variety of ways. Pleasures often promote survival and reproductive behaviour, and some also reduce [[w:Stress|stress]], stimulate [[w:Personal growth|personal growth]] and promote [[w:Health|health]] and [[w:Well being|well-being]] (Esch & Stefano, 2004). But pleasure-seeking can get out of control. [[../Addiction/|Addiction]] and motivational toxicity can be the negative outcomes of pleasure-seeking gone terribly wrong (Hyman, 2007). What makes an addict capable of lying, stealing and cheating to get their hands on a drug? How can they take advantage of other people and cause such pain and destruction to their families? Why do they apologise and promise to stay sober only to quickly get drunk again? Are addicts morally weak individuals who choose their drug over everything else, or are they sufferers of a chronic disease that robs them of their self-control? This chapter will explore the neurobiological structures and processes that underlie [[w:Substance addiction|substance addictions]] and the associated drug-obsessed thoughts and compulsive, destructive behaviours. Addictions come in many forms and involve a variety of brain regions and [[w:Neurotransmitters|neurotransmitters]]. Merely touching on research in all these areas would fill many books, therefore this chapter will focus mainly on drug and alcohol addiction and the role of the neurotransmitter [[w:Dopamine|dopamine]] in the brain’s [[w:Reward system|reward system]]. == Motivation, addiction and motivational toxicity == [[File:Alcohol desgracia.jpg|thumb|250px|right|Alcoholism is a family disease. It affects not only the drinker but those who care about him or her as well.]] '''[[Motivation]]''' is a collection of fundamental brain processes that underlie the two basic behavioural drives – pleasure and pain. ''Appetitive motivation'' directs us to pursue particular pleasant rewards while ''aversive motivation'' steers us away from potentially negative or dangerous situations. Ultimately, motivation is what directs our attention and triggers the initiation, persistence and intensity of our daily behaviours, which create our experiences, that collectively become our lives (Esch & Stefano, 2004). '''[[w:Addiction|Drug addiction]]''' is a chronic disorder characterised by compulsive drug-seeking behaviours despite obvious negative health and social consequences and regardless of whether or not the drug provides pleasure (Niehaus, Cruz-Bermudez & Kauer, 2009). [[w:Withdrawal|Withdrawal]] avoidance has long been considered the primary motive underlying drug addiction, suggesting that drug-seeking behaviours are a series of voluntary, planned actions that require flexible adaptability to changing conditions (Hyman, 2007). However, many experts now consider addiction to be a brain disease resulting from long-lasting neural changes and involving a complex interaction of biological and environmental factors (Volkow, Fowler & Want, 2003). Modern research identifies a reduced capacity for voluntary behavioural control due to the effects of long-term substance abuse on neural circuits (Bakalar, 2004). '''Motivational toxicity''' refers to the powerful drug-obsessed drive that takes control of a drug addict’s thoughts and behaviours at the expense of all other things. The toxic motivation of chronic, long-term drug abuse steals behavioural control away from the brain’s natural reward system and commandeers the associated cognitions into the endless pursuit for drug use (Esch & Stefano, 2004). Motivational toxicity is the combination of the over-valuing of the abused drug, reduced sensitivity to [[w:Natural rewards|natural rewards]], impaired inhibitory control and disrupted cognitive functioning (Volkow et al., 2004). == The reward system and the pleasure pathways == In 1954, Olds and Milner famously enabled rats to electrically self-stimulate certain parts of their brains. Their research revealed particular areas in the brain that the rats persistently self-stimulated to the exclusion of all other behaviours. These areas are now known as the brain’s pleasure centres (Adinoff, 2004). Olds and Milner’s technique has been refined over the years and is now known as [[w:Intracranial self stimulation|intracranial self stimulation]]. It has been widely used to identify the neurobiological reward pathways and structures (Schmitz, 2005). {{Hide in print| [[File:Dog with rawhide chew toy.jpg|250px|thumb|left|"Oh man, this is my kind of reward!"]]}} [[File:Chocolate Cupcakes with Raspberry Buttercream detail.jpg|250px|thumb|left|A delicious "natural" (non-drug) reward]] The reward system is a complex structure of neuronal circuits in the [[w:Central nervous system|central nervous system]] that are intricately connected with the [[w:Limbic system|limbic system]] and the [[w:Frontal cortex|frontal cortex]] to guide and govern behaviour (Esch & Stefano, 2004). [[w:Human evolution|Evolution]] has refined this system into complex biological mechanisms that promote goal-seeking by enabling us to create internal mental images of our desired rewards, assign them a value for comparison against other alternatives, learn automatic sequences of goal-attaining behaviours, suppress competing distractions and inhibit disruptive impulses (Hyman, 2007). These neural processes prioritise our needs and desires into a motivational hierarchy designed to promote survival, health and well-being (Esch & Stefano, 2004). The basic premise of the reward system lies in the principles of [[w:Operant conditioning|operant conditioning]]. Rewards are the tools that control our motivation and guide our behaviour, when our actions are pleasantly rewarded they are more likely to be repeated (Schmitz, 2005). Getting the reward reinforces, strengthens and improves the efficiency of the behaviour (Hyman, 2007). Pleasure and addiction may be two sides of a coin but they follow the same neural route - the pleasure pathways. As will be discussed below, these pathways are the [[w:Mesolimbic dopamine pathway|mesolimbic dopamine]] tracts which are made up of the projected axons from dopamine neurons situated in the [[w:Ventral tegmental area|ventral tegmental area]] of the [[w:Midbrain|midbrain]] (Esch & Stefano, 2004). ===== Natural reinforcers ===== Natural rewards are the pleasant feelings of enjoyment and arousal that are experienced in relation to behaviours or stimuli that promote survival, health and well-being (Adinoff, 2004). These reinforcers are controlled by biological feedback mechanisms to help regulate the motivational hierarchy. For example, we eat when we are hungry and we stop eating when we feel full (Esch & Stefano, 2004). These reinforcers are prioritised by their fluctuating value and [[w: Saliency|saliency]] in order to maintain an optimal level of [[w:Homeostasis|homeostasis]] (Volkow, Fowlwer & Want, 2003). Rewards trigger the release of the neurotransmitter dopamine in the brain resulting in the feeling of pleasure and promoting learning and memory (Hyman, 2007). ===== Drug rewards ===== Natural and drug rewards act at the same brain sites and both trigger the release of dopamine. Chemical substances however, directly and strongly stimulate the CNS without any controlling mechanism to keep them in check (Bakalar, 2004). This direct and unregulated activation of the CNS earns the drug a fixed and super-potent reward value which contrasts with the fluctuating relative value of natural rewards (Esch & Stefano, 2004). Chronic drug use also results in phasic dopamine levels, a faster, more intense release followed by a rapid decrease, which strengthens the motivational properties of the drug well beyond that of natural rewards (Volkow & Li, 2004). Drug-induced dopamine bursts also enhance [[w:Conditioned learning|conditioned learning]], supporting the growing connections between drug use and drug-predicting internal and environmental cues (Volkow, Fowler & Want, 2003). ===== Drug use vs. drug addiction ===== What are the differences between drug use and drug addiction? Researchers have compared laboratory rats with differential access to cocaine self-administration to explore this question. As expected, their findings indicate that addicted rats self-administer higher levels of cocaine than non-addicted rats. Additionally, when the cocaine is taken away, addicted rats are more motivated to get their dose, take increased risks in the process, persist longer with drug-seeking behaviours and have a higher prevalence of relapse (Ahmed & Koob, 2005). == Drug effects == [[File:Neurons big1.jpg|450px|thumb|Neurons are made up of a cell body, axons and dendrites. The synapse is a small junction where chemical messengers pass from the axon of a pre-synaptic neuron to receptors on a post-synaptic neuron.]] Long-term use of [[w:Alcohol|alcohol]], [[w:Cannabis|cannabis]], [[w:Opiates|opiates]] and [[w:Psychostimulants|psychostimulants]] have all been associated with neuronal injury, dysfunction and loss of brain volume (Yucel & Lubman, 2007). The maladaptive neuronal changes incurred by substances of abuse correlate with the dysfunctional behaviours of addiction (Niehaus, Cruz-Bermudez & Kauer, 2009). These substances disrupt brain function and cause physical changes that redirect the neural circuits involved in planning, decision making, inhibitory control and attention to compulsively respond to drug related stimuli, creating an incredibly strong motivational drive (motivational toxicity) to seek and use drugs even when they no longer provide pleasure (Adinoff, 2004). ===== Neural sensitisation ===== The drugs' ability to directly and intensely stimulate the CNS results in the assignment of a super-potent reward value that far outranks the value of natural reinforcers, reducing the ability of other stimuli to compete for the addicts attention (Bakalar, 2004). This is the concept of [[w:Sensitization|neural sensitisation]] which results in the excessive ''wanting'' of the drug, the motivation to get the drug as opposed to ''liking'' the effect it produces (Robinson & Berridge, 2003). It is important to note the difference between ''liking'' and ''wanting'', as wanting can occur without liking. '''Liking''' refers to the pleasurable feelings experienced when a desired reward has been procured. '''Wanting''' refers to the motivational salience that makes the reward desirable, wanting transforms a reinforcer from a sensory representation into a desired reward capable of capturing attention and motivating behaviour (Zhang et al., 2009). ===== Conditioned learning ===== Addicts also demonstrate an attentional bias towards drug-related stimuli. Drug-cues activate memories, sensations and emotions generating responses strong enough to induce cravings and relapse (Gardini, Caffarra & Venneri, 2009). Over-learning of drug-stimuli associations is due to the enhanced activation of the memory and learning circuits from drug-induced dopamine surges. This strengthened learning supports the development of behaviour sequences that are automatically elicited by drug-stimuli, further increasing the motivational salience of the drug, and links affective states to drug-use to form future drug effect expectations (Volkow, Fowler & Wang, 2003). Conditioned learning is an important aspect of addiction, however problems arise with the ability to activate the neural motivational systems, as learning by itself does not generate compulsive behaviours. Robinson and Berridge (2003) suggest that the abnormalities of addiction do not lie in the conditioned learning itself, but in the response of the brain's motivational systems to the conditioned drug cues. {{Hide in print| == Test yourself == <quiz display=simple> {Motivational toxicity is the drive behind addiction behaviours. |type="()"} + TRUE. - FALSE. {Drug rewards have the same motivational salience as natural rewards. |type="()"} - TRUE. + FALSE. </quiz> }} == Brain structures and functions == Modern [[w:FRMI|fMRI]] research has revealed that drug use and associated cravings activate a complex pattern of interconnected brain areas that receive direct innervations from dopamine neurons. These areas include the ventral tegmental area of the midbrain, the [[w:Nucleus accumbens|nucleus accumbens]], [[w:Orbitofrontal cortex|orbitofrontal cortex]], [[w:Anterior cingulate|anterior cingulate]], [[w:Prefrontal cortex|prefrontal cortex]], [[w:Amygdala|amygdala]] and [[w:Hippocampus|hippocampus]]. These regions, and their involvement in rewards, motivation, memory, learning and cognitive control, mediate our everyday behavioural choices by formulating comparisons, expectations and associations (Volkow, Fowler & Want, 2003). Modern researchers hypothesise that drugs of abuse may create an 'addiction signal' that begins in the ventral tegmental area and is passed along to the nucleus accumbens and other areas of the frontal cortex and limbic system by mechanisms similar to [[w:Long term potentiation|long term potentiation]] (LTP) or depression (LTD). We will explore LTP and LTD in the next section (Niehaus, Cruz-Bermudez & Kauer, 2009). <!-- Check out this [http://www.youtube.com/watch?v=jneJUgo_tFQ| youtube clip] on addiction and the reward system. --> == Neural structures of the reward system == {| class="wikitable" |- | '''Ventral tegmental area''' || The pleasure pathways start here in the midbrain. This is the home of dopamine neurons whose projected axons create the mesolimbic system that innervates and interconnects other regions. Natural and drug rewards stimulate the VTA to start the cycle of dopamine release. Reward-driven behaviours are processed to provide information for modifying future behaviours (Niehaus, Cruz-Bermudez & Kauer, 2009). |- | '''Frontal and prefrontal cortex''' || The large frontal lobes consist of many specialised sections with specific roles. Overall, the frontal cortex coordinates and processes the information that determines appropriate behaviour (Esch & Stefano, 2004). The prefrontal cortex houses our conscious goals, is involved in decision making, inhibitory control and reward values (Volkow, Fowler & Wang, 2003). |- | '''Nucleus accumbens''' || The primary region for the pleasurable experience of rewards, or the sensation of liking. Here, the motivational saliency (how important something is compared to other things) of an expected reward is calculated in relation to the current situation and previous experiences (Volkow, Fowler & Wang, 2003). This area plays a critical role in priming future drug use and is involved in the higher order sensory and motor processes that initiate goal-directed behaviours, regardless of the type of reinforcer (Adinoff, 2004). |- | '''Orbitofrontal cortex''' || A major component of the reward system, the OFC houses several functions that if disrupted produce addiction behaviours. Reward values are compared to alternative options, taking into consideration recent experiences and internal needs. Appropriate courses of action are determined for unpredictable situations or when there is limited information available. The OFC also assesses the relevance and value of behavioural options, the familiarity of situations and outcome expectancies and provides behavioural stop signals (Adinoff, 2004). A critical area for drive and motivation, conditioned responses, learning stimulus-reinforcement associations, inhibiting emotional responses, compulsive behaviours, [[w:Perseveration|perseveration]] and protecting reward-association behaviours from extinction (Volkow et al., 2004). Disrupting the OFC can lead to impulsivity, decision making and [[w:Obsessive compulsive disorder|obsessive-compulsive disorders]] (Adinoff, 2004). |- | '''Anterior cingulate''' || Inhibitory control over behaviour, performance monitoring, consideration of available choices, error detection, prediction of potential conflicts between intended actions and their outcomes (Yucel & Lubman, 2007). Also responsible for emotional self-control, focused problem solving and the ability to adapt to changing conditions under the influence of current motivation and mood (Adinoff, 2004). Disrupting this region impairs ability to detect errors in behaviour reducing capacity to control actions in the face of negative outcomes (Yucel & Lubman, 2007). Also associated with emotion and attention disorders (Adinoff, 2004). |- | '''Amygdala''' || Consolidates memories of emotionally arousing events, assesses whether events are pleasant or unpleasant, and subsequently whether they should be repeated or avoided. Assigns reward values to stimuli and is involved in fear conditioning in new situations (Adinoff, 2004). |- | '''Hippocampus''' || Formation of new memories of personal experiences ([[w:Episodic memory|episodic memory]]) and learning new factual information. Damage to the hippocampus can result in amnesia and is implicated in [[w:Alzheimer's disease|Alzheimer's disease]] (Adinoff, 2004). |- | '''Dorsolateral prefrontal cortex''' || Home of our [[w:Working memory|working memory]], conscious management of chunks of current information enables us to sequence events, and plan and select goals (Adinoff, 2004). |} == Dopamine and the pleasure pathways == Neurotransmitters are chemical messengers released from one neuron, across the synapse (the tiny gap between cells) to be taken up by receptors on other neurons (Volkow et al., 2004). The activation of the [[w:Mesolimbic pathway|mesolimbic pathways]] releases dopamine into the synapses, producing feelings of pleasure and enabling the determination and recording of motivational saliency (reward value) for the particular reinforcer (Adinoff, 2004). Dopamine is naturally released at a slow, steady rate to consistently regulate mood and drive motivation (Esch & Stefano, 2004). Unexpected, new or better than anticipated experiences trigger the release of dopamine to enhance motivation encoding, new learning, information processing, reward prediction and reinforcement. Addictive drugs trigger a flood of dopamine release that creates the euphoric sensation of a drug high and strongly facilitates drug learning (Volkow et al., 2004). Drug-induced higher levels of stimulation are adapted to by the resetting of reward values that further enhance the value of the drug and decrease sensitivity to natural reinforcers. Simultaneously the over-activated motivational and memory circuits start to take over from the frontal cortex, reducing inhibition functions and creating a positive feedback loop that results in compulsive drug-taking (Volkow, Fowler & Wang, 2003). The phasic dopamine levels induced by drugs eventually result in chronically reduced levels of dopamine and [[w:Dopamine receptors|dopamine receptors]] in the mesolimbic system and reward circuitry, this renders the addict hyper-sensitive to the substance and strengthens the motivation to get and use the drug (Bakalar, 2004). [[File:Dopamine Pathways.png|400px|left|thumb|The dopamine pathways originate in the midbrain and spread out to innervate the striatum, hippocampus, amygdala (not shown), nucleus accumbens and frontal cortex]] ===== Dopamine receptors and reward deficiency syndrome ===== There are two main families and five sub-types of dopaminergic receptors. D2 dopamine receptors are the ones most commonly associated with addiction (Adinoff, 2004). The amount of dopamine receptors that an individual has is influenced by both genetic and environmental factors. A person who has more receptors is likely to be less affected by drugs of abuse as there will be less dopamine free-floating at the synapse, this may offer a higher level of protection against addiction. Conversely, a person with less receptors may be predisposed to use drugs as a means of compensating for decreased reward circuit stimulation (Volkow et al, 2004). Reward deficiency syndrome (RDS) refers to chronically low levels of dopamine and dopamine D2 receptors, a common factor in a variety of addictive behaviours including drugs, [[w:Obesity|obesity]], [[w:Compulsive gambling|compulsive gambling]] and [[w:Sex addiction|sex addiction]]. RDS reduces the level of pleasure experienced from normal rewarding stimuli, such as a good book or friendly social interaction. Binging on addictive substances increases dopamine release and may be a form of self-medication to compensate for an under-active reward system. Unfortunately binges are quickly followed by the down-regulation of dopamine D2 receptors returning levels to their chronically low state and creating a vicious cycle of craving and compulsion (Fortuna & Smelson, 2008). ===== Synaptic plasticity and long term potentiation ===== Various drugs exert different effects, but all of them alter the strength of the synapses stimulating VTA dopamine neurons, suggesting that all addictive substances may alter brain reward circuitry through the same mechanisms early in addiction (Niehaus, Cruz-Bermudez & Kauer, 2009). [[w:Synaptic plasticity|Synaptic plasticity]] refers to the ability to alter the synaptic connections between neurons by a mechanism known as long term potentiation or depression (LTP or LTD), a long-lasting increase or decrease in synaptic transmission hypothesised to underlie information storage in the brain (Kauer & Malenka, 2007). After drug use, changes in synaptic plasticity are equivalent to changes in behaviours. More importantly, sometimes when the synaptic changes are reversed, some of the behavioural effects are reversed along with them. It is theorised that addictive drugs use LTP or LTD to alter dopamine synapses in the reward circuits. This either promotes or blocks plasticity of learning and memory circuits, hence influencing the over-learning and long-term storage of the reward-related memories involved in addiction (Niehaus, Cruz-Bermudez & Kauer, 2009). == Non-chemical addictions == [[File:Gambling chips.jpg|thumb|right|Gambling is an impulse-control disorder that can become an addiction]] Both substance-related and [[w:Impulse control disorders|impulse-control disorders]] can be considered addictions. Although there are notable differences between the various disorders many appear to share common pathways. [[../Gambling|Pathological gamblers]] in search of a particular state of arousal stimulate the same dopaminergic reward and pleasure pathways as drug addictions, and demonstrate similar forms of tolerance and dependence. Like chemical addictions, pathological gamblers demonstrate increased rates of comorbid [[w:Mood disorders|mood disorders]], [[w:Attention deficit hyperactivity disorder|attention deficit/hyperactivity disorder]], substance-related disorders, and [[w:Personality disorders|personality disorders]] (Schmitz, 2005). Binge-eating disorder, also known as [[w:Food addiction|food addiction]], is defined by a lack of control over eating larger than normal amounts of food. Research has suggested similar decreases in dopamine D2 receptors may lead food addicts to seek out tasty rewards. Also in line with drug addictions, research recently identified the nucleus accumbens as the main site for the sweet tooth, sparking suggestions that food binges are a form of addictive behaviour that increase levels of dopamine (Fortuna & Smelson, 2008). {{Hide in print| == Test yourself == <quiz display=simple> {Phasic dopamine eventually results in chronically decreased levels of dopamine. |type="()"} + TRUE. - FALSE. {Reversing changes to synaptic plasticity also reverses addiction. |type="()"} - TRUE. + FALSE. </quiz> }} == Vulnerability to addiction == [[File:Santiago Rusinol Before the Morphine.jpg|thumb|right|A woman addicted to morphine. Painting by Santiago Rusiño, last decade of the 19th century.]] Even with an extremely addictive drug like [[w:Cocaine|cocaine]], only about 15% of users will become addicted. So what makes those people vulnerable to addiction? Although genetics are estimated to account for 40 to 60% of addiction vulnerability, other well known influential factors include stress, mental illness, social status and environmental factors (Volkow & Li, 2004). Increased risk of substance abuse and addiction may also be related to a decreased sensitivity to natural rewards, genetic or stress related deficiencies in D2 dopamine receptors, reduced activity of dopamine circuits or increased sensitivity to drug-stimuli by learning or motivation circuits (Volkow, Fowler & Wang, 2003). Adolescents have a particularly high susceptibility to addiction as their developing brains will not reach full maturity until into their 30’s. Research consistently shows higher levels of disruption and increased addiction due to the vulnerability of teenage brains to the neurotoxic effects of drugs (Yucel & Lubman, 2007). == Opponent-process theories of drug addiction == Drug-induced over-activation of the reward system eventually results in chronically decreased responsivity. According to [[w:Opponent process theory|opponent-process theories]], this occurs due to an anti-reward system that counter-balances the natural reward system to maintain a narrow range for reward thresholds and an optimal level of homeostasis (Ahmed & Koob, 2005). The concept of opponent-processes is a common feature in biological systems but has not been as extensively researched in relation to addiction. Researchers suggest that the anti-reward process may manifest as withdrawal symptoms and involve [[w:Stress hormones|stress hormones]] such as corticotrophin releasing factor (CRF), [[w:Norepinephrine|norepinephrine]] and [[w:Dynorphin|dynorphin]] (Koob & Le Moal, 2005). The general concept of opponent-process theories is that an ''a''-process moves a biological system out of its normal range, closely followed by a slower ''b''-process designed to gradually reinstate stability. In relation to drug use, initial drug consumption may boost the reward system responsivity well outside of normal range, the closely following opposing ''b''-process works to stabilise the reward threshold resulting in the symptoms of withdrawal. During drug-binge use, the anti-reward ''b''-process may not have sufficient time to successfully return the system to normal before the next drug hit delivers another surge. By this unbalanced process the drug may eventually reset the reward threshold, producing tolerance and gradually increasing the required dosage strength (Ahmed & Koob, 2005). Robinson and Berridge (2003) identify limitations of the opponent-process theories. They note that often the symptoms of withdrawal are simply not potent enough to motivate drug-seeking behaviours. Additionally, withdrawal symptoms generally peak within a few days and then taper off, hence the short-term nature of withdrawal is an insufficient explanation for the long term potential to relapse. == Addiction as a form of pleasure-seeking == Some researchers disagree with the chronic-disease model of addiction. Foddy and Savulescu (2007) suggest that addiction is neither a brain disease nor a moral condition but an unusually strong, socially unacceptable pleasure-seeking preference that has been labelled a disease in order to allow the medical treatment of behaviours that challenge [[w:Social norms|social norms]]. Foddy and Savulescu compare drug addiction to the the nineteenth century disorder of [[w:Drapetomania|drapetomania]], which caused slaves to run away from their masters, and people to mountain climb, suggesting that research is devoted to drug addiction over mountain climbing due to the sociopolitical context of the twenty-first century. Opiate-addicted returning Vietnam soldiers are offered as evidence that many addicts simply stop using or mature out of their addiction. Hence, in the view of Foddy and Savulescu, addiction is nothing more than a strong drive to experience pleasure. == Reversing motivational toxicity == Despite the increased value of drugs, the stronger and well-learned drug cues and the damaged neural control mechanisms, addictions can be overcome. Recovery requires a drug-free environment, strong support system, and the understanding that multiple relapses are likely and should be accepted as a temporary setback rather than a personal failure (Hyman, 2007). Effective treatments need to adopt multi-modal methods combining pharmacological, cognitive and behavioural therapies targeted at decreasing drug reward values, increasing natural reward values, weakening learned drug responses, extinguishing drug-cue associations and differential reinforcement of other behaviours (Volkow, Fowler & Wang, 2003). There are two main pharmacological interventions used when treating addictions. The first type of medications interfere with the reinforcing effects of the drug by triggering aversive responses or interfering with their binding, the drug-induced dopamine increase, the postsynaptic response, or the drug's ability to reach the brain. The second type compensate for the long-term adaptations of addiction by decreasing the drugs motivational value, enhancing the motivational saliency of natural rewards, or interfering with conditioned responses, stress and withdrawal symptoms (Volkow & Li, 2004). [[File:Sand Garden at Tofukuji Temple.jpg|300px|left|thumb|Meditation promotes mindfulness - remaining in the present moment, watching your thoughts without reaction or judgement.]] Alternative and adjunctive treatments may help reduce symptoms and support pharmacological and behavioural therapies. Sensory stimulation, for instance music, may influence the limbic and reward pathways involved in feelings of pleasure and happiness, thus helping to reduce [[w:Anxiety|anxiety]], [[w:Depression|depression]] and stress (Esch & Stefano, 2004). Auricular acupuncture stimulates branches of the cranial nerves that affect the limbic system, resulting in the production of the calming neurotransmitter [[w:Serotonin|serotonin]]. Research demonstrates this may help improve sleep, confidence, concentration, energy and physical and psychological well being, and reduce anxiety, cravings and stress (Handley, 2009). The Buddhist mindfulness practice of [[w:Vipassana meditation|Vipassana meditation]] teaches objective, self-observation without reaction. Research by the University of Washington found Vipassana mediation to be a low-cost adjunctive treatment that helps to reduce substance use and associated psychiatric symptoms, and increase [[w:Optimism|optimism]] and substance-use related [[w:Locus of control|locus of control]] (Monti, 2007). Designing effective preventative methods requires early onset drug use research, increasing adolescent education and early identification of comorbid conditions and characteristic behavioural problems such as impulsivity and emotional dysregulation (Yucel & Lubman, 2007). == Questions to make you think == #Can you define motivational toxicity? #What is the difference between appetitive and aversive motivation? #Do you consider addiction a disease? Why or why not? #Can you describe the reward system? #How is liking different to wanting? #What processes does dopamine facilitate? #Can you define synaptic plasticity and long term potentiation? #What methods would you use to treat an addiction? == Summary == Motivational toxicity is the potent maladaptive drive that directs an addict to acquire and use their drug (Esch & Stefano, 2004). Compulsive addiction behaviours are the direct result of neural damage that is caused by substances of abuse (Adinoff, 2004). Addictive drugs directly and intensely stimulate the brain reward system and dopaminergic pathways causing physical damage and disruption to learning, memory, motivation, reward and cognitive control circuits (Bakalar, 2004). The reward system is a collection of neural structures originating in the midbrain and incorporating the frontal lobes and limbic system,. The system is innervated and interconnected by the mesolimbic dopamine pathway (Volkow, Fowler & Wang, 2003). Stimulating the dopaminergic neurons triggers dopamine release which produces feelings of pleasure, codes motivational saliency and enhances learning (Volkow & Li, 2004). Continuous drug stimulation of the pleasure pathways incurs phasic bursts of dopamine that lead to neuronal damage and eventually result in chronically reduced levels of dopamine and dopaminergic receptors (Fortuna & Smelson, 2008). Neural sensitisation, synaptic plasticity and long-term potentiation are suggested mechanisms for drug-induced neural disruption (Kauer & Malenka, 2007). Addiction theories have moved away from being predominantly withdrawal-avoidance based towards chronic disease and opponent-process models. Opponent-process theories suggest an anti-reward system attempts to counter-balance the drug effect and restabilise homeostasis after drug use (Ahmed & Koob, 2005). Not all researchers agree. Foddy and Savulescu (2007) claim that addiction is merely extreme pleasure-seeking using socially unacceptable behaviours. Common neural pathways, structures and processes have been identified in substance abuse disorders and impulse control disorders indicating shared foundations (Schmitz, 2005). However, not all people that experiment with illicit drugs will become addicted. Vulnerability is influenced by genetics, age, stress, mental health, social status and environmental factors (Volkow & Li, 2004). Addiction and motivational toxicity can be overcome with time, support, understanding and multi-modal treatment strategies (Hyman, 2007). Combining pharmacological, behavioural, cognitive and adjunctive therapies enhances recovery by targeting multiple levels of neural dysfunction, cognitive control and mental and physical health and well-being (Handley, 2009). Like other chronic conditions, drug addiction has known risk factors, a typical course and outcome, equivalent relapse and recovery rates, physiological underpinnings and a genetic influence. Treating drug addiction as a disease reduces the associated stigma, shame and moral devaluation, and offers families and individuals hope and understanding for recovery (Volkow & Li, 2004). {{RoundBoxTop}} {Hide in print| <div style="text-align:center;"> ;The Serenity Prayer God grant me the serenity To accept the things I cannot change,<br> Courage to change the things I can, <br>And wisdom to know the difference. </div>{{RoundBoxBottom}} }} == Glossary == '''Addiction''' - Chronic brain disease characterised by compulsive behaviour and poor inhibition. *'''Amygdala''' - Fear conditioning area of the limbic system. *'''Anterior cingulate''' - Performance monitoring and behaviour control. *'''Appetitive motivation''' - Drive that directs us towards pleasurable events. *'''Aversive motivation''' - Drive that steers us away from aversive events. *'''Drapetomania''' - Nineteenth century mental disorder that caused slaves to flee captivity, now considered a pseudoscience. *'''Dopamine''' - Neurotransmitter critical to mood, reward, motivation and learning *'''Dopaminergic receptor''' - Five subtypes of D1 and D2 receptors take up dopamine at the synapse. *'''Dorsolateral prefrontal cortex''' - Working memory. *'''Frontal lobe''' - Executive functions. *'''Hippocampus''' - Memory, implicated in Alzheimers disease. *'''Homeostasis''' - The optimal, stable, internal condition. *'''Liking''' - Pleasurable feeling in response to attaining a desired reward. *'''Long term potentiation''' - A long lasting increase in synaptic transmission. *'''Motivational toxicity''' - Strong, maladaptive drive to procure drugs in addiction. *'''Nucleus accumbens''' - Pleasure centre and motivational saliency. *'''Opponent processes''' - Concept of paired processes in biological systems, b-process counter-balances a-process to maintain homeostasis. *'''Orbitofrontal cortex''' - Compares alternatives to determine behaviour. *'''Pleasure pathways''' - Tracts of dopamine axons that innervate reward system structures. *'''Prefrontal cortex''' - Conscious goals and decision making. *'''Reward deficiency syndrome''' - Reduced amount of dopamine and D2 dopamine receptors in the reward circuits. *'''Reward system''' - motivational brain system that guides behaviours. *'''Saliency''' - Stands out in comparison to alternatives, grabs our attention. *'''Synapse''' - Tiny junction where messages pass between neurons. *'''Synaptic plasticity''' - Ability of a synapse to change in strength. *'''Ventral tegmental area''' - Start of the pleasure pathways, processes reward-driven behaviours. *'''Wanting''' - Motivational drive to procur a desired reward. {{Hide in print| ==See also== * [[../Addiction/]] (Textbook chapter) * [[../Gambling/]] (Textbook chapter) }} == References == <div style="padding-left: 2em; text-indent: -2em"> Adinoff, B. (2004). Neurobiologic processes in drug reward and addiction. ''Harvard Review of Psychiatry, 12,'' 305-317. doi:10.1080/10673220490910844 Ahmed, S. H., & Koob, G. F. (2005). Transition to drug addiction: A negative reinforcement model based on allostatic decrease in reward function. ''Psychopharmacology, 180'', 473-490. doi:10.1007/s00213-005-2180-z Bakalar, J. B. (2004). The addicted brain. ''Harvard Mental Health Letter, 21,'' 1-4. Retrieved from http://web.ebscohost.com.ezproxy1.canberra.edu.au Esch, T., & Stefano, G. G. (2004). The neurobiology of pleasure, reward processes, addiction and their health implications. ''Neuroendocrinology Letters, 25,'' 235-251. Foddy, B., & Savulescu, J. (2007). Addiction is not an affliction: addictive desires are merely pleasure-oriented desires. ''The American Journal of Bioethics, 7'', 29-32. Retrieved from http://web.ebscohost.com.ezproxy1.canberra.edu.au Fortuna, J. L., & Smelson, D. A. (2008). The phenomenon of drug craving. ''Journal of Psychoactive Drugs, 40,'' 255-261. Retrieved from http://web.ebscohost.com.ezproxy1.canberra.edu.au Gardini, S., Caffarra, P., & Venneri, A. (2009). Decreased drug-cue-induced attentional bias in individuals with treated and untreated drug dependence. ''Acta Neuropsychiatrica, 21'', 179-185. doi:10.1111/j.1601-5215.2009.00389.x Handley, A. (2009). Straight to the point. ''Nursing Standard, 23'', 23-24. Retrieved from http://web.ebscohost.com.ezproxy1.canberra.edu.au Hyman, S. E. (2007). The neurobiology of addiction: Implications for voluntary control of behaviour. ''The American Journal of Bioethics, 7'', 8-11. doi:10.1080/15265160601063969 Kauer, J. A., & Malenka, R. C. (2007) Synaptic plasticity and addiction. ''Nature Reviews Neuroscience, 8'', 844-858. doi:10.1038/nrn2234 Koob, G. F., & Le Moal, M. (2005). Plasticity of reward neurocircuitry and the dark side of drug addiction. ''Nature Neuroscience, 8'', 1442-1444. Retrieved from http://web.ebscohost.com.ezproxy1.canberra.edu.au Monti, P. M. (2007). Mindfulness meditation promising as SUD treatment. ''The Brown University Digest of Addiction Theory and Application,'' 2-3. Retrieved from http://web.ebscohost.com.ezproxy1.canberra.edu.au Niehaus, J. L., Cruz-Bermudez, N. D., & Kauer, J. A. (2009). Plasticity of addiction: A mesolimbic dopamine short-circuit? ''The American Journal on Addictions, 18'', 259-271. doi:10.1080/10550490902925946 Robinson, T. E., & Berridge, K. C. (2003). Addiction. ''Annual Review of Psychology, 54'', 25-53. doi:10.1146/annurev.psych.54.101601.145237 Schmitz, J. M. (2005). The interface between impulse-control disorders and addictions: Are pleasure pathway responses shared neurobiological substrates. ''Sexual Addiction & Compulsivity, 12'', 149-168. doi:10.1080/10720160500203641 Volkow, N. D., Fowler, J. S., Wang, G. J., & Swanson, J. M. (2004). Dopamine in drug abuse and addiction: results from imaging studies and treatment implications. ''Molecular Psychiatry, 9'', 557-569. doi:10.1038/sj.mp.4001507 Volkow, N. D., Fowler, J. S., & Wang, G. (2003). The addicted human brain: insights from imaging studies. ''The Journal of Clinical Investigation, 111'', 1444-1452. doi:10.1172/JCI200318533 Volkow, N. D., & Li, T. (2004). Drug addiction: the neurobiology of behaviour gone awry. ''Nature Reviews, 5'', 963-971. Retrieved from http://web.ebscohost.com.ezproxy1.canberra.edu.au Yucel, M., & Lubman, D. I. (2007). Neurocognitive and neuroimaging evidence of behavioural dysregulation in human drug addiction: implications for diagnosis, treatment and prevention. ''Drug and Alcohol Review, 26'', 33-39. doi:10.1080/09595230601036978 Zhang, J., Berridge, K. C., Tindell, A. J., Smith, K. S., & Aldridge, J. W. (2009). A neural computational model of incentive salience. ''Computational Biology, 5'', 1-13. doi:10.1371/journal.pcbi.1000437 </div> {{Hide in print| == External links == * Alcoholics Anonymous Australia - http://www.aa.org.au/ * Alcohol and Drug Foundation ACT - http://www.adfact.org/index.htm * Centre for Addiction and Mental Health (CAMH) - http://www.camh.net/ * Directions ACT - http://www.directionsact.com/ * Narconon Australia - http://www.getoffdrugs.com.au/Addiction2.htm * National Institute on Alcohol Abuse and Alcoholism (NIAAA) - http://www.niaaa.nih.gov/ * Salvation Army - http://www.salvos.org.au/need-help/drugs-and-alcohol/ * The Sober Recovery Community - http://www.soberrecovery.com/forums/ * Wildmind Buddhist Meditation - http://www.wildmind.org/ }} [[Category:Motivation and emotion/Book/Addiction]] kzl15ainy9z06i74sa0nceb7h0wgy2s 0 117734 2410909 2326310 2022-08-02T05:47:41Z Jtneill 10242 wikitext text/x-wiki {{title|Addiction and motivation:<br>What motivates us to pursue unwanted addictive behaviour and how can this be stopped?}} {{MECR|1=http://www.youtube.com/watch?v=g8QBhxpQkP8&hd=1}} {{MEB}} __TOC__ ==Overview== In our daily lives, we often hear the term "[[w:addiction|addiction]]". People use it in reference to a lot of things, from the most serious addictions such as cocaine or heroin, to the more mild habits, such as shopping, food, or even internet sites such as [[w:facebook|Facebook]]. To some extent, even the mild habits may be acquired and governed through similar processes, and those people may well be experiencing mild addictions to those substances or activities. [[File:Alcohol desgracia.jpg|thumb|200px|right|Serious addictions can have a major impact on an individual's life]] While mild addictions may not be life threatening, they can be annoying, difficult, or time consuming. So, where do we draw the motivation for these addictions, mild or otherwise, from? Why is it that we can spend so much time playing video games, and yet never find the drive to do any homework? Why are we motivated to continue smoking, even though we know it is doing damage to our bodies? And, more importantly, how can we control and remove the addiction, so as to use our time and resources in a more efficient and effective way? This chapter defines addiction as a psychological or physical dependence on a substance or activity. The diagnostic and statistical manual of mental disorders (2000) describes behavioural and substance addictions separately, with the most relevant criteria for the purposes of this topic being that of substance abuse. In this chapter, research concerning serious addictions will be reviewed so as to provide a guideline for how to control and manage the addictions that we experience in our everyday lives, from the mildest the to the most serious. The term "stimulus" will be used to encompass both addictive substance and addictive behaviour. {| cellpadding="10" cellspacing="5" style="width50%; background-color: #000099; margin-left: auto; margin-right: auto" | style="width: 30%; background-color: #808080; border: #000099; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" <!--rowspan="2"--> | {{center top}}{{big2|'''DSM-IV-TR (2000) Definition of substance abuse'''}}</br> <big>"The essential feature of Substance Abuse is a maladaptive pattern of substance use manifested by recurrent and significant adverse consequences related to the repeated use of substances."</big>{{center bottom}} |} ==Origins of addiction== Throughout our lives, certain events occur that require discretion on our part before we engage in them. Once we have experienced a type of event, such as social networking, we make a judgement on it as it whether or not we liked the activity, and whether we should continue to do similar activities in the future. We tend to persist with behaviors that rewards us with positive outcomes, and avoid behaviors that lead us to less favourable consequences. To a large degree, this is the underlying principle from which addictions arise. These principles will be outlined with reference to a number of different theories. ===Social learning theory=== One of the most well known processes of addiction acquisition is the concept of social learning. The theory states that as behaviours are modelled they are learned by others, regardless of how ethical these actions are (Bandura, Ross & Ross, 1961). This was shown in a study by Bandura (1961) where children observed an adult violently attack a bobo doll. Subsequently, the study found that children who were exposed to the violent modelling were more likely to attack the doll themselves. It has also been asserted that the quality of the relationship with the model will contribute towards the likelihood of engaging in the behaviour (Borsari & Carey, 2006). The effect is similar in the process of addiction (Wilson, 1987). If we observe an individual smoking a cigarette, or using Facebook, we are more likely to perceive that modelled behaviour as acceptable. As a result, we are more likely to engage in that behaviour, marking the beginning stages of the addiction process. The social setting also has an impact on the likelihood of a behaviour occurring. The process, termed "social reinforcement", governs how acceptable an individual perceives an action to be, based on their current environment (Borsari & Carey, 2006). For example, you may be more likely to smoke in a designated smoking area than in someone's car, based on the feedback you have received for this action in the past. As a result, the social setting plays almost as critical a role in the development of an addiction as the peer group. ===Conditioning principles=== According to various theories of operant conditioning, behavioural responses to stimuli arise as a result of reinforcement and punishment, which can occur when the behavioural consequence is being added (positive) or removed (negative). In terms of processes of addiction, all of these operant principles play important roles. Positive reinforcement is the first part of the undesirable path towards addiction. Positive reinforcement can be defined as an attractive stimulus given to an individual as a result of an action (Watkins, Koob & Markou, 2000). As such, when an individual initially begins to take an addictive substance, they will likely experience a euphoric sensation, immediately rewarding them for consumption (Watkins, Koob & Markou, 2000). It can be expected, then, that this reward leads to an increased likelihood of the behaviour being performed in the future. In a sense, this part of the addiction cycle is not problematic, the problem arises when the action loses any benefit, as the individual builds up a tolerance (Tiffany, 1990). When this happens, more of the substance needs to be taken in order to achieve the same sensation. At this point, addiction becomes a more serious concern (Tiffany, 1990). It is common knowledge that when an individual builds up a dependence to a substance, and subsequently ceases to take it, [[w:withdrawal|withdrawal]] occurs (Wikler, 1980). What is less well known, is that similar symptoms have been exhibited in cases of behaviour cessation, where substances were not present (Cunningham-Williams, Gattis, Dore, Shi & Spitznagel, 2009). Research has shown that individuals who attempt to give up gambling can experience restlessness, irritability, anger, guilt and disappointment (Cunningham-Williams et al., 2009). Additionally, though less commonly, physiological symptoms, such as abdominal pain and diarrhea, may also arise (Custer, 1982; Rosenthal & Lesievr, 1992). Given that no substance is involved in this example, it can be argued that certain types of highly addictive activities may produce a dependence effect similar to that found in substance addiction. In any case, withdrawal symptoms are highly unpleasant. Not surprisingly, individuals often aim to avoid these aversive consequences, which can be achieved by avoiding the substance altogether. But if a dependence has already been formed, how does the individual avoid the unwanted consequences of cessation? It is this exact tendency that, unfortunately, allows negative reinforcement to be so highly effective in fostering an addiction (Baker, Piper, McCarthy, Majeskie & Fiore, 2004). Individuals are rewarded for perpetuating the addiction through the absences of undesirable consequences (Baker et al., 2004). Consequently, the dependence to the substance or behaviour is strengthened, and a vicious cycle ensues. So how does all of this research affect us? Initially, our first experience with an addictive stimulus may be the most positive, with the effect gradually declining overtime (Tiffany, 1990). For example, an Internet addiction, while it may be fun initially to browse the web, as usage reaches addiction levels the fun will wear off, becoming replaced with feelings of restlessness and unease when the behaviour has to stop (Young, 1999). As such, the focus now shifts from enjoying the behaviour to avoiding the cost of cessation, an effect similar to withdrawal avoidance (Young, 1999). ===Opponent-process theory=== In a similar process to the learning principles, the opponent-process theory of motivation can be applied to an understanding addiction formation (Ettenberg, 2004). The essential feature of the opponent-process theory is the idea that experiencing an extreme high or low (termed "the A process"), our bodies attempt to suppress this by adjusting the emotion in the opposite direction ("the B process"), to a small degree (Solomon & Corbit, 1973). As such, once the A process has worn off, the B process remains, leaving us in a more aroused or depressed state than we would expect to be (Solomon & Corbit, 1973). In terms of addiction, this process displays how the stimulus creates unpredictably greater emotional states. When the individual has been without the stimulus for a period of time, they will achieve greater highs when using it (Solomon & Corbit, 1973). When the individual is using the stimulus, they will experience greater lows when it is not in use (Solomon & Corbit, 1973). While this not a massive effect, it does facilitate the development of the addiction to some extent (Ettenberg, 2004). ===Environment=== Psychologically, we are predisposed to pick up on cues that we have come to associate with other stimuli, especially in the case of addictions (Kuntze et al., 2001). Cues, such as sounds, smells or the sight of a particular setting are all able to elicit a particular addiction-related response. Indeed, these responses can range from a craving for the stimulus, to withdrawal symptoms and mood swings (Kuntze et al., 2001). In this way, the environment itself can foster the addiction process, as the learned responses to the particular environment may trigger these temptation effects, and lead to a compulsion to continue the adverse pattern of behaviours. ===Neurological basis=== A number of neurological structures are involved in the process of attaining and reinforcing an addiction. The manner in which some of these brain structures are affected by addiction will be briefly outlined here. </br> {| cellpadding="10" cellspacing="5" style="float: right;width:30%; background-color: #000066; margin-left: auto; margin-right: auto" | style="width: 30%; background-color: #CCFFFF; border: #660000; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" <!--rowspan="2"--> | {{center top}}{{center bottom}}{{big2|'''Brain Structures Associated with Addiction'''}} * Basal Ganglia * Amygdala * Prefrontal Cortex * Nucleus Accumbens |} ====Basal ganglia==== The [[basal ganglia]], an area of the brain associated with procedural learning, increases production and transmission of [[dopamine]] in response to addiction (Kalivas & Volkow, 2005). Given the role of dopamine in valuing incentive and reward (Berridge & Robinson, 1998), this brain structure is one of the most important areas of the brain in facilitating addiction. ====Amygdala==== As the primary function of the [[amygdala]] is to govern emotional learning and memory (Everitt, Cardinal, Parkinson & Robbins, 2003), it plays a critical role in the formation of behavioural responses through its activation in response to motivationally relevant stimuli (Kalivas & Volkow, 2005). ====Prefrontal cortex==== Similar to the amygdala, the [[prefrontal cortex]] is activated in response to a motivationally relevant event, particularly in response to the likelihood of a reward (Berns, McClure, Pagnoni & Montague, 2001). However, this structure governs the intensity of the response, or indeed if a response will occur at all (Bush et al., 2002). </br> ====Nucleus accumbens==== The [[nucleus accumbens]] is responsible for the attribution of learned associations between environment and events (Bassareo & Di Chiara, 1999), as well as governing a "liking" response to certain stimuli (Peciña & Berridge, 2005). </br> [[File:PET - Human Addiction.jpg|thumb|250px|left|PET scans of a addiction (top) and non-addiction (bottom)]] ==How to resolve addiction== Once the addiction has taken hold, the outlook may appear bleak. Not the case. Treatment options are always available, and with some self-control and regulation, breaking the addiction cycle is a distinct possibility. But which treatment will prove to be the most effective, and indeed will this treatment continue to be effective in the long term? With respect to practicality, the following treatment options are examined in-depth. ====Prevention==== As is the case with much of physical and psychological well-being, prevention is a very effective cure (McAlister, Perry, Killen, Slinkard & Maccoby, 1980). Of course, while it may be easy in theory to resist the temptation of an addiction, in reality it is a very difficult task. As such, in order to form a prevention strategy, the various factors which influence individuals towards the onset of an addiction must be examined. One publicly familiar cause of addiction is social pressure, and more specifically, peer pressure. It is very difficult to say "no" to a group of friends as they compel you to try a substance, particularly when some of these friends are highly respected by you or others (McAlister, 1979). As a result, a number of government programs have been initiated, with the express aim of educating individual (in particular children) on how to resist this type of social pressure, often with surprisingly good results (McAlister, Perry, Killen, Slinkard & Maccoby, 1980). This suggests that one method of prevention is to simply to learn how to say no, and refuse the addictive behaviour before it takes hold. ====Behaviour modification==== It has been suggested that conditioning principles are, to some extent, a large contributor in the addiction development process (Watkins, Koob & Markou, 2000). Fortunately, these conditioning principles have also been shown to work in the opposite direction, facilitating treatment and extinction of adverse addictive behaviours (Chóliz, 2010). A number of studies (Petry, 2006; Silverman, 2004) have suggested a specific operant conditioning schedule designed to reward the cessation of addictive behaviours. Essentially, this treatment proposes that the individual be rewarded, usually monetarily, when they present evidence of abstinence from the behaviour for a certain predetermined period of time. In this way, the individual is rewarded for not engaging in the drug behaviour, thereby facilitating the process of overcoming the addiction. In the past, this treatment has proved to be effective, across differing types of addictions (Higgins, Silverman, 1999). Indeed, this approach has even been shown to reduce usage of substances as addictive as cocaine, with one study finding over half of the participants remaining abstinent for almost the entire time-frame of the experiment (Silverman, 2004). Importantly, these studies found that the optimal level of behaviour modification occurred when the reward became greater overtime{{explain}}, and incentives such as "start-up bonuses" for initiating the abstinence behaviour had no additional benefit (Silverman, 2004; Silverman et al., 1998). The research on behaviour modification has a number of implications for those of us wishing to give up an addiction. Firstly, it is not necessary to use willpower alone. It is both acceptable and ideal to reward ourselves for our own good behaviour. If a day without a cigarette is managed, reward yourself with a nice dinner. If you get through a whole week without using Facebook, buy yourself something you've wanted for a while. But in using this strategy, it is important to remember that the rewards must be consistent (Silverman, 2004), so don't forget to treat yourself if you achieve your goal. ====Aversion therapy==== Given the conditioning principles at play during the formation of an addiction, one factor is clear. In most cases, the immediate response to performing an addictive behaviour is positive reinforcement (Chóliz, 2010). That is, the individual immediately feels good about taking a substance or performing and adverse behaviour, even if that same individual feels guilt or regret moments later. The power of the immediacy of the reinforcement has been shown the play a large role in facilitating the addiction (Chóliz, 2010). As such, the fundamental goal of aversion therapy is to pair aversive and unpleasant stimuli with the desired stimuli, in order to replace this immediate reinforcement with punishment (Cannon, Baker, Gino & Nathan, 1986). An effective treatment, aversion therapy has been shown to reduce cravings for substances as addictive as cocaine (Bordnick, Elkins, Orr, Walters & Thyer, 2004), and is often used in the treatment of alcohol (Howard, 2001). Generally, aversion therapy would need to be administered by a registered and trained health professional, due to the involvement of drugs and other potentially hazardous techniques (Cannon & Baker 1981). However, simplified versions are available for everyday use. For example, specific chemicals designed to be painted on fingernails can help to remove nail biting behaviours as a result of their repulsive taste. In any case, as long as the stimulus is undesirable, and presented simultaneously with the desired stimulus, a negative association can occur, leading to a reduction in craving responses to the addictive stimulus. ===Environment=== As human beings, we pick up and respond to various cues we have associated with particular stimuli, and this is especially true in the case of addictions (Kuntze et al., 2001). [[File:Bar-P1030319.jpg|thumb|250px|left|A setting, such a bar, can elicit various behavioural responses]] In this way, there are two possible resolution pathways for overcoming cue-elicited cravings: Desensitizing ourselves to these cues, or changing the environment. Desensitisation has proven to be an effective method of eliminating the cue-elicited behaviours (Kuntze et al., 2001). As the individual is repeatedly exposed to the cues associated with the stimulus, over time the effect of these cues is lessened as the association tends towards extinction (Kuntze et al., 2001). While this does not resolve the addiction entirely, the removal of elicited responses will contribute significantly towards controlling cravings. Secondly, if the environmental pressures are too much to handle, a change of social environment may be necessary. Given the role of cues in various settings (Kuntze et al., 2001), and social pressure (McAlister, 1979), removing the source of both of these will help significantly in overcoming an addiction. This may not be easy, as it would mean that contact with friends needs to be broken, but in the long-term it may prove to be a highly effective method. ==Summary== The processes of acquisition and extinction of an addiction are relatively straightforward in theory, even if they are more difficult to implement or maintain in reality. The origins of addiction have been explored, shedding some light on how we acquire the motivation to pursue an addictive behaviour or substance, rather than focusing our attention on more positive and healthy activities. Social learning theory was examined, and the process of peer-influence and modelling showed how adverse behaviours can be acquired if modelled by the correct individual (Borsari & Carey, 2006). Secondly, insight into the underlying learning principles suggested some processes by which addiction arises, notably the immediate positive reinforcement of the sensation (Chóliz, 2010), and the punishment of abstinence and withdrawal (Young, 1999). The opponent-process theory showed how these processes can be magnified, and the effect of the learning processes are increased as a result (Ettenberg, 2004). The effect of environmental cues was also shown to foster the addiction process (Kuntze et al., 2001). To complete the examination of addiction origins, a brief overview of brain structures was given, with sections of note being the basal ganglia, amygdala, prefrontal cortex and nucleus accumbens. Once the process of acquisition was examined, the methods of breaking an addiction were also considered. Prevention was one method, which can be achieved by learning how to resist social pressure (McAlister, Perry, Killen, Slinkard & Maccoby, 1980). Secondly, some techniques of behaviour modification could be of use, such as rewarding oneself for more desired behaviour patterns (Silverman, 2004). Aversion therapy can also assist, as pairing the cues normally associated with addictive stimuli, with aversive stimuli can lead to a reduction in cravings overall (Cannon, Baker, Gino & Nathan, 1986). Finally, a change in environment, or desensitization to associated cues (Kuntze et al., 2001) might prove to be useful in controlling and breaking the addiction. It is clear that while the process of addiction can be easily acquired, it is not impossible to break once it has been established, especially if appropriate measures are taken. In this way, as long as we utilize the treatment process that is right for us, anywhere from the most serious addiction to the smallest habit can be kept under control. ==See also== *[[w:Addiction|Addiction]] *[[Motivation_and_emotion/Book/Alcohol|Alcohol]] *[[w:Behavioral_addiction|Behavioral addiction]] *[[Motivation_and_emotion/Book/Gambling|Gambling]] *[[w:Substance_dependence|Substance dependence]] ==References== {{Hanging indent| American Psychiatric Association. (2000). ''Diagnostic and statistical manual of mental disorders'' (Revised 4th ed.). Washington, DC: Author. Baker, T. B., Piper, M. E., McCarthy, D. E., Majeskie, M. R., & Fiore, M. C. (2004). Addiction motivation reformulated: An affective processing model of negative reinforcement. ''Psychological Review'', ''111'', 33-51. doi:10.1037/0033-295X.111.1.33 Bandura, A., Ross, D., & Ross, S. A. (1961). Transmission of aggression through imitation of aggressive models. ''Journal of Abnormal and Social Psychology'', ''63'', 575–582. Bassareo, V., & Di Chiara, G. (1999). Differential responsiveness of dopamine transmission to food-stimuli in nucleus accumbens shell/core compartments. ''Neuroscience'', ''89'', 637-641. Berns, G. S., McClure, S. M., Pagnoni, G., & Montague, P. (2001). Predictability modulates human brain response to reward. ''The Journal of Neuroscience'', ''21'', 2793-2798. Berridge, K., & Robinson, T. (1998). What is the role of dopamine in reward: hedonic impact, reward learning, or incentive salience?. ''Brain Research. Brain Research Reviews'', ''28'', 309-369. Bordnick, P. S., Elkins, R. L., Orr, T., Walters, P., & Thyer, B. A. (2004). Evaluating the relative effectiveness of three aversion therapies designed to reduce craving among cocaine abusers. ''Behavioral Interventions'', ''19'', 1-24. doi:10.1002/bin.146 Borsari, B., & Carey, K. (2006). How the quality of peer relationships influences college alcohol use. ''Drug & Alcohol Review'', 25(4), 361-370. Bush, G., Vogt, B., Holmes, J., Dale, A., Greve, D., Jenike, M., & Rosen, B. (2002). Dorsal anterior cingulate cortex: a role in reward-based decision making. ''Proceedings of The National Academy of Sciences of The United States of America'', ''99'', 523-528. Cannon, D. S., & Baker, T. B. (1981). Emetic and electric shock alcohol aversion therapy: Assessment of conditioning. ''Journal of Consulting And Clinical Psychology'', ''49'', 20-33. doi:10.1037/0022-006X.49.1.20 Cannon, D. S., Baker, T. B., Gino, A., & Nathan, P. E. (1986). Alcohol-aversion therapy: Relation between strength of aversion and abstinence. ''Journal of Consulting And Clinical Psychology'', ''54'', 825-830. doi:10.1037/0022-006X.54.6.825 Chóliz, M. (2010). Experimental analysis of the game in pathological gamblers: Effect of the immediacy of the reward in slot machines. ''Journal of Gambling Studies'', ''26'', 249-256. doi:10.1007/s10899-009-9156-6 Cunningham-Williams, R., Gattis, M., Dore, P., Shi, P., & Spitznagel, E. (2009). Towards DSM-V: considering other withdrawal-like symptoms of pathological gambling disorder. ''International Journal of Methods In Psychiatric Research'', ''18'', 13-22.</br> Custer R.L. (1982) An overview of compulsive gambling. In: ''Addictive Disorders Update: Alcoholism, Drug Abuse, Gambling'' (eds Carone P.A., Yoles S.F., Kiefer S.N., Krinsky L.), p. 120, Human Sciences Press.</br> Ettenberg, A. (2004). Opponent process properties of self-administered cocaine. ''Neuroscience And Biobehavioral Reviews'', ''27'', 721-728. doi:10.1016/j.neubiorev.2003.11.009 Everitt, B., Cardinal, R., Parkinson, J., & Robbins, T. (2003). Appetitive behavior: impact of amygdala-dependent mechanisms of emotional learning. ''Annals of The New York Academy of Sciences'', 985233-250. Higgins, S. T, & Silverman, K. (Eds.). (1999). ''Motivating behavior change among ilicit drug abusers: Research on contingency management interventions.'' Washington, DC: American Psychological Association. Howard, M. (2001). Pharmacological aversion treatment of alcohol dependence. I. Production and prediction of conditioned alcohol aversion. ''The American Journal of Drug And Alcohol Abuse'', ''27'', 561-585. Kalivas, P. W., & Volkow, N. D. (2005). The neural basis of addiction: A pathology of motivation and choice. ''The American Journal of Psychiatry'', ''162'', 1403-1413. doi:10.1176/appi.ajp.162.8.1403 Kuntze, M. F., Stoermer, R., Mager, R., Roessler, A., Mueller-Spahn, F., & Bullinger, A. H. (2001). Immersive virtual environments in cue exposure. ''Cyberpsychology & Behavior'', ''4'', 497-501. McAlister, A. (1979). Adolescent smoking: Onset and prevention. ''Pediatrics'', ''63'', 650. McAlister, A., Perry, C., Killen, J., Slinkard, L., & Maccoby, N. (1980). Pilot study of smoking, alcohol and drug abuse prevention. ''American Journal of Public Health'', ''70'', 719. Peciña, S., & Berridge, K. (2005). Hedonic hot spot in nucleus accumbens shell: where do mu-opioids cause increased hedonic impact of sweetness?. ''The Journal of Neuroscience'', ''25'', 11777-11786. Petry, N. M. (2006). Contingency management treatments. ''British Journal of Psychiatry'', ''189'', 97-98. doi:10.1192/bjp.bp.106.022293 Rosenthal, R.J., Lesievr, H.R. (1992). Self reported withdrawal symptoms and pathological gambling. ''American Journal of Addictions'', ''1'', 150–154. doi: 10.1111/j.1521-0391.1992.tb00020.x</br> Silverman, K. (2004). Exploring the limits and utility of operant conditioning in the treatment of drug addiction. ''The Behavior Analyst'', ''27'', 209-230. Silverman, K., Wong, C. J., Umbricht-Schneiter, A., Montoya, I. D., Schuster, C. R., & Preston, K. L. (1998). Broad beneficial effects of reinforcement for cocaine abstinence among methadone patients. ''Journal of Consulting and Clinical Psychology'', ''66'', 811-824. Solomon, R. L., & Corbit, J. D. (1973). An opponent-process theory of motivation: II. Cigarette addiction. ''Journal Of Abnormal Psychology'', ''81'', 158-171. doi:10.1037/h0034534 Tiffany, S. T. (1990). A cognitive model of drug urges and drug-use behavior: Role of automatic and nonautomatic processes. ''Psychological Review'', ''97'', 147-168. doi:10.1037/0033-295X.97.2.147 Watkins, S. S., Koob, G. F., & Markou, A. (2000). Neural mechanisms underlying nicotine addiction: acute positive reinforcement and withdrawal. ''Nicotine & Tobacco Research'', ''2'', 19-37. doi:10.1080/14622200050011277 </br> Wikler, A. (1980). ''Opioid dependence: Mechanisms and treatment.'' New York: Plenum.</br> Wilson, G. (1987). Cognitive Processes in Addiction. ''British Journal of Addiction'', ''82'', 343-353. Young, K. S. (1999). Internet addiction: Symptoms, evaluation, and treatment. In L. VandeCreek & T. L. Jackson (Eds.), ''Innovations in Clinical Practice'', ''17'', Sarasota, FL: Professional Resource Press. }} [[Category:Motivation and emotion/Book/Addiction]] fdvuwt1p4d4leyj19nro7040th7lt6q 0 139384 2410783 2410400 2022-08-01T13:26:58Z Young1lim 21186 /* Adder */ wikitext text/x-wiki {{nocat}} == Adder == * Binary Adder Architecture Exploration ( [[Media:adder.20131113.pdf |pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20211108.pdf |pdf]] || || [[Media:adder.rca.20140313.pdf |pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf |pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.1.A.CLA.20211106.pdf |pdf]] || || [[Media:adder.cla.20140313.pdf |pdf]] || |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf |pdf]] || || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf |pdf]] || || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20211111.pdf |pdf]] || || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf |pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf |pdf]] || || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf |pdf]], [[Media:adder.cca.20140313.pdf |pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf |pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf |pdf]] || || [[Media:adder.ksa.20140409.pdf |pdf]] || |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf |pdf]] || || || |- || '''9. Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA.20220801.pdf |pdf]] || || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf |pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf |pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf |pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf |pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:bshift.20131105.pdf |bshfit.vhdl]], [[Media:bshift.makefile.20131109.pdf |bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf |pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf |pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf |pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf |pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf |pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf |Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf |H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf |pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Computer architecture]] 2vxmlsyroy2wlppc1vd6d4eklz3izj3 0 150646 2410932 2369640 2022-08-02T06:35:40Z Marshallsumter 311529 /* Interferograms */ wikitext text/x-wiki [[Image:2012 LZ1.jpg|thumb|right|250px|This image is of asteroid 2012 LZ1 by the Arecibo Observatory in Puerto Rico using the Arecibo Planetary Radar. Credit: Arecibo Observatory.{{tlx|fairuse}}]] '''Radar astronomy''' is used to detect and study astronomical objects that reflect radio rays. The image at right is of asteroid 2012 LZ1 using the Arecibo Planetary Radar. '''Radar astronomy''' is a technique of observing nearby astronomical objects by reflecting microwaves off target objects and analyzing the echoes. Radar astronomy differs from [[Radiation astronomy/Radios|radio astronomy]] in that the latter is a passive observation and the former an active one. The radar transmission may either be pulsed or continuous. Radar directly measures the distance to the object (and how fast it is changing). The combination of optical and radar observations normally allows the prediction of orbits at least decades, and sometimes centuries, into the future. The maximum range of astronomy by radar is very limited, and is confined to the [[Solar System|solar system]]. This is because the signal strength drops off very steeply with distance to the target, the small fraction of incident flux that is reflected by the target, and the limited strength of transmitters.<ref name="Hey">{{ cite book | author = J.S. Hey | title = The Evolution of Radio Astronomy | publisher = Paul Elek (Scientific Books) | series = Histories of Science Series | volume = 1 | date = 1973 | location = | pages = | url = | doi = | isbn = }}</ref> It is also necessary to have a relatively good ephemeris of the target before observing it. {{clear}} ==Radars== '''Radio waves''' are a type of electromagnetic radiation with wavelengths in the electromagnetic spectrum longer than infrared light. Radio waves have frequencies from 300 [Gigahertz] GHz to as low as 3 [Kilohertz] kHz, and corresponding [to] wavelengths from 1 millimeter to 100 kilometers. {| class="wikitable" |+ '''Radar frequency bands''' |- style="background:#ccc;" !Band name!!Frequency range!!Wavelength range!!Notes |- |[[w:High frequency|HF]]||3–30 [[w:Megahertz|MHz]]||10–100 [[w:metre|m]]||Coastal radar systems, [[w:over-the-horizon radar|over-the-horizon radar]] (OTH) radars; 'high frequency' |- |[[w:VHF|VHF]]||30–300&nbsp;MHz||1–10 m||Very long range, ground penetrating; 'very high frequency' |- |P||<&nbsp;300&nbsp;MHz||>&nbsp;1&nbsp;m||'P' for 'previous', applied retrospectively to early radar systems; essentially HF + VHF |- |[[w:UHF|UHF]]||300–1000&nbsp;MHz||0.3–1 m||Very long range (e.g. [[w:Ballistic Missile Early Warning System|ballistic missile early warning]]), ground penetrating, foliage penetrating; 'ultra high frequency' |- |[[w:L band|L]]||1–2 [[w:Gigahertz|GHz]]||15–30 [[w:centimetre|cm]]||Long range air traffic control and [[w:Surveillance|surveillance]]; 'L' for 'long' |- |[[w:S band|S]]||2–4&nbsp;GHz||7.5–15&nbsp;cm||Moderate range surveillance, Terminal air traffic control, long-range weather, marine radar; 'S' for 'short' |- |[[w:C band|C]]||4–8&nbsp;GHz||3.75–7.5&nbsp;cm||Satellite transponders; a compromise (hence 'C') between X and S bands; weather; long range tracking |- |[[w:X band|X]]||8–12&nbsp;GHz||2.5–3.75&nbsp;cm||[[w:Missile|Missile]] guidance, [[w:marine radar|marine radar]], weather, medium-resolution mapping and ground surveillance; in the USA the narrow range 10.525&nbsp;GHz ±25&nbsp;MHz is used for airport radar; short range tracking. Named X band because the frequency was a secret during WW2. |- ||[[w:Ku band|K_u]]||12–18&nbsp;GHz||1.67–2.5&nbsp;cm||High-resolution, also used for satellite transponders, frequency under K band (hence 'u') |- |[[w:K band|K]]||18–24&nbsp;GHz||1.11–1.67&nbsp;cm||From [[w:German language|German]] ''kurz'', meaning 'short'; limited use due to absorption by [[w:water vapor|water vapour]], so K_u and K_a were used instead for surveillance. K-band is used for detecting clouds by meteorologists, and by police for detecting speeding motorists. K-band radar guns operate at 24.150 ± 0.100&nbsp;GHz. |- |[[w:Ka band|K_a]]||24–40&nbsp;GHz||0.75–1.11&nbsp;cm||Mapping, short range, airport surveillance; frequency just above K band (hence 'a') Photo radar, used to trigger cameras which take pictures of license plates of cars running red lights, operates at 34.300 ± 0.100&nbsp;GHz. |- |mm||40–300&nbsp;GHz||1.0–7.5&nbsp;[[w:millimetre|mm]] ||[[w:Millimetre band|Millimetre band]], subdivided as below. The frequency ranges depend on waveguide size. Multiple letters are assigned to these bands by different groups. These are from Baytron, a now defunct company that made test equipment. |- |[[w:V band|V]]||40–75&nbsp;GHz||4.0–7.5&nbsp;mm || Very strongly absorbed by atmospheric oxygen, which resonates at 60&nbsp;GHz. |- |[[w:W band|W]]||75–110&nbsp;GHz||2.7–4.0&nbsp;mm||Used as a visual sensor for experimental autonomous vehicles, high-resolution meteorological observation, and imaging. |} ==Planetary sciences== {{main|Planetary sciences}} [[Image:Topography of Mars and radar section.jpg|thumb|right|250px|A: Topography of Mars has major features identified. B: Topography shows the MRO/SHARAD ground tracks for orbits 6830 (a-a′), 7219 (b-b′), and 3672 (c-c′). LDAs crossed by these tracks are labeled. Credit: John W. Holt, Ali Safaeinili, Jeffrey J. Plaut, James W. Head, Roger J. Phillips, Roberto Seu, Scott D. Kempf, Prateek Choudhary, Duncan A. Young, Nathaniel E. Putzig, Daniela Biccari, Yonggyu Gim.]] [[Image:F3.large.jpg|thumb|left|250px|Results are for SHARAD orbit 6830 (line a-a′). A: Simulated surface echoes (clutter) are in one-way travel time. B: SHARAD data is in one-way travel time. Vertical arrows identify echoes not consistent with surface clutter simulation and also confirmed in adjacent tracks. C: Radar data is converted to depth assuming a water-ice composition. Credit: John W. Holt, Ali Safaeinili, Jeffrey J. Plaut, James W. Head, Roger J. Phillips, Roberto Seu, Scott D. Kempf, Prateek Choudhary, Duncan A. Young, Nathaniel E. Putzig, Daniela Biccari, Yonggyu Gim.]] [[Image:SHARAD orbits 7219 and 3672.jpg|thumb|right|250px|Results are for SHARAD orbits 7219 (left) and 3672 (right). Credit: John W. Holt, Ali Safaeinili, Jeffrey J. Plaut, James W. Head, Roger J. Phillips, Roberto Seu, Scott D. Kempf, Prateek Choudhary, Duncan A. Young, Nathaniel E. Putzig, Daniela Biccari, Yonggyu Gim.]] "Radar waves penetrate the surface and pass through materials that do not severely attenuate or scatter them. Reflections arise from interfaces with dielectric contrasts. [Shallow radar] SHARAD has penetrated the ∼2-km-thick polar layered deposits in both the north and south, detecting many internal reflectors (17, 18). Smaller targets can be more challenging because SHARAD's antenna pattern is broad, resulting in surface reflections up to a few tens of kilometers away from the suborbital point in rugged areas, versus only a few kilometers in smooth, flat areas. These off-nadir echoes can appear at time delays similar to those arising from subsurface interfaces, so steps are required to avoid misinterpreting this surface clutter as subsurface echoes. Synthetic-aperture data processing is used to improve along-track resolution to ∼300 m, greatly reducing along-track clutter and focusing the surface and subsurface features. We used the known topography of the surface and the radar geometry to model cross-track clutter together with nadir surface echoes [...]. Comparisons of radar sounding data with these synthetic surface echoes and the examination of possible surface echo sources in imagery (19) were undertaken for all cases [...]; such a procedure is a necessary part of radar sounding data interpretation in high-relief environments."<ref name=Holt>{{ cite journal |author=John W. Holt, Ali Safaeinili, Jeffrey J. Plaut, James W. Head, Roger J. Phillips, Roberto Seu, Scott D. Kempf, Prateek Choudhary, Duncan A. Young, Nathaniel E. Putzig, Daniela Biccari, Yonggyu Gim |title=Radar Sounding Evidence for Buried Glaciers in the Southern Mid-Latitudes of Mars |journal=Science |month=November |year=2008 |volume=322 |issue=5905 |pages=1235-8 |url=http://www.sciencemag.org/content/322/5905/1235.full |arxiv= |bibcode= |doi=10.1126/science.1164246 |pmid= |accessdate=2014-09-28 }}</ref> The "Shallow Radar (SHARAD) (15) on the Mars Reconnaissance Orbiter (MRO) to probe the internal structure of several LDAs surrounding massifs on the eastern rim of the Hellas impact basin [first image at the right] where more than 90 LDA complexes flank steep topography (2, 6, 16). The southernmost LDA we studied (LDA-2, [figure at the upper right] has multiple lobes that coalesce to form a continuous deposit extending more than 20 km outward from a massif along ∼170 km of its margins."<ref name=Holt/> "Examination of radar data from SHARAD orbit 6830 where it crosses multiple [lobate debris aprons] LDAs in the eastern Hellas region [...] shows that the only radar reflections not matching simulated surface echoes occur where the spacecraft passes over each LDA [...]; therefore, these echoes are interpreted as arising from within or beneath the LDAs. In one case (LDA-2A), surface clutter is predicted near the terminus of the LDA, where it may obscure portions of a subsurface reflector that clearly extends farther inward below the LDA. LDA-2A and LDA-2B [image at the lower right] show evidence for multiple, closely spaced subsurface reflectors indicating the presence of at least one thin (∼70 m assuming a water-ice composition), distinct deposit below thicker deposits (up to 800 m)."<ref name=Holt/> {{clear}} ==Theoretical radar astronomy== Here's a couple of [[Definitions/Theory#Theoretical definition|theoretical definition]]s: '''Def.''' the branch of astronomy that uses radar to map the surfaces of planetary bodies in the solar system is called '''radar astronomy'''. '''Def.''' reflective and observational astronomy over radio wavelengths is called '''radar astronomy'''. ==Aerometeors== {{main|Radiation astronomy/Aerometeors}} [[Image:June 2012 Radar image of Derecho.jpg|thumb|right|250px|This is a composite radar image of the June 2012 United States Derecho event. Credit: G. Carbin, NWS/Storm Prediction Center.]] On the right is a composite of hourly radar images. These wind gusts averaged ~75 mph over about 450 miles. This is referred to as the Derecho event. {{clear}} ==Opticals== {{main|Radiation astronomy/Opticals|Optical astronomy}} By [[Radiation astronomy/Opticals|optical astronomy]], optical observations measure very accurately where an object appears on the sky, but cannot measure the distance accurately at all. ==Hydromorphology== [[Image:"Quill" satellite radar image of flooded Eel River outflow current.JPG|thumb|right|250px|''Quill'' satellite radar image is shown of the flooded Eel River outflow current. Credit: [[commons:User:Oldteched|oldteched]], U. S. National Reconnaissance Office, and U. S. Geological Survey.]] On the right, the ''Quill'' satellite radar image is shown of the flooded Eel River outflow current debris field in gray. "Colored content is USGS-derived base map. Gray overlay is derived from a "Quill" satellite radar image made during the December 1964 flooding of California's Eel River. Accurate registration of the overlay onto the map is demonstrated by the excellent match of the stream-valley features in each."<ref name=Oldteched>{{ cite book |author=[[commons:User:Oldteched|oldteched]] |title=''File:"Quill" satellite radar image of flooded Eel River outflow current.JPG'' |publisher=U. S. National Reconnaissance Office |location= |date=December 1964 |url=https://commons.wikimedia.org/wiki/File:%22Quill%22_satellite_radar_image_of_flooded_Eel_River_outflow_current.JPG |accessdate=2016-09-20 }}</ref> {{clear}} ==Ice streams== [[Image:FRicestreams.jpg|thumb|right|250px|Radarsat image is of ice streams flowing into the Filchner-Ronne Ice Shelf. Credit: [[w:User:Polargeo|Polargeo]].]] On the right is a radarsat image of ice streams flowing into the Filchner-Ronne ice shelf. Annotations outline the Rutford ice stream. {{clear}} ==Glaciers== {{main|Rocks/Glaciers|Glaciers}} [[Image:Alfred Ernest Ice Shelf.jpg|thumb|right|250px|This is a radar image of Alfred Ernest Ice Shelf on Ellesmere Island, taken by the ERS-1 satellite. Credit: NASA.]] On the right is a radar image of Alfred Ernest Ice Shelf on Ellesmere Island, taken by the ERS-1 satellite. {{clear}} ==Glaciology== {{main|Rocks/Glaciers/Glaciology|Glaciology}} [[Image:A-62 iceberg connected to Fimbul Ice Shelf at Queen Maud Land.jpg|thumb|right|250px|Iceberg A 62 was connected to the Fimbul Ice Shelf by a mere 800-metre-wide bridge. Credit: DLR - German Space Agency.]] "A small island obstructs the constant flow of the ice shelf on Queen Maud Land – it is the lighter area at the bottom left of the image [on the right]. From September 2010 until it broke off, Iceberg A 62 was connected to the Fimbul Ice Shelf by a mere 800-metre-wide bridge. Two fissures in the ice from different sides of the bridge approached one another until the break occurred. The images transmitted by the radar satellite TerraSAR-X over a long period of time should enable researchers to achieve a better understanding of how icebergs calve."<ref name=DLR>{{ cite book |author=DLR |title=The iceberg breaks free |publisher=Deutsches Zentrum für Luft- und Raumfahrt |location=Cologne |date=20 January 2011 |url=http://www.dlr.de/media/en/desktopdefault.aspx/tabid-4986/8423_read-18675/8423_page-2 |accessdate=2016-09-20 }}</ref> {{clear}} ==Craters== {{main|Radiation astronomy/Craters}} [[Image:Haughton impact crater radar image.jpg|thumb|right|250px|Synthetic aperture radar image is of the Haughton impact crater. Credit: [[w:User:Denni|Denni]].]] [[Image:Iturralde Crater PIA03359 cropped.jpg|thumb|left|250px|A topographic map from Shuttle Radar Topography Mission of Iturralde Crater. Credit: NASA/JPL.]] [[Image:Lost Hills Subsidence interferogram.jpg|thumb|right|250px|This figure shows a comparison of interferograms from four different years mapping the rapid ground subsidence over the Lost Hills oil field in California. Credit: NASA/JPL-Caltech.]] On the right is a synthetic aperture radar image of the Haughton impact crater on Devon Island, Nunavut, Canada. On the left is a topographic map from Shuttle Radar Topography Mission data of Iturralde Crater, an unconfirmed impact crater in Bolivia. "This figure shows a comparison of interferograms from four different years mapping the rapid ground subsidence over the Lost Hills oil field in California. Lost Hills is located about 60 km (40 miles) northwest of Bakersfield in the San Joaquin Valley. The oil field is about 1.5 km (1mile) wide and 6 km (3.5 miles) long."<ref name=Lavoie2001>{{ cite book |author=Sue Lavoie |title=PIA03458: Lost Hills, California Interferogram |publisher=NASA/JPL |location=Pasadena, California USA |date=17 October 2001 |url=http://photojournal.jpl.nasa.gov/catalog/PIA03458 |accessdate=2016-09-27 }}</ref> "Each interferogram was created using pairs of images taken by synthetic aperture radar that have been combined to measure surface deformation or changes that may have occurred in the time between when data for the two images were taken. The images were collected by the European Space Agency's Remote Sensing satellites (ERS-1 and ERS-2) in two months of each year shown (1995, 1996, 1998 and 1999) and were combined to produce these image maps of the apparent surface deformation, or changes."<ref name=Lavoie2001/> "The interferometric measurements that show the changes, primarily vertical subsidence of the surface, are rendered in color with purple indicating no motion and the brightest red showing rapid subsidence. The white areas are where the radar measurements could not be obtained, mostly in the agricultural fields around the oil fields where plant growth or plowing altered the radar properties of the surface."<ref name=Lavoie2001/> "These radar data show that parts of the oil field were subsiding unusually rapidly, more than 3 centimeters (1.2 inches) a month, in 1995 and 1996. They also reveal that while the ground subsidence rate decreased in the center part of the oil field, it increased in the northern part between 1995 and 1996 and 1998 and 1999."<ref name=Lavoie2001/> {{clear}} ==Active faults== [[Image:Los Angeles JPLLandsat.jpg|thumb|right|250px|This perspective image of Los Angeles and the Transverse Ranges is from NASA's Shuttle Radar Topography Mission (SRTM). Credit: NASA/JPL/NIMA.]] "Earthquake faults commonly lie between the mountains and the lowlands. The San Andreas fault, the largest fault in California, likewise divides the very rugged San Gabriel Mountains from the low-relief Mojave Desert, thus forming a straight topographic boundary between the top center and lower right corner of the image. We present two versions of this perspective image from NASA's Shuttle Radar Topography Mission (SRTM): one with and one without a graphic overlay that maps faults that have been active in Late Quaternary times (white lines). The fault database was provided by the U.S. Geological Survey."<ref name=Lavoie2002>{{ cite book |author=Sue Lavoie |title=PIA03376: Perspective View, SRTM / Landsat, Los Angeles, Calif |publisher=NASA/JPL |location=Pasadena, California USA |date=23 December 2002 |url=http://photojournal.jpl.nasa.gov/catalog/PIA03376 |accessdate=2016-09-27 }}</ref> "The Landsat image used here was acquired on May 4, 2001, about seven weeks before the summer solstice, so natural terrain shading is not particularly strong. It is also not especially apparent given a view direction (northwest) nearly parallel to the sun illumination (shadows generally fall on the backsides of mountains). Consequently, topographic shading derived from the SRTM elevation model was added to the Landsat image, with a false sun illumination from the left (southwest). This synthetic shading enhances the appearance of the topography."<ref name=Lavoie2002/> "Landsat has been providing visible and infrared views of the Earth since 1972. SRTM elevation data matches the 30-meter (98-foot) resolution of most Landsat images and substantially helps in analyzing the large and growing Landsat image archive. This Landsat 7 Thematic Mapper image was provided to the SRTM project by the United States Geological Survey, Earth Resources Observation Systems (EROS) Data Center, Sioux Falls, S.D."<ref name=Lavoie2002/> "Elevation data used in this image was acquired by the SRTM aboard the Space Shuttle Endeavour, launched on Feb. 11, 2000. SRTM used the same radar instrument that comprised the Spaceborne Imaging Radar-C/X-Band Synthetic Aperture Radar (SIR-C/X-SAR) that flew twice on the Space Shuttle Endeavour in 1994. SRTM was designed to collect 3-D measurements of the Earth's surface. To collect the 3-D data, engineers added a 60-meter (approximately 200-foot) mast, installed additional C-band and X-band antennas, and improved tracking and navigation devices. The mission is a cooperative project between NASA, the National Imagery and Mapping Agency (NIMA) of the U.S. Department of Defense and the German and Italian space agencies."<ref name=Lavoie2002/> "Size: View width 134 kilometers (83 miles); view distance 150 kilometers (93 miles) Location: 34.3 degrees North latitude, 118.4 degrees West longitude Orientation: View west-northwest, 1.8 X vertical exaggeration Image Data: Landsat Bands 3, 2+4, 1 as red, green, blue, respectively Original Data Resolution: SRTM 1 arcsecond (30 meters or 98 feet), Landsat 30 meters (98 feet) Graphic Data: earthquake faults active in Late Quaternary times Date Acquired: February 2000 (SRTM), May 4, 2001 (Landsat)."<ref name=Lavoie2002/> {{clear}} ==Transform faults== [[Image:Alpine Fault SRTM.jpg|thumb|center|400px|Shaded and colored image from the Shuttle Radar Topography Mission elevation model of New Zealand's Alpine Fault, Northwest is towards the top. Credit: NASA/JPL/NGA.]] "The Alpine fault runs parallel to, and just inland of, much of the west coast of New Zealand's South Island. This view was created from the near-global digital elevation model produced by the Shuttle Radar Topography Mission (SRTM) and is almost 500 kilometers (just over 300 miles) wide. Northwest is toward the top. The fault is extremely distinct in the topographic pattern, nearly slicing this scene in half lengthwise."<ref name=NASAJPLNGA>{{ cite book |author=NASA/JPL/NGA |title=PIA06661: Alpine Fault, New Zealand, SRTM Shaded Relief and Colored Height |publisher=NASA/JPL/NGA |location=Pasadena, California USA |date=6 January 2005 |url=http://photojournal.jpl.nasa.gov/catalog/PIA06661 |accessdate=2014-12-02 }}</ref> "In a regional context, the Alpine fault is part of a system of faults that connects a west dipping subduction zone to the northeast with an east dipping subduction zone to the southwest, both of which occur along the juncture of the Indo-Australian and Pacific tectonic plates. Thus, the fault itself constitutes the major surface manifestation of the plate boundary here. Offsets of streams and ridges evident in the field, and in this view of SRTM data, indicate right-lateral fault motion. But convergence also occurs across the fault, and this causes the continued uplift of the Southern Alps, New Zealand's largest mountain range, along the southeast side of the fault."<ref name=NASAJPLNGA/> "Two visualization methods were combined to produce this image: shading and color coding of topographic height. The shade image was derived by computing topographic slope in the northwest-southeast (image top to bottom) direction, so that northwest slopes appear bright and southeast slopes appear dark. Color coding is directly related to topographic height, with green at the lower elevations, rising through yellow and tan, to white at the highest elevations."<ref name=NASAJPLNGA/> "Elevation data used in this image were acquired by the Shuttle Radar Topography Mission aboard the Space Shuttle Endeavour, launched on Feb. 11, 2000. SRTM used the same radar instrument that comprised the Spaceborne Imaging Radar-C/X-Band Synthetic Aperture Radar (SIR-C/X-SAR) that flew twice on the Space Shuttle Endeavour in 1994. SRTM was designed to collect 3-D measurements of the Earth's surface. To collect the 3-D data, engineers added a 60-meter (approximately 200-foot) mast, installed additional C-band and X-band antennas, and improved tracking and navigation devices."<ref name=NASAJPLNGA/> The Alpine fault is "495 kilometers (307 miles) by 162 kilometers (100 miles). Location: 43.2 degrees South latitude, 170.5 degrees East longitude. Orientation: Northwest toward the top".<ref name=NASAJPLNGA/> {{clear}} ==Volcanoes== [[Image:Ambae island 3D pic.jpg|thumb|right|250px|Ambae island, Vanuatu, image was acquired by the Shuttle Radar Topography Mission on 2000-02-11 aboard the shuttle Endeavour. Credit: NASA/JPL/NGA.]] [[Image:Klyuchevskaya sopka rad.jpg|thumb|left|250px|This image was taken using space radar of Kliuchevskoi Volcano, Russia. Credit: NASA/JPL-Caltech.]] "The recently active volcano Mt. Manaro is the dominant feature in this shaded relief image [on the right] of Ambae Island, part of the Vanuatu archipelago located 1400 miles northeast of Sydney, Australia. About 5000 inhabitants, half the island's population, were evacuated in early December from the path of a possible lahar, or mud flow, when the volcano started spewing clouds of steam and toxic gases 10,000 feet into the atmosphere."<ref name=Lavoie2000>{{ cite web |author=Sue Lavoie |title=PIA06675: Ambae Island, Vanuatu (South Pacific) |publisher=NASA/JPL |location=Pasadena, California USA |date=February 2000 |url=http://photojournal.jpl.nasa.gov/catalog/PIA06675 |accessdate=2016-09-20 }}</ref> "Last active in 1996, the 1496 meter (4908 ft.) high Hawaiian-style basaltic shield volcano features two lakes within its summit caldera, or crater. The ash and gas plume is actually emerging from a vent at the center of Lake Voui (at left), which was formed approximately 425 years ago after an explosive eruption."<ref name=Lavoie2000/> "Two visualization methods were combined to produce the image: shading and color coding of topographic height. The shade image was derived by computing topographic slope in the northwest-southeast direction, so that northwest slopes appear bright and southeast slopes appear dark. Color coding is directly related to topographic height, with green at the lower elevations, rising through yellow and tan, to white at the highest elevations."<ref name=Lavoie2000/> "Elevation data used in this image were acquired by the Shuttle Radar Topography Mission [SRTM] aboard the Space Shuttle Endeavour, launched on Feb. 11, 2000. SRTM used the same radar instrument that comprised the Spaceborne Imaging Radar-C/X-Band Synthetic Aperture Radar (SIR-C/X-SAR) that flew twice on the Space Shuttle Endeavour in 1994. SRTM was designed to collect 3-D measurements of the Earth's surface. To collect the 3-D data, engineers added a 60-meter (approximately 200-foot) mast, installed additional C-band and X-band antennas, and improved tracking and navigation devices."<ref name=Lavoie2000/> "Location: 15.4 degree south latitude, 167.9 degrees east longitude; Orientation: North toward the top, Mercator projection; Size: 36.8 by 27.8 kilometers (22.9 by 17.3 miles); Image Data [is a] shaded and colored SRTM elevation model"<ref name=Lavoie2000/> On the left is a space radar image of Klyuchevskaya sopka. "This is an image of the area of the Kliuchevskoi volcano, Kamchatka, Russia, which began to erupt on September 30, 1994. Kliuchevskoi is the blue triangular peak in the center of the image, towards the left edge of the bright red area that delineates bare snow cover. The image was acquired by the Spaceborne Imaging Radar-C/X-Band Synthetic Aperture Radar (SIR-C/X-SAR) aboard the space shuttle Endeavour on its 88th orbit on October 5, 1994. The image shows an area approximately 75 kilometers by 100 kilometers (46 miles by 62 miles) that is centered at 56.07 degrees north latitude and 160.84 degrees east longitude. North is toward the bottom of the image. The radar illumination is from the top of the image. The Kamchatka volcanoes are among the most active volcanoes in the world. The volcanic zone sits above a tectonic plate boundary, where the Pacific plate is sinking beneath the northeast edge of the Eurasian plate. The Endeavour crew obtained dramatic video and photographic images of this region during the eruption [...]. The colors in this image were obtained using the following radar channels: red represents the L-band, HH (horizontally transmitted and received) channel; green represents the L-band, HV (horizontally transmitted and vertically received) channel; blue represents the C-band, HV (horizontally transmitted and vertically received) channel. In addition to Kliuchevskoi, two other active volcanoes are visible in the image. Bezymianny, the circular crater above and to the right of Kliuchevskoi, contains a slowly growing lava dome. Tolbachik is the large volcano with a dark summit crater near the upper right edge of the red snow covered area. The Kamchatka River runs from right to left across the bottom of the image. The 1994 eruption of Kliuchevskoi included massive ejections of gas, vapor and ash, which reached altitudes of 15,000 meters (50,000 feet). Melting snow mixed with volcanic ash triggered mud flows on the flanks of the volcano. Paths of these flows can be seen as thin lines in various shades of blue and green on the north flank in the center of the image."<ref name=Richardson>{{ cite web |author=Annie Richardson and Al Wong |title=Photograph Identification: P-44823 |publisher=NASA/JPL |location=Pasadena, California USA |date=5 October 1994 |url=http://southport.jpl.nasa.gov/cdrom/sirced03/cdrom/DATA/LOCATION/ASIAMIDE/KLIUCHEV/KLIUCHE2.HTM |accessdate=2016-09-27 }}</ref> {{clear}} ==Mercury== {{main|Rocks/Rocky objects/Mercury}} [[Image:Permanently Shadowed Polar Craters.jpg|thumb|right|250px|This is a radar image of Mercury's north polar region acquired by the Arecibo Observatory in 1991 combined with images from MESSENGER. Credit: NASA/Johns Hopkins University Applied Physics Laboratory/Carnegie Institution of Washington/National Astronomy and Ionosphere Center, Arecibo Observatory.]] [[Image:Merc fig2sm.jpg|right|thumb|250px|This is a radar image of Mercury's north pole. Credit: NASA.]] Radar astronomy of Mercury improved the value for the distance from the earth including the rotational period, libration,[and surface mapping, especially of the polar regions. In "1991, [...] the Arecibo radio telescope in Puerto Rico detected unusually radar-bright patches at Mercury's poles, spots that reflected radio waves in the way one would expect if there were water ice [denoted in the image at right by yellow areas]. Many of these patches corresponded to the location of large impact craters mapped by the Mariner 10 spacecraft in the 1970s."<ref name=Dunbar>{{ cite book |author=Brian Dunbar |title=MESSENGER Finds New Evidence for Water Ice at Mercury's Poles |publisher=NASA |location=Washington, DC USA |date=November 29, 2012 |url=http://www.nasa.gov/mission_pages/messenger/media/PressConf20121129.html |accessdate=2013-10-24 }}</ref> "Images from the spacecraft's Mercury Dual Imaging System taken in 2011 and earlier this year confirmed that radar-bright features at Mercury's north and south poles are within shadowed regions on Mercury's surface, findings that are consistent with the water-ice hypothesis."<ref name=Dunbar/> "The neutron data indicate that Mercury's radar-bright polar deposits contain, on average, a hydrogen-rich layer more than tens of centimeters thick beneath a surficial layer 10 to 20 centimeters thick that is less rich in hydrogen".<ref name=Lawrence>{{ cite book |author=David Lawrence |title=MESSENGER Finds New Evidence for Water Ice at Mercury's Poles |publisher=NASA |location=Washington, DC USA |date=November 29, 2012 |url=http://www.nasa.gov/mission_pages/messenger/media/PressConf20121129.html |accessdate=2013-10-24 }}</ref> "The buried layer has a hydrogen content consistent with nearly pure water ice."<ref name=Lawrence/> "These reflectance anomalies are concentrated on poleward-facing slopes and are spatially collocated with areas of high radar backscatter postulated to be the result of near-surface water ice".<ref name=Neumann>{{ cite book |author=Gregory Neumann |title=MESSENGER Finds New Evidence for Water Ice at Mercury's Poles |publisher=NASA |location=Washington, DC USA |date=November 29, 2012 |url=http://www.nasa.gov/mission_pages/messenger/media/PressConf20121129.html |accessdate=2013-10-24 }}</ref> "Correlation of observed reflectance with modeled temperatures indicates that the optically bright regions are consistent with surface water ice."<ref name=Neumann/> MESSENGER's Mercury Laser Altimeter (MLA) data "show that the spatial distribution of regions of high radar backscatter is well matched by the predicted distribution of thermally stable water ice".<ref name=Paige>{{ cite book |author=David Paige |title=MESSENGER Finds New Evidence for Water Ice at Mercury's Poles |publisher=NASA |location=Washington, DC USA |date=November 29, 2012 |url=http://www.nasa.gov/mission_pages/messenger/media/PressConf20121129.html |accessdate=2013-10-24 }}</ref> Water ice strongly reflects radar, and observations by the 70 m [Goldstone Deep Space Communications Complex] Goldstone telescope and the [Very Large Array] VLA in the early 1990s revealed that there are patches of very high radar reflection near the poles.<ref name=Slade>{{ cite journal | author=M. A. Slade, B. J. Butler, D. O. Muhleman | year=1992 | title=Mercury radar imaging – Evidence for polar ice | journal=Science | volume=258 | issue=5082 | pages=635–640 | doi=10.1126/science.258.5082.635 | pmid=17748898 |bibcode = 1992Sci...258..635S }}</ref> While ice is not the only possible cause of these reflective regions, astronomers believe it is the most likely.<ref name=Williams>{{ cite book | author=David R. Williams | date=June 2, 2005 | url=http://nssdc.gsfc.nasa.gov/planetary/ice/ice_mercury.html | title=Ice on Mercury | publisher=NASA Goddard Space Flight Center | accessdate=2008-05-23 }}</ref> {{clear}} ==Venus== [[Image:Venus2 mag big.png|thumb|right|250px|Using an imaging radar technique, the Magellan spacecraft was able to lift the veil from the face of Venus and produce this spectacular high resolution image of the planet's surface. Red, in this false-color map, represents mountains, while blue represents valleys. Credit: Magellan Team, JPL, NASA.]] [[Image:Venus globe.jpg|thumb|left|250px|This is a false color image of Venus produced from a global radar view of the surface by the Magellan probe while radar imaging between 1990-1994. Credit: NASA.]] The first un-ambiguous detection of Venus was made by the Jet Propulsion Laboratory (JPL) on 10 March 1961. A correct measurement of the AU soon followed. "The advantages of radar in planetary astronomy result from (1) the observer's control of all the attributes of the coherent signal used to illuminate the target, especially the wave form's time/frequency modulation and polarization; (2) the ability of radar to resolve objects spatially via measurements of the distribution of echo power in time delay and Doppler frequency; (3) the pronounced degree to which delay-Doppler measurements constrain orbits and spin vectors; and (4) centimeter-to-meter wavelengths, which easily penetrate optically opaque planetary clouds and cometary comae, permit investigation of near-surface macrostructure and bulk density, and are sensitive to high concentrations of metal or, in certain situations, ice."<ref name=Ostro>{{ cite journal |author=Steven J. Ostro |title=Planetary radar astronomy |journal=Reviews of Modern Physics |month=October-December |year=1993 |volume=65 |issue=4 |pages=1235-79 |url=http://rmp.aps.org/abstract/RMP/v65/i4/p1235_1 |arxiv= |bibcode= |doi=10.1103/RevModPhys.65.1235 |pmid= |accessdate=2012-02-09 }}</ref> The radar image at left shows that just beneath the cloud layers is a rocky object. {{clear}} ==Earth== {{main|Rocks/Rocky objects/Earth}} [[Image:Aorounga crater.png|thumb|right|250px|The concentric ring structure is the Aorounga impact crater. Credit: NASA.]] [[Image:Antarctica2.jpg|thumb|right|250px|This is a shaded relief map of Antarctica developed from RADARSAT Synthetic Aperture Radar data. Credit: U.S. National Oceanic & Atmospheric Administration.]] Numerous airborne and spacecraft radars have mapped the entire planet, for various purposes. One example is the Shuttle Radar Topography Mission, which mapped the entire Earth at 30 m resolution. "The impact of an asteroid or comet several hundred million years ago left scars in the landscape that are still visible in this spaceborne radar image [on the right] of an area in the Sahara Desert of northern Chad. The concentric ring structure is the Aorounga impact crater, with a diameter of about 17 kilometers (10.5 miles). The original crater was buried by sediments, which were then partially eroded to reveal the current ring-like appearance. The dark streaks are deposits of windblown sand that migrate along valleys cut by thousands of years of wind erosion. The dark band in the upper right of the image is a portion of a proposed second crater."<ref name=Wong/> "Radar imaging is a valuable tool for the study of desert regions because the radar waves can penetrate thin layers of dry sand to reveal details of geologic structure that are invisible to other sensors. The image was acquired by the Spaceborne Imaging Radar-C/X-band Synthetic Aperture Radar (SIR-C/X-SAR) on April 18 and 19, 1994, onboard the space shuttle Endeavour. The area shown is 22 kilometers by 28 kilometers (14 miles by 17 miles) and is centered at 19.1 degrees north latitude, 19.3 degrees east longitude. North is toward the upper right. The colors are assigned to different radar frequencies and polarizations as follows: red is L-band, horizontally transmitted and received; green is C-band, horizontally transmitted and received; and blue is C-band, horizontally transmitted, vertically received."<ref name=Wong>{{ cite book |author=Al Wong |title=Aorounga Impact Crater, Chad |publisher=NASA/JPL |location=Pasadena, California, USA |date=20 March 1996 |url=http://southport.jpl.nasa.gov/cdrom/sirced03/cdrom/DATA/LOCATION/AFRICA/SAHARA/POSTSIRC/CHAD.HTM |accessdate=2016-09-15 }}</ref> At right is a shaded relief map of Antarctica developed from RADARSAT Synthetic Aperture Radar data. RADARSAT is a Canadian satellite. {{clear}} ==Moon== [[Image:Moon South Pole.jpg|thumb|left|250px|alt=Twenty degrees of latitude of the Moon's disk, completely covered in the overlapping circles of craters. The illumination angles are from all directions, keeping almost all the crater floors in sunlight, but a set of merged crater floors right at the south pole are completely shadowed.|Mosaic image of the lunar south pole as taken by ''Clementine'': note permanent polar shadow. Credit: NASA/JPL-Caltech.]] [[Image:The Moon's North Pole.jpg|thumb|right|250px|The Moon's north pole is shown during the summer. Credit: NASA/GSFC/Arizona State University.]] [[Image:Nsmoonpole.s.jpeg|thumb|left|250px|This is a radar image of the Moon's south pole made with the Arecibo Planetary Radar. Credit: N. J. Stacy.]] [[Image:Radar Image of the Lunar South Pole from Jean-Luc Margot's PhD thesis.jpg|thumb|right|250px|Goldstone radar image of the craters at the Moon's south pole. Credit: [[commons:User:JeanLucMargot|JeanLucMargot]].]] [[Image:GRAIL's gravity map of the moon.jpg|thumb|right|250px|This image shows the variations in the lunar gravity field as measured by NASA's Gravity Recovery and Interior Laboratory (GRAIL) during the primary mapping mission from March to May 2012. Credit: NASA/JPL-Caltech/MIT/GSFC.]] [[Image:Moon at 850 microns.gif|thumb|right|250px|Thermal emission from the Moon is shown at 850 microns. Credit: NRAO.]] [[Image:Moon at 70 cm radar.jpg|thumb|right|250px|This is an Arecibo/GBT λ = 70 cm bistatic radar image of the lunar north pole. Credit: Arecibo/GBT.]] The moon is comparatively close and was detected by radar, soon after the invention of the technique, in 1946.<ref name=Mofensen>{{cite journal |author=J. Mofensen |title=Radar Echoes from the Moon |journal=Nature, Electronics |month=February, April |year=1946 |volume=157, 19 |issue=3379 |pages=129, 92-8 |bibcode=1946Natur.157R.129. |doi=10.1038/157129b0 |url= }}</ref><ref>Z. Bay, "Reflection of microwaves from the moon," Hung. Acta Phys., vol. 1, pp. 1-22; April, 1946.</ref> Measurements included surface roughness and later mapping of shadowed regions near the poles. "Clementine orbited the Moon in 1994 for 71 days, mapping the Moon globally in 11 wavelengths and measuring its topography by laser ranging. [... The] bistatic radar experiment (so-called because the spacecraft transmitted while we listened to the echoes on Earth) found evidence in the dark areas near the south pole of the Moon for material with high circular polarization ratio [CPR]".<ref name=Spudis>{{ cite book | author=P. Spudis |title = Ice on the Moon |url = http://www.thespacereview.com/article/740/1 |publisher = The Space Review |date = November 6, 2006 |accessdate =12 April 2007 }}</ref> "Meanwhile, astronomers on Earth began publishing results questioning the Clementine and Lunar Prospector [1998-2000] results. With the giant Arecibo radiotelescope, radar images were taken from the Earth. They found radar reflections with high CPR lying in both permanent darkness and in sunlit areas. Ice is not stable in sunlight, so they postulated that all high CPR is caused by surface roughness; if any ice is at the lunar poles, it must be in a finely disseminated form, invisible to radar mapping."<ref name=Spudis/> The experiment from Clemintine "was bistatic, i.e., the transmitter and receiver were in different places. Bistatic radar has the advantage of observing reflections through the phase angle, the angle between transmitted and received radio rays [...]. This phase dependence is important. It’s similar to the effect one gets from looking at a bicycle reflector at just the right angle: at certain angles, the internal planes in the transparent plastic align and a very bright reflection is seen. Similarly, in both radio and visible wavelengths on the Moon, we see an “opposition surge”, an apparent increase in brightness looking directly down from the sun (zero phase). Clementine orbited the Moon such that we could observe its phase dependence [...] and we specifically looked for this “opposition surge”, called the Coherent Backscatter Opposition Effect (CBOE). CBOE is particularly valuable to identify ice on planetary surfaces."<ref name=Spudis/> "Clementine transmitted right circular polarized (RCP) radio and we listened on Earth in both right- and left-circular polarized (LCP) channels. The ratio of power received in these two channels is called the circular polarization ratio (CPR). The dry, equatorial Moon has CPR less than one, but the icy satellites of Jupiter all have CPR greater than one. We know these objects have surfaces of water ice; in this case, the ice acts as a radio-transparent media in which waves penetrate the ice, are scattered and reflected multiple times, and returned such that some of the waves are received in the same polarization sense as they are sent—they have CPR greater than unity"<ref name=Spudis/> "The problem with CPR alone is that we can also get high values from very rough surfaces, such as a rough, blocky lava flow, which has angles that form many small corner reflectors. In this case, a radio wave could hit a rock face (changing RCP into LCP) and then bounce over to another rock face (changing the LCP back into RCP) and hence to the receiver [...]. This “double-bounce” effect also creates high CPR in that “same sense” reflections could mimic the enhanced CPR one gets from ice targets."<ref name=Spudis/> At lower right is an image using the Goldstone DSS-14 antenna as a transmitter and the DSS-13 as a receiver, a form of radar interferometry. The cross for the south pole in the Arecibo image is in the Shackleton crater of the Goldstone image. "Very precise microwave measurements between two spacecraft, named Ebb and Flow, were used to map gravity with high precision and high spatial resolution. The field shown resolves blocks on the surface of about 12 miles (20 kilometres) and measurements are three to five orders of magnitude improved over previous data. Red corresponds to mass excesses and blue corresponds to mass deficiencies. The map shows more small-scale detail on the far side of the moon compared to the nearside because the far side has many more small craters."<ref name=Greicius>{{ cite web |author=Tony Greicius |title=GRAIL's Gravity Map of the Moon |publisher=NASA/JPL-Caltech/MIT/GSFC |location= |date=December 6, 2012 |url=http://www.nasa.gov/mission_pages/grail/multimedia/zuber4.html |accessdate=2012-12-15 }}</ref> "Twin NASA probes orbiting Earth's moon have generated the highest resolution gravity field map of any celestial body. The new map, created by the Gravity Recovery and Interior Laboratory (GRAIL) mission, is allowing scientists to learn about the moon's internal structure and composition in unprecedented detail. Data from the two washing machine-sized spacecraft also will provide a better understanding of how Earth and other rocky planets in the solar system formed and evolved."<ref name=Greicius/> "The gravity field map reveals an abundance of features never before seen in detail, such as tectonic structures, volcanic landforms, basin rings, crater central peaks and numerous simple, bowl-shaped craters. Data also show the moon's gravity field is unlike that of any terrestrial planet in our solar system."<ref name=PRNewswire>{{ cite book |author=PR Newswire |title=NASA Twin Spacecraft Create Most Accurate Gravity Map Of Moon |publisher= |location= |date=December 5, 2012 |url=http://news.yahoo.com/nasa-twin-spacecraft-create-most-accurate-gravity-map-173600085.html |accessdate=2012-12-15 }}</ref> ""What this map tells us is that more than any other celestial body we know of, the moon wears its gravity field on its sleeve," said GRAIL Principal Investigator Maria Zuber of the Massachusetts Institute of Technology in Cambridge. "When we see a notable change in the gravity field, we can sync up this change with surface topography features such as craters, rilles or mountains.""<ref name=PRNewswire/> "According to Zuber, the moon's gravity field preserves the record of impact bombardment that characterized all terrestrial planetary bodies and reveals evidence for fracturing of the interior extending to the deep crust and possibly the mantle. This impact record is preserved, and now precisely measured, on the moon. The probes revealed the bulk density of the moon's highland crust is substantially lower than generally assumed. This low-bulk crustal density agrees well with data obtained during the final Apollo lunar missions in the early 1970s, indicating that local samples returned by astronauts are indicative of global processes."<ref name=PRNewswire/> ""With our new crustal bulk density determination, we find that the average thickness of the moon's crust is between 21 and 27 miles (34 and 43 kilometres), which is about 6 to 12 miles (10 to 20 kilometres) thinner than previously thought," said Mark Wieczorek, GRAIL co-investigator at the Institut de Physique du Globe de Paris. "With this crustal thickness, the bulk composition of the moon is similar to that of Earth. This supports models where the moon is derived from Earth materials that were ejected during a giant impact event early in solar system history.""<ref name=PRNewswire/> "The map was created by the spacecraft transmitting radio signals to define precisely the distance between them as they orbit the moon in formation. As they fly over areas of greater and lesser gravity caused by visible features, such as mountains and craters, and masses hidden beneath the lunar surface, the distance between the two spacecraft will change slightly."<ref name=PRNewswire/> ""We used gradients of the gravity field in order to highlight smaller and narrower structures than could be seen in previous datasets," said Jeff Andrews-Hanna, a GRAIL guest scientist with the Colorado School of Mines in Golden. "This data revealed a population of long, linear gravity anomalies, with lengths of hundreds of kilometres, crisscrossing the surface. These linear gravity anomalies indicate the presence of dikes, or long, thin, vertical bodies of solidified magma in the subsurface. The dikes are among the oldest features on the moon, and understanding them will tell us about its early history.""<ref name=PRNewswire/> At fourth right is an image of the Moon using its thermal emission at 850 microns. "The Moon and planets are not detectable by reflected solar radiation at radio wavelengths. However, they all emit thermal radiation, and Jupiter is a strong nonthermal source as well. If the Sun were suddenly switched off, the planets would remain radio sources for a long time, slowly fading as they cooled. At first glance, the λ = 0.85 mm radio image of the Moon [at second right] looks familiar, but there are differences from the visible Moon."<ref name=Djorgovski>{{ cite book |author=S.G. Djorgovski |display-authors=etal |title=A Tour of the Radio Universe |publisher=National Radio Astronomy Observatory |location= |date= |url=http://www.cv.nrao.edu/course/astr534/Tour.html |accessdate=2014-03-16 }}</ref> "The darker right edge of the Moon is not being illuminated by the Sun, but it still emits radio waves because it does not cool to absolute zero during the lunar night. A subtler point is that the radio emission is not produced at the visible surface; it emerges from a layer about ten wavelengths thick. As a result, monthly temperature variations of the Moon decrease with increasing wavelength. These wavelength-dependent temperature variations encode information about the conductivity and heat capacity of the rocky and dusty outer layers of the Moon."<ref name=Djorgovski/> "Radar images like the one [at fifth right] were recently used to search for water ice trapped in cold craters near the lunar poles."<ref name=Djorgovski/> The ESA Lunar Lander Mission ''Lunar Dust Environment and Plasma Package'': "Observe radio spectrum (with an additional goal to prepare for future radiation astronomy activities.)"<ref name=Thomsen>{{ cite journal |author=B. Gardini |title=ESA strategy for human exploration and the Lunar Lander Mission |journal=Memorie della Societa Astronomica Italiana |date=2011 |volume=82 |issue= |pages=422-29 |url=http://adsabs.harvard.edu/full/2011MmSAI..82..422G |arxiv= |bibcode=2011MmSAI..82..422G |doi= |pmid= |accessdate=5 May 2019 }}</ref> {{clear}} ==Mars== [[Image:PIA13164 North Polar Cap Cross Section, Annotated Version.jpg|thumb|right|300px|This image shows a cross-section of a portion of the north polar ice cap of Mars. Credit: NASA.]] [[Image:Martian Ionosphere MARSIS release 4A.jpg|thumb|left|250px|This plot, called an ionogram, shows data from sounding Mars' ionosphere with the Mars Advanced Radar for Subsurface and Ionospheric Sounding (MARSIS). Credit: ASI/NASA/ESA/Univ. of Rome/Univ. of Iowa/JPL.]] [[Image:PIA09076.jpg|thumb|right|250px|This image is a SHARAD subsurface radargram of Mars. Credit: NASA/JPL-Caltech/ASI/University of Rome/Washington Universtiy in St. Louis.]] Mapping of surface roughness [has been performed] from Arecibo Observatory. The Mars Express mission carries a ground-penetrating radar. The image at right "shows a cross-section of a portion of the north polar ice cap of Mars, derived from data acquired by the Mars Reconnaissance Orbiter's Shallow Radar (SHARAD), one of six instruments on the spacecraft. The data depict the region's internal ice structure, with annotations describing different layers. The ice depicted in this graphic is approximately 2 kilometers (1.2 miles) thick and 250 kilometers (155 miles) across. White lines show reflection of the radar signal back to the spacecraft. Each line represents a place where a layer sits on top of another. Scientists study how thick the pancake-like layers are, where they bulge and how they tilt up or down to understand what the surface of the ice sheet was like in the past as each new layer was deposited."<ref name=BatteryIncluded>{{ cite book |author=[[commons:User:BatteryIncluded|BatteryIncluded]] |title=''File:PIA13164 North Polar Cap Cross Section, Annotated Version.jpg'' |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=June 14, 2013 |url=https://commons.wikimedia.org/wiki/File:PIA13164_North_Polar_Cap_Cross_Section,_Annotated_Version.jpg |accessdate=2013-10-24 }}</ref> The image at left, "called an ionogram, shows data from sounding Mars' ionosphere with the Mars Advanced Radar for Subsurface and Ionospheric Sounding (MARSIS). The horizontal axis is the frequency of the pulse. The left vertical axis is the time delay after transmitting the pulse, with time increasing downward. The right vertical axis is a conversion of time delay to distance, showing the apparent range to the reflection point. The intensity of the received signal at any given frequency and apparent range is indicated by the color, with dark blue being the least intense and green being the most intense."<ref name=Thompson>{{ cite book |author=Thomas Thompson |title=Radar Ionogram with Oblique Ionospheric Echo |publisher=NASA |location=Pasadena, California USA |date=September 2, 2013 |url=http://mars.jpl.nasa.gov/express/gallery/martianterrain/MARSIS_release_4.html |accessdate=2013-10-24 }}</ref> "The green echo at an apparent range of about 800 kilometers (497 miles) from 2.5 to 5.5 megahertz is the reflected signal from the surface of Mars. The curved bright green feature with an apparent range varying from about 600 to 750 kilometers (373 to 466 miles) at frequencies from about 0.7 to 1.8 megahertz is the echo from the top side of the ionosphere. A second echo of the ionosphere, at an apparent range of about 100 kilometers (62 miles) is labeled "Oblique ionospheric echo." Such echoes are believed to come from distorted structures in the ionosphere caused by the magnetic fields in the crust of Mars."<ref name=Thompson/> "MARSIS is an instrument on the European Space Agency's Mars Express orbiter."<ref name=Thompson/> At lower right is a "radargram from the Shallow Subsurface Radar instrument (SHARAD)".<ref name=Lavoie2006>{{ cite book |author=Sue Lavoie |title=PIA09076: Interpreting Radar View near Mars' South Pole, Orbit 1334 |publisher=NASA |location=Pasadena, California USA |date=December 13, 2006 |url=http://photojournal.jpl.nasa.gov/catalog/PIA09076 |accessdate=2013-10-24 }}</ref> The "Shallow Subsurface Radar instrument (SHARAD) on NASA's Mars Reconnaissance Orbiter [radargram] is shown in the upper-right panel and reveals detailed structure in the polar layered deposits of the south pole of Mars."<ref name=Lavoie2006/> "The sounding radar collected the data presented here during orbit 1334 of the mission, on Nov. 8, 2006."<ref name=Lavoie2006/> "The horizontal scale in the radargram is distance along the ground track. It can be referenced to the ground track map shown in the lower right. The radar traversed from about 75 to 85 degrees south latitude, or about 590 kilometers (370 miles). The ground track map shows elevation measured by the Mars Orbiter Laser Altimeter on NASA's Mars Global Surveyor orbiter. Green indicates low elevation; reddish-white indicates higher elevation. The traverse proceeds up onto a plateau formed by the layers."<ref name=Lavoie2006/> "The vertical scale on the radargram is time delay of the radar signals reflected back to Mars Reconnaissance Orbiter from the surface and subsurface. For reference, using an assumed velocity of the radar waves in the subsurface, time is converted to depth below the surface at one place: about 1,500 meters (5,000 feet) to one of the deeper subsurface reflectors. The color scale varies from black for weak reflections to white for strong reflections."<ref name=Lavoie2006/> "The middle panel shows mapping of the major subsurface reflectors, some of which can be traced for a distance of 100 kilometers (60 miles) or more. The layers are not all horizontal and the reflectors are not always parallel to one another. Some of this is due to variations in surface elevation, which produce differing velocity path lengths for different reflector depths. However, some of this behavior is due to spatial variations in the deposition and removal of material in the layered deposits, a result of the recent climate history of Mars."<ref name=Lavoie2006/> "The Shallow Subsurface Radar was provided by the Italian Space Agency (ASI). Its operations are led by the University of Rome and its data are analyzed by a joint U.S.-Italian science team."<ref name=Lavoie2006/> {{clear}} ==Asteroids== {{main|Rocks/Rocky objects/Asteroids}} [[Image:Toutatis.jpg|thumb|right|250px|This is a Goldstone radar image of asteroid 4179 Toustatis. Credit: Steve Ostro, JPL.]] The image at the top of the page is of asteroid 2012 LZ1. "On Sunday, June 10, a potentially hazardous asteroid thought to have been 500 meters (0.31 miles) wide was discovered by Siding Spring Observatory in New South Wales, Australia. Fortunately for us, asteroid 2012 LZ1 drifted safely by, coming within 14 lunar distances from Earth on Thursday, June 14."<ref name=Neill>{{ cite book |author=Ian O'Neill |title=Asteroid 2012 LZ1 Just Got Supersized |publisher=Discovery Communications, LLC |location= |date=June 22, 2012 |url=http://news.discovery.com/space/asteroid-2012-lz1-just-got-supersized-120622.htm |accessdate=2013-10-24 }}</ref> "Asteroid 2012 LZ1 is actually bigger than thought… in fact, it is quite a lot bigger. 2012 LZ1 is one kilometer wide (0.62 miles), double the initial estimate."<ref name=Neill/> Asteroid "2012 LZ1′s surface is really dark, reflecting only 2-4 percent of the light that hits it — this contributed to the underestimated initial optical observations. Looking for an asteroid the shade of charcoal isn’t easy."<ref name=Neill/> “This object turned out to be quite a bit bigger than we expected, which shows how important radar observations can be, because we’re still learning a lot about the population of asteroids”.<ref name=Howell>{{ cite book |author=Ellen Howell |title=Asteroid 2012 LZ1 Just Got Supersized |publisher=Discovery Communications, LLC |location= |date=June 22, 2012 |url=http://news.discovery.com/space/asteroid-2012-lz1-just-got-supersized-120622.htm |accessdate=2013-10-24 }}</ref> “The sensitivity of our radar has permitted us to measure this asteroid’s properties and determine that it will not impact the Earth at least in the next 750 years”.<ref name=Nolan>{{ cite book |author=Mike Nolan |title=Asteroid 2012 LZ1 Just Got Supersized |publisher=Discovery Communications, LLC |location= |date=June 22, 2012 |url=http://news.discovery.com/space/asteroid-2012-lz1-just-got-supersized-120622.htm |accessdate=2013-10-24 }}</ref> The extremely accurate astrometry provided by radar is critical in long-term predictions of asteroid-Earth impacts, as illustrated by the object 99942 Apophis. At right is a Goldstone radar image of the asteroid 4179 Toutatis on November 26, 1996. The "images were recorded at NASA's Deep Space Network 70-meter and 34-meter radio/radar antennas in Goldstone, CA, and the 305-meter Arecibo Radio Telescope in Puerto Rico."<ref name=Savage>{{ cite book |author=Don Savage and Jane Platt |title=Images of Asteroid 4179 Toutatis |publisher=NASA |location=Washington, DC USA |date=November 27, 1996 |url=http://neo.jpl.nasa.gov/images/toutatis.html |accessdate=2013-10-24 }}</ref> "It's amazing that the shape of Toutatis can be determined so accurately from ground-based observations".<ref name=Jong>{{ cite book |author=Eric De Jong |title=Images of Asteroid 4179 Toutatis |publisher=NASA |location=Washington, DC USA |date=November 27, 1996 |url=http://neo.jpl.nasa.gov/images/toutatis.html |accessdate=2013-10-24 }}</ref> "This technology will provide us with startling, close-up views of thousands of asteroids that orbit near the Earth."<ref name=Jong/> "We used the computer to mathematically create a three- dimensional model of the surface and rotation of Toutatis".<ref name=Hudson>{{ cite book |author=Scott Hudson |title=Images of Asteroid 4179 Toutatis |publisher=NASA |location=Washington, DC USA |date=November 27, 1996 |url=http://neo.jpl.nasa.gov/images/toutatis.html |accessdate=2013-10-24 }}</ref> "It's as though we put a clay model in space and molded it until it matched the appearance of the actual asteroid."<ref name=Hudson/> "The video is of particular interest as Toutatis nears Earth and makes its closest approach on Friday, Nov. 29, when it will pass by at a distance of 3.3 million miles (5.3 million kilometers), or about 14 times the distance from the Earth to the Moon. In 2004, Toutatis will pass only four lunar distances from Earth, closer than any known Earth- approaching object expected to pass by in the next 60 years."<ref name=Savage/> "Toutatis poses no significant threat to Earth, at least for a few hundred years".<ref name=Ostro1996>{{ cite book |author=Steven Ostro |title=Images of Asteroid 4179 Toutatis |publisher=NASA |location=Washington, DC USA |date=November 27, 1996 |url=http://neo.jpl.nasa.gov/images/toutatis.html |accessdate=2013-10-24 }}</ref> "The discovery that we live in an asteroid swarm is important for the future of humanity".<ref name=Ostro1996/> "These leftover debris from planetary formation can teach us a good deal about the formation of our Solar System. Asteroids also contain valuable minerals and many are the cheapest possible destinations for space missions."<ref name=Ostro1996/> {{clear}} ==16 Psyche== [[Image:16Psyche (Lightcurve Inversion).png|right|thumb|250px|A three-dimensional model of 16 Psyche was computed using light curve inversion techniques. Credit: Josef Ďurech, Vojtěch Sidorin, Astronomical Institute of the Charles University.]] [[Image:AnimatedOrbitOf16Psyche.gif|left|thumb|250px|This is an animation of the revolution of 16 Psyche. Credit: [[commons:User:WilyD|WilyD]].]] [[Image:Imke radar image of 16 Psyche.jpg|right|thumb|250px|Asteroid 16 Psyche displays significant variations in radar albedo with rotation. Credit: Imke de Pater.]] A "huge, metallic asteroid named 16 Psyche [...] resides" in the asteroid belt.<ref name=Wall>{{ cite book |author=Mike Wall |title=NASA Unveils 2 New Missions to Study Truly Strange Asteroids |publisher=Space.com |location= |date=4 January 2017 |url=http://www.space.com/35221-nasa-discovery-missions-asteroids-psyche-lucy.html |accessdate=2017-01-09 }}</ref> 16 Psyche is "a 130-mile-wide (210 kilometers) metallic asteroid that may be the core of an ancient, Mars-size planet. Violent collisions billions of years ago might have stripped away the layers of rock that once lay atop this metallic object."<ref name=Tanton>{{ cite book |author=Lindy Elkins-Tanton |title=NASA Unveils 2 New Missions to Study Truly Strange Asteroids |publisher=Space.com |location= |date=4 January 2017 |url=http://www.space.com/35221-nasa-discovery-missions-asteroids-psyche-lucy.html |accessdate=2017-01-09 }}</ref> "16 Psyche is the only known object of its kind in the solar system, and this is the only way humans will ever visit a core. We learn about inner space by visiting outer space."<ref name=Tanton/> The "asteroid Psyche displays significant variations in radar and optical albedo with rotation."<ref name=Pater>{{ cite book |author=Imke de Pater |title=Prof. Imke de Pater Confirms Asteroid 16 Psyche to be the Largest Metal Asteroid in the Main Belt |publisher= |location= |date=27 October 2016 |url=http://eps.berkeley.edu/news/prof-imke-de-pater-confirms-asteroid-16-psyche-be-largest-metal-asteroid-main-belt |accessdate=2017-01-09 }}</ref> "16 Psyche [is] the largest M-class asteroid in the main belt."<ref name=Pater/> "18 radar imaging and 6 continuous wave runs in November and December 2015, [were] combined [...] with 16 continuous wave runs from 2005 and 6 recent adaptive-optics (AO) images (Drummond et al., 2016) to generate a three-dimensional shape model of Psyche."<ref name=Pater/> The "shape model has dimensions 279 × 232 × 189 km (± 10%), D_eff = 226 ± 23 km, and is 6% larger than, but within the uncertainties of, the most recently published size and shape model generated from the inversion of lightcurves (Hanus et al., 2013). Psyche is roughly ellipsoidal but displays a mass-deficit over a region spanning 90° of longitude. There is also evidence for two ∼50–70 km wide depressions near its south pole. Their size and published masses lead to an overall bulk density estimate of 4500 ± 1400 kg·m⁻³. Psyche's mean radar albedo of 0.37 ± 0.09 is consistent with a near-surface regolith composed largely of iron-nickel and ∼40% porosity. Its radar reflectivity varies by a factor of 1.6 as the asteroid rotates, suggesting global variations in metal abundance or bulk density in the near surface."<ref name=Pater/> {{clear}} ==Jupiter== {{main|Jupiter}} Between September and November 23, 1963, Jupiter is detected by radar astronomy.<ref name=Pettengill>{{ cite journal |author=Gordon H. Pettengill & Irwin I. Shapiro |title=Radar Astronomy |journal=Annual Review of Astronomy and Astrophysics |month= |year=1965 |volume=3 |issue= |pages=377-410 |url=http://adsabs.harvard.edu//abs/1965ARA&A...3..377P |arxiv= |bibcode=1965ARA&A...3..377P |doi= |pmid= |accessdate=2012-12-25 }}</ref> "The dense atmosphere makes a penetration to a hard surface (if indeed one exists at all) very unlikely. In fact, the JPL results imply a correlation of the echo with Jupiter ... which corresponds to the upper (visible) atmosphere. ... Further observations will be needed to clarify the current uncertainties surrounding radar observations of Jupiter."<ref name=Pettengill/> "Although in 1963 some claimed to have detected echoes from Jupiter, these were quite weak and have not been verified by later experiments."<ref name=Shapiro>{{ cite journal |author=Irwin I. Shapiro |title=Planetary radar astronomy |journal=Spectrum, IEEE |month=March |year=1968 |volume=5 |issue=3 |pages=70-9 |url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5214821 |arxiv= |bibcode= |doi=10.1109/MSPEC.1968.5214821 |pmid= |accessdate=2012-12-25 }}</ref> "A search for radar echoes from Jupiter at 430 MHz during the oppositions of 1964 and 1965 failed to yield positive results, despite a sensitivity several orders of magnitude better than employed by other groups in earlier (1963) attempts at higher frequencies. ... [I]t might be suspected that meteorological disturbances of a random nature were involved, and that the echoes might be returned only in exceptional circumstances. Further support for this point of view may be gleaned from the fact that JPL found positive results for only 1 (centered at 32° System I longitude) of the 8 longitude regions investigated in 1963 (Goldstein 1964) and, in fact, had no success during their observations in 1964 (see comment by Goldstein following Dyce 1965)."<ref name=Dyce>{{ cite journal |author=R. B. Dyce and G. H. Pettengill, and A. D. Sanchez |title=Radar Observations of Mars and Jupiter at 70 cm |journal=The Astronomical Journal |month=August |year=1967 |volume=72 |issue=4 |pages=771-7 |url=http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1967AJ.....72..771D&amp;data_type=PDF_HIGH&amp;whole_paper=YES&amp;type=PRINTER&amp;filetype=.pdf |arxiv= |bibcode=1967AJ.....72..771D |doi=10.1086/110307 |pmid= |accessdate=2012-12-25 }}</ref> ==Titan== [[Image:PIA10008 Seas and Lakes on Titan.jpg|thumb|right|250px|This Cassini false-color mosaic shows all synthetic-aperture radar images to date of Titan's north polar region. Credit: NASA / JPL / USGS.]] [[Image:Liquid lakes on titan.jpg|thumb|right|250px|This is a radar image of the Titan surface taken on July 22, 2006, from the Cassini probe. Credit: NASA/JPL/USGS.]] Radar detection of Titan from Arecibo Observatory, included mapping of Titan's surface. "This Cassini false-color mosaic [at right] shows all synthetic-aperture radar images to date of Titan's north polar region. Approximately 60 percent of Titan's north polar region, above 60 degrees north latitude, is now mapped with radar. About 14 percent of the mapped region is covered by what is interpreted as liquid hydrocarbon lakes."<ref name=Lavoie07>{{ cite book |author=Sue Lavoie |title=PIA10008: Titan's North Polar Region |publisher=NASA/JPL |location=Pasadena, California USA |date=October 11, 2007 |url=http://photojournal.jpl.nasa.gov/catalog/?IDNumber=PIA10008 |accessdate=2013-06-13 }}</ref> "Features thought to be liquid are shown in blue and black, and the areas likely to be solid surface are tinted brown. The terrain in the upper left of this mosaic is imaged at lower resolution than the remainder of the image".<ref name=Lavoie07/> "Most of the many lakes and seas seen so far are contained in this image, including the largest known body of liquid on Titan. These seas are most likely filled with liquid ethane, methane and dissolved nitrogen."<ref name=Lavoie07/> "Many bays, islands and presumed tributary networks are associated with the seas. The large feature in the upper right center of this image is at least 100,000 square kilometers (40,000 square miles) in area, greater in extent than Lake Superior (82,000 square kilometers or 32,000 square miles), one of Earth's largest lakes. This Titan feature covers a greater fraction of the surface, at least 0.12 percent, than the Black Sea, Earth's largest terrestrial inland sea, at 0.085 percent. Larger seas may exist, as it is probable that some of these bodies are connected, either in areas unmapped by radar or under the surface (see PIA08365)."<ref name=Lavoie07/> "Of the 400 observed lakes and seas, 70 percent of their area is taken up by large "seas" greater than 26,000 square kilometers (10,000 square miles)."<ref name=Lavoie07/> In the second image at right is another radar image of Titan's surface. "The existence of oceans or lakes of liquid methane on Saturn's moon Titan was predicted more than 20 years ago. But with a dense haze preventing a closer look it has not been possible to confirm their presence. Until the Cassini flyby of July 22, 2006, that is."<ref name=Piazza>{{ cite book |author=Enrico Piazza |title=Liquid Lakes on Titan |publisher=NASA/JPL |location=Pasadena, California USA |date=January 3, 2007 |url=http://saturn.jpl.nasa.gov/photos/imagedetails/index.cfm?imageId=2432 |accessdate=2013-06-13 }}</ref> "Radar imaging data from the flyby, published this week in the journal Nature, provide convincing evidence for large bodies of liquid. This image, used on the journal's cover, gives a taste of what Cassini saw. Intensity in this colorized image is proportional to how much radar brightness is returned, or more specifically, the logarithm of the radar backscatter cross-section. The colors are not a representation of what the human eye would see."<ref name=Piazza/> "The lakes, darker than the surrounding terrain, are emphasized here by tinting regions of low backscatter in blue. Radar-brighter regions are shown in tan. The strip of radar imagery is foreshortened to simulate an oblique view of the highest latitude region, seen from a point to its west."<ref name=Piazza/> "This radar image was acquired by the Cassini radar instrument in synthetic aperture mode on July 22, 2006. The image is centered near 80 degrees north, 35 degrees west and is about 140 kilometers (84 miles) across. Smallest details in this image are about 500 meters (1,640 feet) across."<ref name=Piazza/> {{clear}} ==Comets== {{main|Radiation astronomy/Comets}} [[Image:HMP Aug19 2011.b.gif|thumb|right|250px|Goldstone radar observations on 2011 August 19 and 20 shown detected echoes from the nucleus and coma of comet 45P/Honda-Mrkos-Pajdusakova (HMP). Credit: Goldstone.]] "Goldstone radar observations on 2011 August 19 and 20 detected echoes from the nucleus and coma of comet 45P/Honda-Mrkos-Pajdusakova (HMP). This is only the fourth Goldstone comet detection and the first since detection of comet 73P/Schwassmann-Wachmann 3 in 2006."<ref name=Goldstone>{{ cite book |author=Goldstone |title=GOLDSTONE RADAR DETECTION OF COMET HONDA-MRKOS-PAJDUSAKOVA |publisher=NASA/JPL |location=Pasasena, California USA |date=19 August 2011 |url=http://echo.jpl.nasa.gov/asteroids/45P_HMP/comet.hmp.html |accessdate=2016-09-15 }}</ref> "The [continuous wave] CW spectrum [on the right] shows the opposite-circular echo from the comet obtained on August 19. The narrow spike is the echo from the nucleus and the broad, low, asymmetric hump is the echo from coma particles. The skew of the coma echo to positive frequencies indicates that most of the coma particles were approaching Earth at the time of the observations."<ref name=Goldstone/> "The Goldstone measurements provided a range correction of 49 km for the nucleus, which significantly improved the orbit and which revealed a systematic bias in many of the optical observations."<ref name=Goldstone/> "This is only the fifteenth comet that has been detected with radar."<ref name=Goldstone/> {{clear}} ==Astrometry== "Asteroid radar astronomy began on 14 June 1968, with the detection of 1566 Icarus from Goldstone (Goldstein 1969) and Haystack (Pettengill et al. 1969)."<ref name=Ostro1991>{{ cite journal |author=Ostro, S. J.; Campbell, D. B.; Chandler, J. F.; Shapiro, I. I.; Hine, A. A.; Velez, R.; Jurgens, R. F.; Rosema, K. D.; Winkler, R.; Yeomans, D. K. |title=Asteroid radar astrometry |journal=Astronomical Journal |month=October |year=1991 |volume=102 |issue=10 |pages=1490-1502 |url=http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1991AJ....102.1490O&link_type=ARTICLE&db_key=AST&high= |arxiv= |bibcode=1991AJ....102.1490O |doi=10.1086/115975 |pmid= |accessdate=2017-07-27 }}</ref> "Radar measurements of echo Doppler frequencies and time delays permit significant refinements of orbital elements and commensurate improvements in the accuracy of prediction ephemerides because these measurements have fine fractional precision and are orthogonal to optical, angular-position measurements."<ref name=Ostro1991/> "Yeomans ''et al.'' (1987) used numerical experiments to explore the extent to which delay/ Doppler astrometry can refine orbit estimates for NEAs. They concluded that radar measurements can reduce ephemeris uncertainties dramatically for asteroids having short optical-data histories. They noted that a few radar observations of a newly discovered NEA could mean the difference between successfully recovering the object during its next close approach and losing it entirely. Even for asteroids with very long astrometric histories and secure orbits, radar measurements can significantly shrink their positional error ellipsoids for at least a decade."<ref name=Ostro1991/> "A typical transmit/receive cycle, or run, consists of signal transmission for a duration close to the roundtrip light time between the radar and the target, i.e., until the first echoes are about to come back, followed by reception of echoes for a similar duration. In continuous wave (cw) observations, one transmits a nearly monochromatic waveform and measures the distribution of echo power as a function of frequency. The resultant echo spectra can be thought of as one-dimensional images, or brightness scans across the target through a slit parallel to the asteroid’s apparent spin vector. In ranging observations, time coding of the waveform permits measurement of the distribution of echo power in time delay (range) as well."<ref name=Ostro1991/> "An asteroid’s apparent radial motion introduces a continuously changing Doppler shift into the echoes. One avoids spectral smear by tuning the receiver according to an ephemeris based on an orbit determined from astrometric asteroid observations."<ref name=Ostro1991/> "In cw experiments, voltage samples of the received signal are Fourier transformed and the results are squared to obtain an estimate of the power spectrum, with the frequency resolution equal to the reciprocal of the time series length, i.e., of the coherence time. The sampling rate is chosen to provide an unaliased bandwidth many times larger than both the ''a priori'' Doppler uncertainty and the echo bandwidth, so ''f''_est can be determined unambiguously from the received power spectrum. Normally, a number of these "single-look" spectra are averaged to improve the spectral estimates."<ref name=Ostro1991/> "In principle, range resolution can be obtained by using a coherent pulsed cw waveform—the transmitter’s carrier-frequency oscillator operates continuously but radio-frequency power is radiated only during intervals that are one delay resolution cell long and occur at intervals called the pulse repetition period (PRP). The PRP is normally much greater than the target’s intrinsic delay dispersion, thereby ensuring that the echo will consist of successive, nonoverlapping range profiles. Fourier transformation of ''N'' time samples taken at the same position (i.e., the same delay relative to τ₀) within each of N successive range profiles yields the echo power spectrum for the corresponding range cell on the target. This spectrum has an unaliased bandwidth ''B'' - l/[(PRP)(NCOH)] and a frequency resolution ''B/N'', where NCOH is the number of code cycles for which voltage samples have been coherently summed prior to Fourier transformation."<ref name=Ostro1991/> ==Locations on Earth== [[Image:France radar.jpg|thumb|right|250px|This is a Shuttle radar image of France. Credit: NASA.]] [[Image:Pakistan Khairabad digital elevation model by TanDEM-X (DLR).jpg|thumb|left|250px|Colour coded TanDEM-X digital elevation model is of Khairabad in northern Pakistan. Credit: DLR.]] The radar image of France on the right is color-coded for relative relief from darker green at or near sea level to white for mountain tops. The border of France has been shaded and given a slight elevation away from the rest of the Earth nearby. On the left is a colour coded TanDEM-X digital elevation model is of Khairabad in northern Pakistan, created on 6 August 2010. {{clear}} ==Paleogeography== [[Image:Kufra-space-radar.jpg|thumb|right|250px|A previously unknown branch of an ancient river, buried under thousands of years of windblown sand, is revealed by radar. Credit: [[commons:User:Porao|Porao]], NASA.]] "The ability of a sophisticated radar instrument to image large regions of the world from space, using different frequencies that can penetrate dry sand cover, produced the discovery in this image: a previously unknown branch of an ancient river, buried under thousands of years of windblown sand in a region of the Sahara Desert in North Africa. This area is near the Kufra Oasis in southeast Libya, centered at 23.3 degrees north latitude, 22.9 degrees east longitude. The image was acquired by the Spaceborne Imaging Radar-C/X-band Synthetic Aperture (SIR- C/X-SAR) imaging radar when it flew aboard the space shuttle Endeavour on its 60th orbit on October 4, 1994. This SIR-C image reveals a system of old, now inactive stream valleys, called "paleodrainage systems.""<ref name=Porao>{{ cite web |author=[[c:User:Porao|Porao]] |title=''File:Kufra-space-radar.jpg'' |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=4 October 1994 |url=https://commons.wikimedia.org/wiki/File:Kufra-space-radar.jpg |accessdate=2016-09-27 }}</ref> {{clear}} ==Impact craters== {{main|Radiation astronomy/Craters}} [[Image:Impact crater in northwestern Greenland.png|right|thumb|300px|The location of the 19-mile-wide (31 kilometers) impact crater in northwestern Greenland is circled in red. Credit: Kurt H. Kjær, Nicolaj K. Larsen, Tobias Binder, Anders A. Bjørk, Olaf Eisen, Mark A. Fahnestock, Svend Funder, Adam A. Garde, Henning Haack, Veit Helm, Michael Houmark-Nielsen, Kristian K. Kjeldsen1, Shfaqat A. Khan, Horst Machguth, Iain McDonald, Mathieu Morlighem, Jérémie Mouginot, John D. Paden, Tod E. Waight, Christian Weikusat, Eske Willerslev and Joseph A. MacGregor, The Natural History Museum of Denmark.{{tlx|fairuse}}]] [[Image:Bed topography based on airborne radar sounding.jpg|left|thumb|300px|Bed topography based on airborne radar sounding from 1997 to 2014 NASA data and 2016 Alfred Wegener Institute (AWI) data. Credit: Kurt H. Kjær, Nicolaj K. Larsen, Tobias Binder, Anders A. Bjørk, Olaf Eisen, Mark A. Fahnestock, Svend Funder, Adam A. Garde, Henning Haack, Veit Helm, Michael Houmark-Nielsen, Kristian K. Kjeldsen1, Shfaqat A. Khan, Horst Machguth, Iain McDonald, Mathieu Morlighem, Jérémie Mouginot, John D. Paden, Tod E. Waight, Christian Weikusat, Eske Willerslev and Joseph A. MacGregor, The Natural History Museum of Denmark.{{tlx|fairuse}}]] Per the image on the right: "In 2015 I was looking at a new map of the bedrock below the Greenland Ice Sheet and discovered a large circular feature under the Hiawatha glacier in northwest Greenland."<ref name=Larsen>{{ cite book |author=Nicolaj K. Larsen |title=A Massive Impact Crater Has Been Detected Beneath Greenland’s Ice Sheet |publisher=Gizmodo |location= |date=14 November 2018 |url=https://gizmodo.com/a-massive-impact-crater-has-been-detected-beneath-green-1830437095 |accessdate=14 November 2018 }}</ref> "You can see the rounded structure at the edge of the ice sheet, especially when flying high enough."<ref name=Paden>{{ cite book |author=John Paden |title=A Massive Impact Crater Has Been Detected Beneath Greenland’s Ice Sheet |publisher=Gizmodo |location= |date=14 November 2018 |url=https://gizmodo.com/a-massive-impact-crater-has-been-detected-beneath-green-1830437095 |accessdate=14 November 2018 }}</ref> "For the most part the crater isn’t visible out the airplane window. It’s funny that until now nobody thought, ‘Hey, what’s that semicircular feature there?’ From the airplane it is subtle and hard to see unless you already know it’s there. Using satellite imagery taken at a low sun angle that accentuates hills and valleys in the ice sheet’s terrain—you can really see the circle of the whole crater in these images."<ref name=Paden/> "It is correct that the crater is not well dated but there’s good evidence that it is geologically young, that is, it formed within the last 2 to 3 million years, and most likely it is as young as the last Ice Age [which ended around 12,000 years ago],” Larsen explained to Gizmodo. “We are currently trying to come up with ideas on how to date the impact. One idea is to drill through the ice and get bedrock samples that can be used for numerical dating."<ref name=Larsen/> On the left is an image of bed "topography based on airborne radar sounding from 1997 to 2014 NASA data and 2016 Alfred Wegener Institute (AWI) data. Black triangles represent elevated rim picks from the radargrams, and the dark purple circles represent peaks in the central uplift. Hatched red lines are field measurements of the strike of ice-marginal bedrock structures. Black circles show location of the three glaciofluvial sediment samples".<ref name=Kjær>{{ cite journal |author=Kurt H. Kjær, Nicolaj K. Larsen, Tobias Binder, Anders A. Bjørk, Olaf Eisen, Mark A. Fahnestock, Svend Funder, Adam A. Garde, Henning Haack, Veit Helm, Michael Houmark-Nielsen, Kristian K. Kjeldsen1, Shfaqat A. Khan, Horst Machguth, Iain McDonald, Mathieu Morlighem, Jérémie Mouginot, John D. Paden, Tod E. Waight, Christian Weikusat, Eske Willerslev and Joseph A. MacGregor |title=A large impact crater beneath Hiawatha Glacier in northwest Greenland |journal=Sciences Advances |date=14 November 2018 |volume=4 |issue=11 |pages=eaar8173 |url=http://advances.sciencemag.org/content/4/11/eaar8173 |arxiv= |bibcode= |doi=10.1126/sciadv.aar8173 |pmid= |accessdate=14 November 2018 }}</ref> "Glaciofluvial sediment from the largest river draining the crater contains shocked quartz and other impact-related grains. Geochemical analysis of this sediment indicates that the impactor was a fractionated iron asteroid, which must have been more than a kilometer wide to produce the identified crater. Radiostratigraphy of the ice in the crater shows that the Holocene ice is continuous and conformable, but all deeper and older ice appears to be debris rich or heavily disturbed."<ref name=Kjær/> {{clear}} ==Rainbands== [[Image:Sturmfront auf Doppler-Radar-Schirm.jpg|250px|thumb|right|Band of thunderstorms is seen on a weather radar display. Credit: NOAA.]] Rainbands are cloud and precipitation areas which are significantly elongated. Rainbands can be stratiform or convective,<ref>Glossary of Meteorology (2009). Rainband. [https://web.archive.org/web/20110606102457/http://amsglossary.allenpress.com/glossary/search?p=1&query=rainband&submit=Search] 2011-06-06 Retrieved on 2008-12-24.</ref> and are generated by differences in temperature. When noted on weather radar imagery, this precipitation elongation is referred to as banded structure.<ref>Glossary of Meteorology (2009). Banded structure. [https://web.archive.org/web/20110606102630/http://amsglossary.allenpress.com/glossary/search?id=banded-structure1] 2011-06-06 Retrieved on 2008-12-24.</ref> Rainbands in advance of warm occluded fronts and warm fronts are associated with weak upward motion,<ref>Owen Hertzman (1988). [http://adsabs.harvard.edu/abs/1988PhDT.......110H Three-Dimensional Kinematics of Rainbands in Midlatitude Cyclones.] Retrieved on 2008-12-24</ref> and tend to be wide and stratiform in nature.<ref>{{cite book |author=Yuh-Lang Lin |title=Mesoscale Dynamics |url=https://books.google.com/books?id=4KXtnQ3bDeEC&pg=PA405 |year=2007 |publisher=Cambridge University Press |isbn=978-0-521-80875-0 |page=405 }}</ref> Rainbands spawned near and ahead of cold fronts can be squall lines which are able to produce tornadoes.<ref>Glossary of Meteorology (2009). Prefrontal squall line. [url=https://web.archive.org/web/20070817224959/http://amsglossary.allenpress.com/glossary/search?id=prefrontal-squall-line1] 2007-08-17 Retrieved on 2008-12-24.</ref> Rainbands associated with cold fronts can be warped by mountain barriers perpendicular to the front's orientation due to the formation of a low-level barrier jet.<ref>J. D. Doyle (1997). The influence of mesoscale orography on a coastal jet and rainband. [https://web.archive.org/web/20120106231443/http://cat.inist.fr/?aModele=afficheN&cpsidt=2721180] 2012-01-06 Retrieved on 2008-12-25.</ref> Bands of thunderstorms can form with sea breeze and land breeze boundaries, if enough moisture is present. If sea breeze rainbands become active enough just ahead of a cold front, they can mask the location of the cold front itself.<ref>A. Rodin (1995). Interaction of a cold front with a sea-breeze front numerical simulations. [https://web.archive.org/web/20110909094659/http://cat.inist.fr/?aModele=afficheN&cpsidt=3570629] 2011-09-09 Retrieved on 2008-12-25.</ref> {{clear}} ==Extratropical cyclones== [[Image:Feb242007 blizzard.gif|thumb|350px|right|A February 24, 2007 radar image of a large extratropical cyclonic storm system at its peak over the central United States. Note the band of thunderstorms along its trailing cold front.]] Rainbands in advance of warm occluded fronts and warm fronts are associated with weak upward motion,<ref>Owen Hertzman (1988). [http://adsabs.harvard.edu/abs/1988PhDT.......110H Three-Dimensional Kinematics of Rainbands in Midlatitude Cyclones.] Retrieved on 2008-12-24</ref> and tend to be wide and stratiform in nature.<ref>Yuh-Lang Lin (2007). [https://books.google.com/books?id=4KXtnQ3bDeEC&pg=PA405&lpg=PA405&source=bl&ots=CL5KD0HLAJ&sig=Gz5bwKi9yu8j25EbXLD3TVNNQ68&hl=en&sa=X&oi=book_result&resnum=10&ct=result Mesoscale Dynamics.] Retrieved on 2008-12-25.</ref> In an atmosphere with rich low level moisture and vertical wind shear,<ref>Richard H. Grumm (2006). 16 November Narrow Frontal Rain band Floods and severe weather [https://web.archive.org/web/20110720032822/http://nws.met.psu.edu/severe/2006/16Nov2006.pdf] 2011-07-20 Retrieved on 2008-12-26.</ref> narrow, convective rainbands known as squall lines generally in the cyclone's warm sector, ahead of strong cold fronts associated with extratropical cyclones.<ref>Glossary of Meteorology (2009). Prefrontal squall line. [https://web.archive.org/web/20070817224959/http://amsglossary.allenpress.com/glossary/search?id=prefrontal-squall-line1] 2007-08-17 Retrieved on 2008-12-24.</ref> Wider rain bands can occur behind cold fronts, which tend to have more stratiform, and less convective, precipitation.<ref>K. A. Browning and Robert J. Gurney (1999). [https://books.google.com/books?id=DO5K1NK_ZewC&pg=PA116&lpg=PA116&source=web&ots=j2xQaps00_&sig=SSYUXEW1H4QDe4UI0K3T0WIW6Tk&hl=en&sa=X&oi=book_result&resnum=9&ct=result#PPA117,M1 Global Energy and Water Cycles.] Retrieved on 2008-12-26.</ref> Within the cold sector north to northwest of a cyclone center, in colder cyclones, small scale, or mesoscale, bands of heavy snow can occur within a cyclone's comma head precipitation pattern with a width of {{convert|20|mi|km|order=flip}} to {{convert|50|mi|km|order=flip}}.<ref>KELLY HEIDBREDER (2007). [http://www.theweatherprediction.com/weatherpapers/023/index.html Mesoscale snow banding.] Retrieved on 2008-12-24.</ref> These bands in the comma head are associated with areas of frontogensis, or zones of strengthening temperature contrast.<ref>David R. Novak, Lance F. Bosart, Daniel Keyser, and Jeff S. Waldstreicher (2002). [http://cstar.cestm.albany.edu/CAP_Projects/Project4/Banded%20Precip/novakWAF.pdf A CLIMATOLOGICAL AND COMPOSITE STUDY OF COLD SEASON BANDED PRECIPITATION IN THE NORTHEAST UNITED STATES.] Retrieved on 2008-12-26.</ref> Southwest of extratropical cyclones, curved flow bringing cold air across the relatively warm Great Lakes can lead to narrow lake effect snow bands which bring significant localized snowfall.<ref>{{cite web |author=B. Geerts |year=1998 |url=http://www-das.uwyo.edu/~geerts/cwx/notes/chap10/lake_effect_snow.html |title=Lake Effect Snow |accessdate=2008-12-24 |publisher=University of Wyoming }}</ref> {{clear}} ==Recent history== [[Image:ADU-1000-3.jpg|thumb|250px|right|This image shows the early planetary radar at Pluton, USSR, 1960. Credit: [[c:User:Rumlin|Rumlin]].{{tlx|free media}}]] The '''recent history''' period dates from around 1,000 b2k to present. The "Arecibo telescope was completed in 1963 at the initiative of Cornell electrical engineering professor William E. Gordon".<ref name=Brand/> At right is an image of the Pluton radar complex used for radar astronomy since 1960. {{clear}} ==Arecibo Observatory== [[Image:Arecibo Observatory Aerial View2.jpg|thumb|right|250px|The Arecibo Radio Telescope, Arecibo, Puerto Rico, at 1000 feet (305 m) across, is the largest dish antenna in the world. Credit: Michael D. Bicay.]] The "Arecibo Observatory in Puerto Rico [is] the world's largest, and most sensitive, single-dish radio telescope."<ref name=Brand>{{ cite book | author=David Brand | title=Astrophysicist Robert Brown, leader in telescope development, named to head NAIC and its main facility, Arecibo Observatory | publisher=Cornell University | date=21 January 2003 | url=http://www.news.cornell.edu/releases/Jan03/NAIC.director.deb.html | accessdate=2008-09-02 }}</ref> "The 1,000-foot-diameter (305 meters) Arecibo telescope [... provides] access to state-of-the-art observing for scientists in [[Radiation astronomy/Radios|radio astronomy]], solar system radar and [[Radiation astronomy/Atmospheres|atmospheric studies]], and the observatory has the unique capability for solar system and ionosphere (the atmosphere's ionized upper layers) radar remote sensing."<ref name=Brand/> It contains the largest curved focusing dish on Earth, giving Arecibo the largest electromagnetic-wave-gathering capacity.<ref name=Castel>{{ cite book | title=Arecibo: Celestial Eavesdropper | author=Frederic Castel | publisher=Space.com | date=8 May 2000 | accessdate=2008-09-02 |url=http://web.archive.org/web/20000619110005/http://www.space.com/scienceastronomy/astronomy/arecibo_profile_000508.html }}</ref> The dish surface is made of 38,778 perforated aluminum panels, each measuring about 3 by 6 feet (1 by 2 m), supported by a mesh of steel cables. The telescope has three radar transmitters, with effective isotropic radiated powers [EIRP] of 20 TW at 2380&nbsp;MHz, 2.5 TW (pulse peak) at 430&nbsp;MHz, and 300 MW at 47&nbsp;MHz. The telescope is a spherical reflector, not a parabolic reflector. To aim the telescope, the receiver is moved to intercept signals reflected from different directions by the spherical dish surface. A parabolic mirror would induce a varying astigmatism when the receiver is in different positions off the focal point, but the error of a spherical mirror is the same in every direction. The receiver is located on a 900-ton platform which is suspended 150&nbsp;m (500&nbsp;ft) in the air above the dish by 18 cables running from three reinforced concrete towers, one of which is 110&nbsp;m (365&nbsp;ft) high and the other two of which are 80&nbsp;m (265&nbsp;ft) high (the tops of the three towers are at the same elevation). The platform has a 93-meter-long rotating bow-shaped track called the azimuth arm on which receiving antennas, secondary and tertiary reflectors are mounted. This allows the telescope to observe any region of the sky within a forty-degree cone of visibility about the local zenith (between −1 and 38 degrees of declination). Puerto Rico's location near the equator allows Arecibo to view all of the planets in the Solar System, though the round trip light time to objects beyond Saturn is longer than the time the telescope can track it, preventing radar observations of more distant objects. {{clear}} ==Goldstone Deep Space Communication Complex== [[Image:Goldstone Deep Space Communication Complex - GPN-2000-000506.jpg|thumb|right|250px|This is an image of the Goldstone Deep Space Communication Complex. Credit: NASA.]] Shown at right are the three "34m (110 ft.) diameter Beam Waveguide antennas located at the Goldstone Deep Space Communications Complex, situated in the Mojave Desert in California. This is one of three complexes which comprise NASA's Deep Space Network (DSN). The DSN provides radio communications for all of NASA's interplanetary spacecraft and is also utilized for radio astronomy and radar observations of the solar system and the universe."<ref name=Hahn>{{ cite book |author=Michael Hahn |title=Goldstone Deep Space Communication Complex |publisher=NASA HQ |location=Washington, DC USA |date=January 1, 1990 |url=http://grin.hq.nasa.gov/ABSTRACTS/GPN-2000-000506.html |accessdate=2013-10-24 }}</ref> {{clear}} ==Synthetic Aperture Radar Satellite== [[Image:NISAR artist concept.jpg|thumb|right|250px|Artist concept is of the Nasa-Isro synthetic aperture radar (NISAR) satellite in orbit. Credit: NASA.]] "Using advanced radar imaging that will provide an unprecedented, detailed view of Earth, the NASA-ISRO Synthetic Aperture Radar, or NISAR, satellite is designed to observe and take measurements of some of the planet's most complex processes, including ecosystem disturbances, ice-sheet collapse, and natural hazards such as earthquakes, tsunamis, volcanoes and landslides."<ref name=Nelson>{{ cite book |author=Jon Nelson |title=NASA-ISRO Synthetic Aperture Radar |publisher=NASA/JPL |location=Pasadena, California USA |date=2 July 2014 |url=http://www.jpl.nasa.gov/missions/nasa-isro-synthetic-aperture-radar-nisar/ |accessdate=2015-03-10 }}</ref> "Data collected from NISAR will reveal information about the evolution and state of Earth's crust, help scientists better understand our planet's processes and changing climate, and aid future resource and hazard management. The mission is a partnership between NASA and the Indian Space Research Organization [ISRO]."<ref name=Nelson/> "Scientific Instrument(s) - L-band (24-centimeter wavelength) Polarimetric Synthetic Aperture Radar [JPL] - S-band (12-centimeter wavelength) Polarimetric Synthetic Aperture Radar [ISRO]"<ref name=Nelson/> {{clear}} ==Interferograms== [[Image:Izmit interferogram.jpg|thumb|right|250px|This image is an interferogram created using pairs of images taken by Synthetic Aperture Radar (SAR). Credit: NASA/JPL-Caltech.]] [[Image:L C band topo interferograms.jpg|thumb|left|250px|This image of Fort Irwin in California's Mojave Desert compares interferometric radar signatures topography. Credit: NASA/JPL-Caltech STS-65 (STS-68) crew.{{tlx|free media}}]] "This image is an interferogram that was created using pairs of images taken by Synthetic Aperture Radar (SAR). The images, acquired at two different times, have been combined to measure surface deformation or changes that may have occurred during the time between data acquisition. The images were collected by the European Space Agency's Remote Sensing satellite (ERS-2) on 13 August 1999 and 17 September 1999 and were combined to produce these image maps of the apparent surface deformation, or changes, during and after the 17 August 1999 Izmit, Turkey earthquake. This magnitude 7.6 earthquake was the largest in 60 years in Turkey and caused extensive damage and loss of life. Each of the color contours of the interferogram represents 28 mm (1.1 inches) of motion towards the satellite, or about 70 mm (2.8 inches) of horizontal motion. White areas are outside the SAR image or water of seas and lakes. The North Anatolian Fault that broke during the Izmit earthquake moved more than 2.5 meters (8.1 feet) to produce the pattern measured by the interferogram. Thin red lines show the locations of fault breaks mapped on the surface. The SAR interferogram shows that the deformation and fault slip extended west of the surface faults, underneath the Gulf of Izmit. Thick black lines mark the fault rupture inferred from the SAR data. Scientists are using the SAR interferometry along with other data collected on the ground to estimate the pattern of slip that occurred during the Izmit earthquake. This then used to improve computer models that predict how this deformation transferred stress to other faults and to the continuation of the North Anatolian Fault, which extends to the west past the large city of Istanbul. These models show that the Izmit earthquake further increased the already high probability of a major earthquake near Istanbul."<ref name=Lavoie1999>{{ cite book |author=Sue Lavoie |title=PIA00557: Izmit, Turkey 1999 Earthquake Interferogram |publisher=NASA/JPL |location=Pasadena, California USA |date=17 September 1999 |url=http://photojournal.jpl.nasa.gov/catalog/PIA00557 |accessdate=2016-09-27 }}</ref> "This image [on the left] of Fort Irwin in California's Mojave Desert compares interferometric radar signatures topography -- data that were obtained by multiple imaging of the same region to produce three-dimensional elevation maps -- as it was obtained on October 7-8, 1994 by the Spaceborne Imaging Radar-C/X-band Synthetic Aperture Radar aboard the space shuttle Endeavour. Data were acquired using the L-band (24 centimeter wavelength) and C-band (6 centimeter wavelength). The image covers an area about 25 kilometers by 70 kilometers (15.5 miles by 43 miles). North is to the lower right of the image. The color contours shown are proportional to the topographic elevation. With a wavelength one-fourth that of the L-band, the results from the C-band cycle through the color contours four times faster for a given elevation change. Detailed comparisons of these multiple frequency data over different terrain types will provide insights in the future into wavelength-dependent effects of penetration and scattering on the topography measurement accuracy. Fort Irwin is an ideal site for such detailed digital elevation model comparisons because a number of high precision digital models of the area already exist from conventional measurements as well as from airborne interferometric SAR data."<ref name=Lavoie1994>{{ cite book |author=Sue Lavoie |title=PIA01759: SPace Radar Image of Fort Irwin, California |publisher=NASA/JPL |location=Pasadena, California USA |date=7 October 1994 |url=http://photojournal.jpl.nasa.gov/catalog/PIA01759 |accessdate=2016-09-27 }}</ref> "Spaceborne Imaging Radar-C and X-band Synthetic Aperture Radar (SIR-C/X-SAR) is part of NASA's Mission to Planet Earth. The radars illuminate Earth with microwaves, allowing detailed observations at any time, regardless of weather or sunlight conditions. SIR-C/X-SAR uses three microwave wavelengths: L-band (24 cm), C-band (6 cm) and X-band (3 cm). The multi-frequency data will be used by the international scientific community to better understand the global environment and how it is changing. The SIR-C/X-SAR data, complemented by aircraft and ground studies, will give scientists clearer insights into those environmental changes which are caused by nature and those changes which are induced by human activity."<ref name=Lavoie1994/> "SIR-C was developed by NASA's Jet Propulsion Laboratory. X-SAR was developed by the Dornier and Alenia Spazio companies for the German space agency, Deutsche Agentur fuer Raumfahrtangelegenheiten (DARA), and the Italian space agency, Agenzia Spaziale Italiana (ASI), with the Deutsche Forschungsanstalt fuer Luft und Raumfahrt e.V.(DLR), the major partner in science, operations and data processing of X-SAR."<ref name=Lavoie1994/> {{clear}} ==Gyrotrons== "Other important applications of gyrotrons are high-power microwave sources include high resolution radar ranging and imaging in atmospheric and planetary science as well as deep-space and specialized satellite communications and RF drivers for next-generation high-gradient linear accelerators".<ref name=Ying>{{ cite journal |author=Liu Ying-hui; Li Hong-fu ; Li Hao ; Wang E-feng ; Xu Yong ; Sun Yu |title=Analysis of RF Field in Open Cavity by Mode-Matching Technique, In: ''Infrared Millimeter Waves and 14th International Conference on Teraherz Electronics'' |publisher=IEEE |location=Shanghai |month=September 18-22, |year=2006 |editor= |pages=78 |url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4222020 |arxiv= |bibcode= |doi=10.1109/ICIMW.2006.368286 |pmid= |isbn=1-4244-0400-2 |accessdate=2013-10-20 }}</ref> ==Aerial gravity gradiometry== [[Image:Tpdc-on-cgg.jpg|right|thumb|300px|Aerial high-resolution gravity gradiometry system is in combination with LIDAR digital terrain mapping, electromagnetics, digital video, and gamma-ray spectrometry. Credit: Mozambique Resources Post.{{tlx|fairuse}}]] The aircraft imaged on the right carried-out aerial high-resolution gravity gradiometry system in combination with LIDAR digital terrain mapping, electromagnetics, digital video, and gamma-ray spectrometry over "onshore areas along the South-Eastern Tanzanian Coastal Basin and the eastern arm of the East African Rift."<ref name=Post>{{ cite web |author=Mozambique Resources Post |title=Africa Oil & Gas: TPDC awards CGG airborne gravity gradiometer surveys onshore Tanzania |publisher=Mozambique Resources Post |location=Tanzania |date=23 July 2015 |url=https://mozambiqueminingpost.wordpress.com/2015/07/23/africa-oil-gas-tpdc-awards-cgg-airborne-gravity-gradiometer-surveys-onshore-tanzania/ |accessdate=2017-10-02 }}</ref> {{clear}} ==Hypotheses== {{main|Hypotheses}} # A radar frequency exists that is reflected back by the Sun. # A control group for radar astronomy would likely contain a standard target that allows maximum radar facility resolution and imaging. ==See also== {{div col|colwidth=20em}} * [[Radiation astronomy/Gravitationals|Gravitational astronomy]] * [[Radiation astronomy/Infrareds|Infrared astronomy]] * [[Radiation astronomy/Microwaves|Microwave astronomy]] * [[Radiation astronomy/Radios|Radio astronomy]] * [[Radiation astronomy/Reds|Red astronomy]] * [[Radiation astronomy/Submillimeters|Submillimeter astronomy]] {{Div col end}} ==References== {{reflist|2}} ==External links== * [http://www.iau.org/ International Astronomical Union] * [http://nedwww.ipac.caltech.edu/ NASA/IPAC Extragalactic Database - NED] * [http://nssdc.gsfc.nasa.gov/ NASA's National Space Science Data Center] * [http://www.osti.gov/ Office of Scientific & Technical Information] * [http://www.questia.com/ Questia - The Online Library of Books and Journals] * [http://online.sagepub.com/ SAGE journals online] * [http://www.adsabs.harvard.edu/ The SAO/NASA Astrophysics Data System] * [http://www.scirus.com/srsapp/advanced/index.jsp?q1= Scirus for scientific information only advanced search] * [http://cas.sdss.org/astrodr6/en/tools/quicklook/quickobj.asp SDSS Quick Look tool: SkyServer] * [http://simbad.u-strasbg.fr/simbad/ SIMBAD Astronomical Database] * [http://simbad.harvard.edu/simbad/ SIMBAD Web interface, Harvard alternate] * [http://nssdc.gsfc.nasa.gov/nmc/SpacecraftQuery.jsp Spacecraft Query at NASA] * [http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/convcoord/convcoord.pl Universal coordinate converter] <!-- footer templates --> {{tlx|Flight resouces}}{{tlx|Geology resources}}{{tlx|Materials science resources}}{{Principles of radiation astronomy}}{{tlx|Technology resources}}{{Sisterlinks|Radar astronomy}} <!-- categories --> [[Category:Astrophysics/Lectures]] [[Category:Astronomy/Lectures]] [[Category:Radiation/Lectures]] [[Category:Radiation astronomy/Lectures]] [[Category:Technology/Lectures]] lgf2jyazzrwos9jtw9b6a2wn02zkf5b 2410939 2410932 2022-08-02T07:24:25Z Marshallsumter 311529 /* Craters */ wikitext text/x-wiki [[Image:2012 LZ1.jpg|thumb|right|250px|This image is of asteroid 2012 LZ1 by the Arecibo Observatory in Puerto Rico using the Arecibo Planetary Radar. Credit: Arecibo Observatory.{{tlx|fairuse}}]] '''Radar astronomy''' is used to detect and study astronomical objects that reflect radio rays. The image at right is of asteroid 2012 LZ1 using the Arecibo Planetary Radar. '''Radar astronomy''' is a technique of observing nearby astronomical objects by reflecting microwaves off target objects and analyzing the echoes. Radar astronomy differs from [[Radiation astronomy/Radios|radio astronomy]] in that the latter is a passive observation and the former an active one. The radar transmission may either be pulsed or continuous. Radar directly measures the distance to the object (and how fast it is changing). The combination of optical and radar observations normally allows the prediction of orbits at least decades, and sometimes centuries, into the future. The maximum range of astronomy by radar is very limited, and is confined to the [[Solar System|solar system]]. This is because the signal strength drops off very steeply with distance to the target, the small fraction of incident flux that is reflected by the target, and the limited strength of transmitters.<ref name="Hey">{{ cite book | author = J.S. Hey | title = The Evolution of Radio Astronomy | publisher = Paul Elek (Scientific Books) | series = Histories of Science Series | volume = 1 | date = 1973 | location = | pages = | url = | doi = | isbn = }}</ref> It is also necessary to have a relatively good ephemeris of the target before observing it. {{clear}} ==Radars== '''Radio waves''' are a type of electromagnetic radiation with wavelengths in the electromagnetic spectrum longer than infrared light. Radio waves have frequencies from 300 [Gigahertz] GHz to as low as 3 [Kilohertz] kHz, and corresponding [to] wavelengths from 1 millimeter to 100 kilometers. {| class="wikitable" |+ '''Radar frequency bands''' |- style="background:#ccc;" !Band name!!Frequency range!!Wavelength range!!Notes |- |[[w:High frequency|HF]]||3–30 [[w:Megahertz|MHz]]||10–100 [[w:metre|m]]||Coastal radar systems, [[w:over-the-horizon radar|over-the-horizon radar]] (OTH) radars; 'high frequency' |- |[[w:VHF|VHF]]||30–300&nbsp;MHz||1–10 m||Very long range, ground penetrating; 'very high frequency' |- |P||<&nbsp;300&nbsp;MHz||>&nbsp;1&nbsp;m||'P' for 'previous', applied retrospectively to early radar systems; essentially HF + VHF |- |[[w:UHF|UHF]]||300–1000&nbsp;MHz||0.3–1 m||Very long range (e.g. [[w:Ballistic Missile Early Warning System|ballistic missile early warning]]), ground penetrating, foliage penetrating; 'ultra high frequency' |- |[[w:L band|L]]||1–2 [[w:Gigahertz|GHz]]||15–30 [[w:centimetre|cm]]||Long range air traffic control and [[w:Surveillance|surveillance]]; 'L' for 'long' |- |[[w:S band|S]]||2–4&nbsp;GHz||7.5–15&nbsp;cm||Moderate range surveillance, Terminal air traffic control, long-range weather, marine radar; 'S' for 'short' |- |[[w:C band|C]]||4–8&nbsp;GHz||3.75–7.5&nbsp;cm||Satellite transponders; a compromise (hence 'C') between X and S bands; weather; long range tracking |- |[[w:X band|X]]||8–12&nbsp;GHz||2.5–3.75&nbsp;cm||[[w:Missile|Missile]] guidance, [[w:marine radar|marine radar]], weather, medium-resolution mapping and ground surveillance; in the USA the narrow range 10.525&nbsp;GHz ±25&nbsp;MHz is used for airport radar; short range tracking. Named X band because the frequency was a secret during WW2. |- ||[[w:Ku band|K_u]]||12–18&nbsp;GHz||1.67–2.5&nbsp;cm||High-resolution, also used for satellite transponders, frequency under K band (hence 'u') |- |[[w:K band|K]]||18–24&nbsp;GHz||1.11–1.67&nbsp;cm||From [[w:German language|German]] ''kurz'', meaning 'short'; limited use due to absorption by [[w:water vapor|water vapour]], so K_u and K_a were used instead for surveillance. K-band is used for detecting clouds by meteorologists, and by police for detecting speeding motorists. K-band radar guns operate at 24.150 ± 0.100&nbsp;GHz. |- |[[w:Ka band|K_a]]||24–40&nbsp;GHz||0.75–1.11&nbsp;cm||Mapping, short range, airport surveillance; frequency just above K band (hence 'a') Photo radar, used to trigger cameras which take pictures of license plates of cars running red lights, operates at 34.300 ± 0.100&nbsp;GHz. |- |mm||40–300&nbsp;GHz||1.0–7.5&nbsp;[[w:millimetre|mm]] ||[[w:Millimetre band|Millimetre band]], subdivided as below. The frequency ranges depend on waveguide size. Multiple letters are assigned to these bands by different groups. These are from Baytron, a now defunct company that made test equipment. |- |[[w:V band|V]]||40–75&nbsp;GHz||4.0–7.5&nbsp;mm || Very strongly absorbed by atmospheric oxygen, which resonates at 60&nbsp;GHz. |- |[[w:W band|W]]||75–110&nbsp;GHz||2.7–4.0&nbsp;mm||Used as a visual sensor for experimental autonomous vehicles, high-resolution meteorological observation, and imaging. |} ==Planetary sciences== {{main|Planetary sciences}} [[Image:Topography of Mars and radar section.jpg|thumb|right|250px|A: Topography of Mars has major features identified. B: Topography shows the MRO/SHARAD ground tracks for orbits 6830 (a-a′), 7219 (b-b′), and 3672 (c-c′). LDAs crossed by these tracks are labeled. Credit: John W. Holt, Ali Safaeinili, Jeffrey J. Plaut, James W. Head, Roger J. Phillips, Roberto Seu, Scott D. Kempf, Prateek Choudhary, Duncan A. Young, Nathaniel E. Putzig, Daniela Biccari, Yonggyu Gim.]] [[Image:F3.large.jpg|thumb|left|250px|Results are for SHARAD orbit 6830 (line a-a′). A: Simulated surface echoes (clutter) are in one-way travel time. B: SHARAD data is in one-way travel time. Vertical arrows identify echoes not consistent with surface clutter simulation and also confirmed in adjacent tracks. C: Radar data is converted to depth assuming a water-ice composition. Credit: John W. Holt, Ali Safaeinili, Jeffrey J. Plaut, James W. Head, Roger J. Phillips, Roberto Seu, Scott D. Kempf, Prateek Choudhary, Duncan A. Young, Nathaniel E. Putzig, Daniela Biccari, Yonggyu Gim.]] [[Image:SHARAD orbits 7219 and 3672.jpg|thumb|right|250px|Results are for SHARAD orbits 7219 (left) and 3672 (right). Credit: John W. Holt, Ali Safaeinili, Jeffrey J. Plaut, James W. Head, Roger J. Phillips, Roberto Seu, Scott D. Kempf, Prateek Choudhary, Duncan A. Young, Nathaniel E. Putzig, Daniela Biccari, Yonggyu Gim.]] "Radar waves penetrate the surface and pass through materials that do not severely attenuate or scatter them. Reflections arise from interfaces with dielectric contrasts. [Shallow radar] SHARAD has penetrated the ∼2-km-thick polar layered deposits in both the north and south, detecting many internal reflectors (17, 18). Smaller targets can be more challenging because SHARAD's antenna pattern is broad, resulting in surface reflections up to a few tens of kilometers away from the suborbital point in rugged areas, versus only a few kilometers in smooth, flat areas. These off-nadir echoes can appear at time delays similar to those arising from subsurface interfaces, so steps are required to avoid misinterpreting this surface clutter as subsurface echoes. Synthetic-aperture data processing is used to improve along-track resolution to ∼300 m, greatly reducing along-track clutter and focusing the surface and subsurface features. We used the known topography of the surface and the radar geometry to model cross-track clutter together with nadir surface echoes [...]. Comparisons of radar sounding data with these synthetic surface echoes and the examination of possible surface echo sources in imagery (19) were undertaken for all cases [...]; such a procedure is a necessary part of radar sounding data interpretation in high-relief environments."<ref name=Holt>{{ cite journal |author=John W. Holt, Ali Safaeinili, Jeffrey J. Plaut, James W. Head, Roger J. Phillips, Roberto Seu, Scott D. Kempf, Prateek Choudhary, Duncan A. Young, Nathaniel E. Putzig, Daniela Biccari, Yonggyu Gim |title=Radar Sounding Evidence for Buried Glaciers in the Southern Mid-Latitudes of Mars |journal=Science |month=November |year=2008 |volume=322 |issue=5905 |pages=1235-8 |url=http://www.sciencemag.org/content/322/5905/1235.full |arxiv= |bibcode= |doi=10.1126/science.1164246 |pmid= |accessdate=2014-09-28 }}</ref> The "Shallow Radar (SHARAD) (15) on the Mars Reconnaissance Orbiter (MRO) to probe the internal structure of several LDAs surrounding massifs on the eastern rim of the Hellas impact basin [first image at the right] where more than 90 LDA complexes flank steep topography (2, 6, 16). The southernmost LDA we studied (LDA-2, [figure at the upper right] has multiple lobes that coalesce to form a continuous deposit extending more than 20 km outward from a massif along ∼170 km of its margins."<ref name=Holt/> "Examination of radar data from SHARAD orbit 6830 where it crosses multiple [lobate debris aprons] LDAs in the eastern Hellas region [...] shows that the only radar reflections not matching simulated surface echoes occur where the spacecraft passes over each LDA [...]; therefore, these echoes are interpreted as arising from within or beneath the LDAs. In one case (LDA-2A), surface clutter is predicted near the terminus of the LDA, where it may obscure portions of a subsurface reflector that clearly extends farther inward below the LDA. LDA-2A and LDA-2B [image at the lower right] show evidence for multiple, closely spaced subsurface reflectors indicating the presence of at least one thin (∼70 m assuming a water-ice composition), distinct deposit below thicker deposits (up to 800 m)."<ref name=Holt/> {{clear}} ==Theoretical radar astronomy== Here's a couple of [[Definitions/Theory#Theoretical definition|theoretical definition]]s: '''Def.''' the branch of astronomy that uses radar to map the surfaces of planetary bodies in the solar system is called '''radar astronomy'''. '''Def.''' reflective and observational astronomy over radio wavelengths is called '''radar astronomy'''. ==Aerometeors== {{main|Radiation astronomy/Aerometeors}} [[Image:June 2012 Radar image of Derecho.jpg|thumb|right|250px|This is a composite radar image of the June 2012 United States Derecho event. Credit: G. Carbin, NWS/Storm Prediction Center.]] On the right is a composite of hourly radar images. These wind gusts averaged ~75 mph over about 450 miles. This is referred to as the Derecho event. {{clear}} ==Opticals== {{main|Radiation astronomy/Opticals|Optical astronomy}} By [[Radiation astronomy/Opticals|optical astronomy]], optical observations measure very accurately where an object appears on the sky, but cannot measure the distance accurately at all. ==Hydromorphology== [[Image:"Quill" satellite radar image of flooded Eel River outflow current.JPG|thumb|right|250px|''Quill'' satellite radar image is shown of the flooded Eel River outflow current. Credit: [[commons:User:Oldteched|oldteched]], U. S. National Reconnaissance Office, and U. S. Geological Survey.]] On the right, the ''Quill'' satellite radar image is shown of the flooded Eel River outflow current debris field in gray. "Colored content is USGS-derived base map. Gray overlay is derived from a "Quill" satellite radar image made during the December 1964 flooding of California's Eel River. Accurate registration of the overlay onto the map is demonstrated by the excellent match of the stream-valley features in each."<ref name=Oldteched>{{ cite book |author=[[commons:User:Oldteched|oldteched]] |title=''File:"Quill" satellite radar image of flooded Eel River outflow current.JPG'' |publisher=U. S. National Reconnaissance Office |location= |date=December 1964 |url=https://commons.wikimedia.org/wiki/File:%22Quill%22_satellite_radar_image_of_flooded_Eel_River_outflow_current.JPG |accessdate=2016-09-20 }}</ref> {{clear}} ==Ice streams== [[Image:FRicestreams.jpg|thumb|right|250px|Radarsat image is of ice streams flowing into the Filchner-Ronne Ice Shelf. Credit: [[w:User:Polargeo|Polargeo]].]] On the right is a radarsat image of ice streams flowing into the Filchner-Ronne ice shelf. Annotations outline the Rutford ice stream. {{clear}} ==Glaciers== {{main|Rocks/Glaciers|Glaciers}} [[Image:Alfred Ernest Ice Shelf.jpg|thumb|right|250px|This is a radar image of Alfred Ernest Ice Shelf on Ellesmere Island, taken by the ERS-1 satellite. Credit: NASA.]] On the right is a radar image of Alfred Ernest Ice Shelf on Ellesmere Island, taken by the ERS-1 satellite. {{clear}} ==Glaciology== {{main|Rocks/Glaciers/Glaciology|Glaciology}} [[Image:A-62 iceberg connected to Fimbul Ice Shelf at Queen Maud Land.jpg|thumb|right|250px|Iceberg A 62 was connected to the Fimbul Ice Shelf by a mere 800-metre-wide bridge. Credit: DLR - German Space Agency.]] "A small island obstructs the constant flow of the ice shelf on Queen Maud Land – it is the lighter area at the bottom left of the image [on the right]. From September 2010 until it broke off, Iceberg A 62 was connected to the Fimbul Ice Shelf by a mere 800-metre-wide bridge. Two fissures in the ice from different sides of the bridge approached one another until the break occurred. The images transmitted by the radar satellite TerraSAR-X over a long period of time should enable researchers to achieve a better understanding of how icebergs calve."<ref name=DLR>{{ cite book |author=DLR |title=The iceberg breaks free |publisher=Deutsches Zentrum für Luft- und Raumfahrt |location=Cologne |date=20 January 2011 |url=http://www.dlr.de/media/en/desktopdefault.aspx/tabid-4986/8423_read-18675/8423_page-2 |accessdate=2016-09-20 }}</ref> {{clear}} ==Craters== {{main|Radiation astronomy/Craters}} [[Image:Haughton impact crater radar image.jpg|thumb|right|250px|Synthetic aperture radar image is of the Haughton impact crater. Credit: [[w:User:Denni|Denni]].{{tlx|free media}}]] [[Image:Iturralde Crater PIA03359 cropped.jpg|thumb|left|250px|A topographic map from Shuttle Radar Topography Mission of Iturralde Crater. Credit: NASA/JPL STS-97 (STS-99) crew.{{tlx|free media}}]] [[Image:Lost Hills Subsidence interferogram.jpg|thumb|right|250px|This figure shows a comparison of interferograms from four different years mapping the rapid ground subsidence over the Lost Hills oil field in California. Credit: NASA/JPL-Caltech.{{tlx|free media}}]] On the right is a synthetic aperture radar image of the Haughton impact crater on Devon Island, Nunavut, Canada. On the left is a topographic map from Shuttle Radar Topography Mission data of Iturralde Crater, an unconfirmed impact crater in Bolivia. "This figure shows a comparison of interferograms from four different years mapping the rapid ground subsidence over the Lost Hills oil field in California. Lost Hills is located about 60 km (40 miles) northwest of Bakersfield in the San Joaquin Valley. The oil field is about 1.5 km (1mile) wide and 6 km (3.5 miles) long."<ref name=Lavoie2001>{{ cite book |author=Sue Lavoie |title=PIA03458: Lost Hills, California Interferogram |publisher=NASA/JPL |location=Pasadena, California USA |date=17 October 2001 |url=http://photojournal.jpl.nasa.gov/catalog/PIA03458 |accessdate=2016-09-27 }}</ref> "Each interferogram was created using pairs of images taken by synthetic aperture radar that have been combined to measure surface deformation or changes that may have occurred in the time between when data for the two images were taken. The images were collected by the European Space Agency's Remote Sensing satellites (ERS-1 and ERS-2) in two months of each year shown (1995, 1996, 1998 and 1999) and were combined to produce these image maps of the apparent surface deformation, or changes."<ref name=Lavoie2001/> "The interferometric measurements that show the changes, primarily vertical subsidence of the surface, are rendered in color with purple indicating no motion and the brightest red showing rapid subsidence. The white areas are where the radar measurements could not be obtained, mostly in the agricultural fields around the oil fields where plant growth or plowing altered the radar properties of the surface."<ref name=Lavoie2001/> "These radar data show that parts of the oil field were subsiding unusually rapidly, more than 3 centimeters (1.2 inches) a month, in 1995 and 1996. They also reveal that while the ground subsidence rate decreased in the center part of the oil field, it increased in the northern part between 1995 and 1996 and 1998 and 1999."<ref name=Lavoie2001/> {{clear}} ==Active faults== [[Image:Los Angeles JPLLandsat.jpg|thumb|right|250px|This perspective image of Los Angeles and the Transverse Ranges is from NASA's Shuttle Radar Topography Mission (SRTM). Credit: NASA/JPL/NIMA.]] "Earthquake faults commonly lie between the mountains and the lowlands. The San Andreas fault, the largest fault in California, likewise divides the very rugged San Gabriel Mountains from the low-relief Mojave Desert, thus forming a straight topographic boundary between the top center and lower right corner of the image. We present two versions of this perspective image from NASA's Shuttle Radar Topography Mission (SRTM): one with and one without a graphic overlay that maps faults that have been active in Late Quaternary times (white lines). The fault database was provided by the U.S. Geological Survey."<ref name=Lavoie2002>{{ cite book |author=Sue Lavoie |title=PIA03376: Perspective View, SRTM / Landsat, Los Angeles, Calif |publisher=NASA/JPL |location=Pasadena, California USA |date=23 December 2002 |url=http://photojournal.jpl.nasa.gov/catalog/PIA03376 |accessdate=2016-09-27 }}</ref> "The Landsat image used here was acquired on May 4, 2001, about seven weeks before the summer solstice, so natural terrain shading is not particularly strong. It is also not especially apparent given a view direction (northwest) nearly parallel to the sun illumination (shadows generally fall on the backsides of mountains). Consequently, topographic shading derived from the SRTM elevation model was added to the Landsat image, with a false sun illumination from the left (southwest). This synthetic shading enhances the appearance of the topography."<ref name=Lavoie2002/> "Landsat has been providing visible and infrared views of the Earth since 1972. SRTM elevation data matches the 30-meter (98-foot) resolution of most Landsat images and substantially helps in analyzing the large and growing Landsat image archive. This Landsat 7 Thematic Mapper image was provided to the SRTM project by the United States Geological Survey, Earth Resources Observation Systems (EROS) Data Center, Sioux Falls, S.D."<ref name=Lavoie2002/> "Elevation data used in this image was acquired by the SRTM aboard the Space Shuttle Endeavour, launched on Feb. 11, 2000. SRTM used the same radar instrument that comprised the Spaceborne Imaging Radar-C/X-Band Synthetic Aperture Radar (SIR-C/X-SAR) that flew twice on the Space Shuttle Endeavour in 1994. SRTM was designed to collect 3-D measurements of the Earth's surface. To collect the 3-D data, engineers added a 60-meter (approximately 200-foot) mast, installed additional C-band and X-band antennas, and improved tracking and navigation devices. The mission is a cooperative project between NASA, the National Imagery and Mapping Agency (NIMA) of the U.S. Department of Defense and the German and Italian space agencies."<ref name=Lavoie2002/> "Size: View width 134 kilometers (83 miles); view distance 150 kilometers (93 miles) Location: 34.3 degrees North latitude, 118.4 degrees West longitude Orientation: View west-northwest, 1.8 X vertical exaggeration Image Data: Landsat Bands 3, 2+4, 1 as red, green, blue, respectively Original Data Resolution: SRTM 1 arcsecond (30 meters or 98 feet), Landsat 30 meters (98 feet) Graphic Data: earthquake faults active in Late Quaternary times Date Acquired: February 2000 (SRTM), May 4, 2001 (Landsat)."<ref name=Lavoie2002/> {{clear}} ==Transform faults== [[Image:Alpine Fault SRTM.jpg|thumb|center|400px|Shaded and colored image from the Shuttle Radar Topography Mission elevation model of New Zealand's Alpine Fault, Northwest is towards the top. Credit: NASA/JPL/NGA.]] "The Alpine fault runs parallel to, and just inland of, much of the west coast of New Zealand's South Island. This view was created from the near-global digital elevation model produced by the Shuttle Radar Topography Mission (SRTM) and is almost 500 kilometers (just over 300 miles) wide. Northwest is toward the top. The fault is extremely distinct in the topographic pattern, nearly slicing this scene in half lengthwise."<ref name=NASAJPLNGA>{{ cite book |author=NASA/JPL/NGA |title=PIA06661: Alpine Fault, New Zealand, SRTM Shaded Relief and Colored Height |publisher=NASA/JPL/NGA |location=Pasadena, California USA |date=6 January 2005 |url=http://photojournal.jpl.nasa.gov/catalog/PIA06661 |accessdate=2014-12-02 }}</ref> "In a regional context, the Alpine fault is part of a system of faults that connects a west dipping subduction zone to the northeast with an east dipping subduction zone to the southwest, both of which occur along the juncture of the Indo-Australian and Pacific tectonic plates. Thus, the fault itself constitutes the major surface manifestation of the plate boundary here. Offsets of streams and ridges evident in the field, and in this view of SRTM data, indicate right-lateral fault motion. But convergence also occurs across the fault, and this causes the continued uplift of the Southern Alps, New Zealand's largest mountain range, along the southeast side of the fault."<ref name=NASAJPLNGA/> "Two visualization methods were combined to produce this image: shading and color coding of topographic height. The shade image was derived by computing topographic slope in the northwest-southeast (image top to bottom) direction, so that northwest slopes appear bright and southeast slopes appear dark. Color coding is directly related to topographic height, with green at the lower elevations, rising through yellow and tan, to white at the highest elevations."<ref name=NASAJPLNGA/> "Elevation data used in this image were acquired by the Shuttle Radar Topography Mission aboard the Space Shuttle Endeavour, launched on Feb. 11, 2000. SRTM used the same radar instrument that comprised the Spaceborne Imaging Radar-C/X-Band Synthetic Aperture Radar (SIR-C/X-SAR) that flew twice on the Space Shuttle Endeavour in 1994. SRTM was designed to collect 3-D measurements of the Earth's surface. To collect the 3-D data, engineers added a 60-meter (approximately 200-foot) mast, installed additional C-band and X-band antennas, and improved tracking and navigation devices."<ref name=NASAJPLNGA/> The Alpine fault is "495 kilometers (307 miles) by 162 kilometers (100 miles). Location: 43.2 degrees South latitude, 170.5 degrees East longitude. Orientation: Northwest toward the top".<ref name=NASAJPLNGA/> {{clear}} ==Volcanoes== [[Image:Ambae island 3D pic.jpg|thumb|right|250px|Ambae island, Vanuatu, image was acquired by the Shuttle Radar Topography Mission on 2000-02-11 aboard the shuttle Endeavour. Credit: NASA/JPL/NGA.]] [[Image:Klyuchevskaya sopka rad.jpg|thumb|left|250px|This image was taken using space radar of Kliuchevskoi Volcano, Russia. Credit: NASA/JPL-Caltech.]] "The recently active volcano Mt. Manaro is the dominant feature in this shaded relief image [on the right] of Ambae Island, part of the Vanuatu archipelago located 1400 miles northeast of Sydney, Australia. About 5000 inhabitants, half the island's population, were evacuated in early December from the path of a possible lahar, or mud flow, when the volcano started spewing clouds of steam and toxic gases 10,000 feet into the atmosphere."<ref name=Lavoie2000>{{ cite web |author=Sue Lavoie |title=PIA06675: Ambae Island, Vanuatu (South Pacific) |publisher=NASA/JPL |location=Pasadena, California USA |date=February 2000 |url=http://photojournal.jpl.nasa.gov/catalog/PIA06675 |accessdate=2016-09-20 }}</ref> "Last active in 1996, the 1496 meter (4908 ft.) high Hawaiian-style basaltic shield volcano features two lakes within its summit caldera, or crater. The ash and gas plume is actually emerging from a vent at the center of Lake Voui (at left), which was formed approximately 425 years ago after an explosive eruption."<ref name=Lavoie2000/> "Two visualization methods were combined to produce the image: shading and color coding of topographic height. The shade image was derived by computing topographic slope in the northwest-southeast direction, so that northwest slopes appear bright and southeast slopes appear dark. Color coding is directly related to topographic height, with green at the lower elevations, rising through yellow and tan, to white at the highest elevations."<ref name=Lavoie2000/> "Elevation data used in this image were acquired by the Shuttle Radar Topography Mission [SRTM] aboard the Space Shuttle Endeavour, launched on Feb. 11, 2000. SRTM used the same radar instrument that comprised the Spaceborne Imaging Radar-C/X-Band Synthetic Aperture Radar (SIR-C/X-SAR) that flew twice on the Space Shuttle Endeavour in 1994. SRTM was designed to collect 3-D measurements of the Earth's surface. To collect the 3-D data, engineers added a 60-meter (approximately 200-foot) mast, installed additional C-band and X-band antennas, and improved tracking and navigation devices."<ref name=Lavoie2000/> "Location: 15.4 degree south latitude, 167.9 degrees east longitude; Orientation: North toward the top, Mercator projection; Size: 36.8 by 27.8 kilometers (22.9 by 17.3 miles); Image Data [is a] shaded and colored SRTM elevation model"<ref name=Lavoie2000/> On the left is a space radar image of Klyuchevskaya sopka. "This is an image of the area of the Kliuchevskoi volcano, Kamchatka, Russia, which began to erupt on September 30, 1994. Kliuchevskoi is the blue triangular peak in the center of the image, towards the left edge of the bright red area that delineates bare snow cover. The image was acquired by the Spaceborne Imaging Radar-C/X-Band Synthetic Aperture Radar (SIR-C/X-SAR) aboard the space shuttle Endeavour on its 88th orbit on October 5, 1994. The image shows an area approximately 75 kilometers by 100 kilometers (46 miles by 62 miles) that is centered at 56.07 degrees north latitude and 160.84 degrees east longitude. North is toward the bottom of the image. The radar illumination is from the top of the image. The Kamchatka volcanoes are among the most active volcanoes in the world. The volcanic zone sits above a tectonic plate boundary, where the Pacific plate is sinking beneath the northeast edge of the Eurasian plate. The Endeavour crew obtained dramatic video and photographic images of this region during the eruption [...]. The colors in this image were obtained using the following radar channels: red represents the L-band, HH (horizontally transmitted and received) channel; green represents the L-band, HV (horizontally transmitted and vertically received) channel; blue represents the C-band, HV (horizontally transmitted and vertically received) channel. In addition to Kliuchevskoi, two other active volcanoes are visible in the image. Bezymianny, the circular crater above and to the right of Kliuchevskoi, contains a slowly growing lava dome. Tolbachik is the large volcano with a dark summit crater near the upper right edge of the red snow covered area. The Kamchatka River runs from right to left across the bottom of the image. The 1994 eruption of Kliuchevskoi included massive ejections of gas, vapor and ash, which reached altitudes of 15,000 meters (50,000 feet). Melting snow mixed with volcanic ash triggered mud flows on the flanks of the volcano. Paths of these flows can be seen as thin lines in various shades of blue and green on the north flank in the center of the image."<ref name=Richardson>{{ cite web |author=Annie Richardson and Al Wong |title=Photograph Identification: P-44823 |publisher=NASA/JPL |location=Pasadena, California USA |date=5 October 1994 |url=http://southport.jpl.nasa.gov/cdrom/sirced03/cdrom/DATA/LOCATION/ASIAMIDE/KLIUCHEV/KLIUCHE2.HTM |accessdate=2016-09-27 }}</ref> {{clear}} ==Mercury== {{main|Rocks/Rocky objects/Mercury}} [[Image:Permanently Shadowed Polar Craters.jpg|thumb|right|250px|This is a radar image of Mercury's north polar region acquired by the Arecibo Observatory in 1991 combined with images from MESSENGER. Credit: NASA/Johns Hopkins University Applied Physics Laboratory/Carnegie Institution of Washington/National Astronomy and Ionosphere Center, Arecibo Observatory.]] [[Image:Merc fig2sm.jpg|right|thumb|250px|This is a radar image of Mercury's north pole. Credit: NASA.]] Radar astronomy of Mercury improved the value for the distance from the earth including the rotational period, libration,[and surface mapping, especially of the polar regions. In "1991, [...] the Arecibo radio telescope in Puerto Rico detected unusually radar-bright patches at Mercury's poles, spots that reflected radio waves in the way one would expect if there were water ice [denoted in the image at right by yellow areas]. Many of these patches corresponded to the location of large impact craters mapped by the Mariner 10 spacecraft in the 1970s."<ref name=Dunbar>{{ cite book |author=Brian Dunbar |title=MESSENGER Finds New Evidence for Water Ice at Mercury's Poles |publisher=NASA |location=Washington, DC USA |date=November 29, 2012 |url=http://www.nasa.gov/mission_pages/messenger/media/PressConf20121129.html |accessdate=2013-10-24 }}</ref> "Images from the spacecraft's Mercury Dual Imaging System taken in 2011 and earlier this year confirmed that radar-bright features at Mercury's north and south poles are within shadowed regions on Mercury's surface, findings that are consistent with the water-ice hypothesis."<ref name=Dunbar/> "The neutron data indicate that Mercury's radar-bright polar deposits contain, on average, a hydrogen-rich layer more than tens of centimeters thick beneath a surficial layer 10 to 20 centimeters thick that is less rich in hydrogen".<ref name=Lawrence>{{ cite book |author=David Lawrence |title=MESSENGER Finds New Evidence for Water Ice at Mercury's Poles |publisher=NASA |location=Washington, DC USA |date=November 29, 2012 |url=http://www.nasa.gov/mission_pages/messenger/media/PressConf20121129.html |accessdate=2013-10-24 }}</ref> "The buried layer has a hydrogen content consistent with nearly pure water ice."<ref name=Lawrence/> "These reflectance anomalies are concentrated on poleward-facing slopes and are spatially collocated with areas of high radar backscatter postulated to be the result of near-surface water ice".<ref name=Neumann>{{ cite book |author=Gregory Neumann |title=MESSENGER Finds New Evidence for Water Ice at Mercury's Poles |publisher=NASA |location=Washington, DC USA |date=November 29, 2012 |url=http://www.nasa.gov/mission_pages/messenger/media/PressConf20121129.html |accessdate=2013-10-24 }}</ref> "Correlation of observed reflectance with modeled temperatures indicates that the optically bright regions are consistent with surface water ice."<ref name=Neumann/> MESSENGER's Mercury Laser Altimeter (MLA) data "show that the spatial distribution of regions of high radar backscatter is well matched by the predicted distribution of thermally stable water ice".<ref name=Paige>{{ cite book |author=David Paige |title=MESSENGER Finds New Evidence for Water Ice at Mercury's Poles |publisher=NASA |location=Washington, DC USA |date=November 29, 2012 |url=http://www.nasa.gov/mission_pages/messenger/media/PressConf20121129.html |accessdate=2013-10-24 }}</ref> Water ice strongly reflects radar, and observations by the 70 m [Goldstone Deep Space Communications Complex] Goldstone telescope and the [Very Large Array] VLA in the early 1990s revealed that there are patches of very high radar reflection near the poles.<ref name=Slade>{{ cite journal | author=M. A. Slade, B. J. Butler, D. O. Muhleman | year=1992 | title=Mercury radar imaging – Evidence for polar ice | journal=Science | volume=258 | issue=5082 | pages=635–640 | doi=10.1126/science.258.5082.635 | pmid=17748898 |bibcode = 1992Sci...258..635S }}</ref> While ice is not the only possible cause of these reflective regions, astronomers believe it is the most likely.<ref name=Williams>{{ cite book | author=David R. Williams | date=June 2, 2005 | url=http://nssdc.gsfc.nasa.gov/planetary/ice/ice_mercury.html | title=Ice on Mercury | publisher=NASA Goddard Space Flight Center | accessdate=2008-05-23 }}</ref> {{clear}} ==Venus== [[Image:Venus2 mag big.png|thumb|right|250px|Using an imaging radar technique, the Magellan spacecraft was able to lift the veil from the face of Venus and produce this spectacular high resolution image of the planet's surface. Red, in this false-color map, represents mountains, while blue represents valleys. Credit: Magellan Team, JPL, NASA.]] [[Image:Venus globe.jpg|thumb|left|250px|This is a false color image of Venus produced from a global radar view of the surface by the Magellan probe while radar imaging between 1990-1994. Credit: NASA.]] The first un-ambiguous detection of Venus was made by the Jet Propulsion Laboratory (JPL) on 10 March 1961. A correct measurement of the AU soon followed. "The advantages of radar in planetary astronomy result from (1) the observer's control of all the attributes of the coherent signal used to illuminate the target, especially the wave form's time/frequency modulation and polarization; (2) the ability of radar to resolve objects spatially via measurements of the distribution of echo power in time delay and Doppler frequency; (3) the pronounced degree to which delay-Doppler measurements constrain orbits and spin vectors; and (4) centimeter-to-meter wavelengths, which easily penetrate optically opaque planetary clouds and cometary comae, permit investigation of near-surface macrostructure and bulk density, and are sensitive to high concentrations of metal or, in certain situations, ice."<ref name=Ostro>{{ cite journal |author=Steven J. Ostro |title=Planetary radar astronomy |journal=Reviews of Modern Physics |month=October-December |year=1993 |volume=65 |issue=4 |pages=1235-79 |url=http://rmp.aps.org/abstract/RMP/v65/i4/p1235_1 |arxiv= |bibcode= |doi=10.1103/RevModPhys.65.1235 |pmid= |accessdate=2012-02-09 }}</ref> The radar image at left shows that just beneath the cloud layers is a rocky object. {{clear}} ==Earth== {{main|Rocks/Rocky objects/Earth}} [[Image:Aorounga crater.png|thumb|right|250px|The concentric ring structure is the Aorounga impact crater. Credit: NASA.]] [[Image:Antarctica2.jpg|thumb|right|250px|This is a shaded relief map of Antarctica developed from RADARSAT Synthetic Aperture Radar data. Credit: U.S. National Oceanic & Atmospheric Administration.]] Numerous airborne and spacecraft radars have mapped the entire planet, for various purposes. One example is the Shuttle Radar Topography Mission, which mapped the entire Earth at 30 m resolution. "The impact of an asteroid or comet several hundred million years ago left scars in the landscape that are still visible in this spaceborne radar image [on the right] of an area in the Sahara Desert of northern Chad. The concentric ring structure is the Aorounga impact crater, with a diameter of about 17 kilometers (10.5 miles). The original crater was buried by sediments, which were then partially eroded to reveal the current ring-like appearance. The dark streaks are deposits of windblown sand that migrate along valleys cut by thousands of years of wind erosion. The dark band in the upper right of the image is a portion of a proposed second crater."<ref name=Wong/> "Radar imaging is a valuable tool for the study of desert regions because the radar waves can penetrate thin layers of dry sand to reveal details of geologic structure that are invisible to other sensors. The image was acquired by the Spaceborne Imaging Radar-C/X-band Synthetic Aperture Radar (SIR-C/X-SAR) on April 18 and 19, 1994, onboard the space shuttle Endeavour. The area shown is 22 kilometers by 28 kilometers (14 miles by 17 miles) and is centered at 19.1 degrees north latitude, 19.3 degrees east longitude. North is toward the upper right. The colors are assigned to different radar frequencies and polarizations as follows: red is L-band, horizontally transmitted and received; green is C-band, horizontally transmitted and received; and blue is C-band, horizontally transmitted, vertically received."<ref name=Wong>{{ cite book |author=Al Wong |title=Aorounga Impact Crater, Chad |publisher=NASA/JPL |location=Pasadena, California, USA |date=20 March 1996 |url=http://southport.jpl.nasa.gov/cdrom/sirced03/cdrom/DATA/LOCATION/AFRICA/SAHARA/POSTSIRC/CHAD.HTM |accessdate=2016-09-15 }}</ref> At right is a shaded relief map of Antarctica developed from RADARSAT Synthetic Aperture Radar data. RADARSAT is a Canadian satellite. {{clear}} ==Moon== [[Image:Moon South Pole.jpg|thumb|left|250px|alt=Twenty degrees of latitude of the Moon's disk, completely covered in the overlapping circles of craters. The illumination angles are from all directions, keeping almost all the crater floors in sunlight, but a set of merged crater floors right at the south pole are completely shadowed.|Mosaic image of the lunar south pole as taken by ''Clementine'': note permanent polar shadow. Credit: NASA/JPL-Caltech.]] [[Image:The Moon's North Pole.jpg|thumb|right|250px|The Moon's north pole is shown during the summer. Credit: NASA/GSFC/Arizona State University.]] [[Image:Nsmoonpole.s.jpeg|thumb|left|250px|This is a radar image of the Moon's south pole made with the Arecibo Planetary Radar. Credit: N. J. Stacy.]] [[Image:Radar Image of the Lunar South Pole from Jean-Luc Margot's PhD thesis.jpg|thumb|right|250px|Goldstone radar image of the craters at the Moon's south pole. Credit: [[commons:User:JeanLucMargot|JeanLucMargot]].]] [[Image:GRAIL's gravity map of the moon.jpg|thumb|right|250px|This image shows the variations in the lunar gravity field as measured by NASA's Gravity Recovery and Interior Laboratory (GRAIL) during the primary mapping mission from March to May 2012. Credit: NASA/JPL-Caltech/MIT/GSFC.]] [[Image:Moon at 850 microns.gif|thumb|right|250px|Thermal emission from the Moon is shown at 850 microns. Credit: NRAO.]] [[Image:Moon at 70 cm radar.jpg|thumb|right|250px|This is an Arecibo/GBT λ = 70 cm bistatic radar image of the lunar north pole. Credit: Arecibo/GBT.]] The moon is comparatively close and was detected by radar, soon after the invention of the technique, in 1946.<ref name=Mofensen>{{cite journal |author=J. Mofensen |title=Radar Echoes from the Moon |journal=Nature, Electronics |month=February, April |year=1946 |volume=157, 19 |issue=3379 |pages=129, 92-8 |bibcode=1946Natur.157R.129. |doi=10.1038/157129b0 |url= }}</ref><ref>Z. Bay, "Reflection of microwaves from the moon," Hung. Acta Phys., vol. 1, pp. 1-22; April, 1946.</ref> Measurements included surface roughness and later mapping of shadowed regions near the poles. "Clementine orbited the Moon in 1994 for 71 days, mapping the Moon globally in 11 wavelengths and measuring its topography by laser ranging. [... The] bistatic radar experiment (so-called because the spacecraft transmitted while we listened to the echoes on Earth) found evidence in the dark areas near the south pole of the Moon for material with high circular polarization ratio [CPR]".<ref name=Spudis>{{ cite book | author=P. Spudis |title = Ice on the Moon |url = http://www.thespacereview.com/article/740/1 |publisher = The Space Review |date = November 6, 2006 |accessdate =12 April 2007 }}</ref> "Meanwhile, astronomers on Earth began publishing results questioning the Clementine and Lunar Prospector [1998-2000] results. With the giant Arecibo radiotelescope, radar images were taken from the Earth. They found radar reflections with high CPR lying in both permanent darkness and in sunlit areas. Ice is not stable in sunlight, so they postulated that all high CPR is caused by surface roughness; if any ice is at the lunar poles, it must be in a finely disseminated form, invisible to radar mapping."<ref name=Spudis/> The experiment from Clemintine "was bistatic, i.e., the transmitter and receiver were in different places. Bistatic radar has the advantage of observing reflections through the phase angle, the angle between transmitted and received radio rays [...]. This phase dependence is important. It’s similar to the effect one gets from looking at a bicycle reflector at just the right angle: at certain angles, the internal planes in the transparent plastic align and a very bright reflection is seen. Similarly, in both radio and visible wavelengths on the Moon, we see an “opposition surge”, an apparent increase in brightness looking directly down from the sun (zero phase). Clementine orbited the Moon such that we could observe its phase dependence [...] and we specifically looked for this “opposition surge”, called the Coherent Backscatter Opposition Effect (CBOE). CBOE is particularly valuable to identify ice on planetary surfaces."<ref name=Spudis/> "Clementine transmitted right circular polarized (RCP) radio and we listened on Earth in both right- and left-circular polarized (LCP) channels. The ratio of power received in these two channels is called the circular polarization ratio (CPR). The dry, equatorial Moon has CPR less than one, but the icy satellites of Jupiter all have CPR greater than one. We know these objects have surfaces of water ice; in this case, the ice acts as a radio-transparent media in which waves penetrate the ice, are scattered and reflected multiple times, and returned such that some of the waves are received in the same polarization sense as they are sent—they have CPR greater than unity"<ref name=Spudis/> "The problem with CPR alone is that we can also get high values from very rough surfaces, such as a rough, blocky lava flow, which has angles that form many small corner reflectors. In this case, a radio wave could hit a rock face (changing RCP into LCP) and then bounce over to another rock face (changing the LCP back into RCP) and hence to the receiver [...]. This “double-bounce” effect also creates high CPR in that “same sense” reflections could mimic the enhanced CPR one gets from ice targets."<ref name=Spudis/> At lower right is an image using the Goldstone DSS-14 antenna as a transmitter and the DSS-13 as a receiver, a form of radar interferometry. The cross for the south pole in the Arecibo image is in the Shackleton crater of the Goldstone image. "Very precise microwave measurements between two spacecraft, named Ebb and Flow, were used to map gravity with high precision and high spatial resolution. The field shown resolves blocks on the surface of about 12 miles (20 kilometres) and measurements are three to five orders of magnitude improved over previous data. Red corresponds to mass excesses and blue corresponds to mass deficiencies. The map shows more small-scale detail on the far side of the moon compared to the nearside because the far side has many more small craters."<ref name=Greicius>{{ cite web |author=Tony Greicius |title=GRAIL's Gravity Map of the Moon |publisher=NASA/JPL-Caltech/MIT/GSFC |location= |date=December 6, 2012 |url=http://www.nasa.gov/mission_pages/grail/multimedia/zuber4.html |accessdate=2012-12-15 }}</ref> "Twin NASA probes orbiting Earth's moon have generated the highest resolution gravity field map of any celestial body. The new map, created by the Gravity Recovery and Interior Laboratory (GRAIL) mission, is allowing scientists to learn about the moon's internal structure and composition in unprecedented detail. Data from the two washing machine-sized spacecraft also will provide a better understanding of how Earth and other rocky planets in the solar system formed and evolved."<ref name=Greicius/> "The gravity field map reveals an abundance of features never before seen in detail, such as tectonic structures, volcanic landforms, basin rings, crater central peaks and numerous simple, bowl-shaped craters. Data also show the moon's gravity field is unlike that of any terrestrial planet in our solar system."<ref name=PRNewswire>{{ cite book |author=PR Newswire |title=NASA Twin Spacecraft Create Most Accurate Gravity Map Of Moon |publisher= |location= |date=December 5, 2012 |url=http://news.yahoo.com/nasa-twin-spacecraft-create-most-accurate-gravity-map-173600085.html |accessdate=2012-12-15 }}</ref> ""What this map tells us is that more than any other celestial body we know of, the moon wears its gravity field on its sleeve," said GRAIL Principal Investigator Maria Zuber of the Massachusetts Institute of Technology in Cambridge. "When we see a notable change in the gravity field, we can sync up this change with surface topography features such as craters, rilles or mountains.""<ref name=PRNewswire/> "According to Zuber, the moon's gravity field preserves the record of impact bombardment that characterized all terrestrial planetary bodies and reveals evidence for fracturing of the interior extending to the deep crust and possibly the mantle. This impact record is preserved, and now precisely measured, on the moon. The probes revealed the bulk density of the moon's highland crust is substantially lower than generally assumed. This low-bulk crustal density agrees well with data obtained during the final Apollo lunar missions in the early 1970s, indicating that local samples returned by astronauts are indicative of global processes."<ref name=PRNewswire/> ""With our new crustal bulk density determination, we find that the average thickness of the moon's crust is between 21 and 27 miles (34 and 43 kilometres), which is about 6 to 12 miles (10 to 20 kilometres) thinner than previously thought," said Mark Wieczorek, GRAIL co-investigator at the Institut de Physique du Globe de Paris. "With this crustal thickness, the bulk composition of the moon is similar to that of Earth. This supports models where the moon is derived from Earth materials that were ejected during a giant impact event early in solar system history.""<ref name=PRNewswire/> "The map was created by the spacecraft transmitting radio signals to define precisely the distance between them as they orbit the moon in formation. As they fly over areas of greater and lesser gravity caused by visible features, such as mountains and craters, and masses hidden beneath the lunar surface, the distance between the two spacecraft will change slightly."<ref name=PRNewswire/> ""We used gradients of the gravity field in order to highlight smaller and narrower structures than could be seen in previous datasets," said Jeff Andrews-Hanna, a GRAIL guest scientist with the Colorado School of Mines in Golden. "This data revealed a population of long, linear gravity anomalies, with lengths of hundreds of kilometres, crisscrossing the surface. These linear gravity anomalies indicate the presence of dikes, or long, thin, vertical bodies of solidified magma in the subsurface. The dikes are among the oldest features on the moon, and understanding them will tell us about its early history.""<ref name=PRNewswire/> At fourth right is an image of the Moon using its thermal emission at 850 microns. "The Moon and planets are not detectable by reflected solar radiation at radio wavelengths. However, they all emit thermal radiation, and Jupiter is a strong nonthermal source as well. If the Sun were suddenly switched off, the planets would remain radio sources for a long time, slowly fading as they cooled. At first glance, the λ = 0.85 mm radio image of the Moon [at second right] looks familiar, but there are differences from the visible Moon."<ref name=Djorgovski>{{ cite book |author=S.G. Djorgovski |display-authors=etal |title=A Tour of the Radio Universe |publisher=National Radio Astronomy Observatory |location= |date= |url=http://www.cv.nrao.edu/course/astr534/Tour.html |accessdate=2014-03-16 }}</ref> "The darker right edge of the Moon is not being illuminated by the Sun, but it still emits radio waves because it does not cool to absolute zero during the lunar night. A subtler point is that the radio emission is not produced at the visible surface; it emerges from a layer about ten wavelengths thick. As a result, monthly temperature variations of the Moon decrease with increasing wavelength. These wavelength-dependent temperature variations encode information about the conductivity and heat capacity of the rocky and dusty outer layers of the Moon."<ref name=Djorgovski/> "Radar images like the one [at fifth right] were recently used to search for water ice trapped in cold craters near the lunar poles."<ref name=Djorgovski/> The ESA Lunar Lander Mission ''Lunar Dust Environment and Plasma Package'': "Observe radio spectrum (with an additional goal to prepare for future radiation astronomy activities.)"<ref name=Thomsen>{{ cite journal |author=B. Gardini |title=ESA strategy for human exploration and the Lunar Lander Mission |journal=Memorie della Societa Astronomica Italiana |date=2011 |volume=82 |issue= |pages=422-29 |url=http://adsabs.harvard.edu/full/2011MmSAI..82..422G |arxiv= |bibcode=2011MmSAI..82..422G |doi= |pmid= |accessdate=5 May 2019 }}</ref> {{clear}} ==Mars== [[Image:PIA13164 North Polar Cap Cross Section, Annotated Version.jpg|thumb|right|300px|This image shows a cross-section of a portion of the north polar ice cap of Mars. Credit: NASA.]] [[Image:Martian Ionosphere MARSIS release 4A.jpg|thumb|left|250px|This plot, called an ionogram, shows data from sounding Mars' ionosphere with the Mars Advanced Radar for Subsurface and Ionospheric Sounding (MARSIS). Credit: ASI/NASA/ESA/Univ. of Rome/Univ. of Iowa/JPL.]] [[Image:PIA09076.jpg|thumb|right|250px|This image is a SHARAD subsurface radargram of Mars. Credit: NASA/JPL-Caltech/ASI/University of Rome/Washington Universtiy in St. Louis.]] Mapping of surface roughness [has been performed] from Arecibo Observatory. The Mars Express mission carries a ground-penetrating radar. The image at right "shows a cross-section of a portion of the north polar ice cap of Mars, derived from data acquired by the Mars Reconnaissance Orbiter's Shallow Radar (SHARAD), one of six instruments on the spacecraft. The data depict the region's internal ice structure, with annotations describing different layers. The ice depicted in this graphic is approximately 2 kilometers (1.2 miles) thick and 250 kilometers (155 miles) across. White lines show reflection of the radar signal back to the spacecraft. Each line represents a place where a layer sits on top of another. Scientists study how thick the pancake-like layers are, where they bulge and how they tilt up or down to understand what the surface of the ice sheet was like in the past as each new layer was deposited."<ref name=BatteryIncluded>{{ cite book |author=[[commons:User:BatteryIncluded|BatteryIncluded]] |title=''File:PIA13164 North Polar Cap Cross Section, Annotated Version.jpg'' |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=June 14, 2013 |url=https://commons.wikimedia.org/wiki/File:PIA13164_North_Polar_Cap_Cross_Section,_Annotated_Version.jpg |accessdate=2013-10-24 }}</ref> The image at left, "called an ionogram, shows data from sounding Mars' ionosphere with the Mars Advanced Radar for Subsurface and Ionospheric Sounding (MARSIS). The horizontal axis is the frequency of the pulse. The left vertical axis is the time delay after transmitting the pulse, with time increasing downward. The right vertical axis is a conversion of time delay to distance, showing the apparent range to the reflection point. The intensity of the received signal at any given frequency and apparent range is indicated by the color, with dark blue being the least intense and green being the most intense."<ref name=Thompson>{{ cite book |author=Thomas Thompson |title=Radar Ionogram with Oblique Ionospheric Echo |publisher=NASA |location=Pasadena, California USA |date=September 2, 2013 |url=http://mars.jpl.nasa.gov/express/gallery/martianterrain/MARSIS_release_4.html |accessdate=2013-10-24 }}</ref> "The green echo at an apparent range of about 800 kilometers (497 miles) from 2.5 to 5.5 megahertz is the reflected signal from the surface of Mars. The curved bright green feature with an apparent range varying from about 600 to 750 kilometers (373 to 466 miles) at frequencies from about 0.7 to 1.8 megahertz is the echo from the top side of the ionosphere. A second echo of the ionosphere, at an apparent range of about 100 kilometers (62 miles) is labeled "Oblique ionospheric echo." Such echoes are believed to come from distorted structures in the ionosphere caused by the magnetic fields in the crust of Mars."<ref name=Thompson/> "MARSIS is an instrument on the European Space Agency's Mars Express orbiter."<ref name=Thompson/> At lower right is a "radargram from the Shallow Subsurface Radar instrument (SHARAD)".<ref name=Lavoie2006>{{ cite book |author=Sue Lavoie |title=PIA09076: Interpreting Radar View near Mars' South Pole, Orbit 1334 |publisher=NASA |location=Pasadena, California USA |date=December 13, 2006 |url=http://photojournal.jpl.nasa.gov/catalog/PIA09076 |accessdate=2013-10-24 }}</ref> The "Shallow Subsurface Radar instrument (SHARAD) on NASA's Mars Reconnaissance Orbiter [radargram] is shown in the upper-right panel and reveals detailed structure in the polar layered deposits of the south pole of Mars."<ref name=Lavoie2006/> "The sounding radar collected the data presented here during orbit 1334 of the mission, on Nov. 8, 2006."<ref name=Lavoie2006/> "The horizontal scale in the radargram is distance along the ground track. It can be referenced to the ground track map shown in the lower right. The radar traversed from about 75 to 85 degrees south latitude, or about 590 kilometers (370 miles). The ground track map shows elevation measured by the Mars Orbiter Laser Altimeter on NASA's Mars Global Surveyor orbiter. Green indicates low elevation; reddish-white indicates higher elevation. The traverse proceeds up onto a plateau formed by the layers."<ref name=Lavoie2006/> "The vertical scale on the radargram is time delay of the radar signals reflected back to Mars Reconnaissance Orbiter from the surface and subsurface. For reference, using an assumed velocity of the radar waves in the subsurface, time is converted to depth below the surface at one place: about 1,500 meters (5,000 feet) to one of the deeper subsurface reflectors. The color scale varies from black for weak reflections to white for strong reflections."<ref name=Lavoie2006/> "The middle panel shows mapping of the major subsurface reflectors, some of which can be traced for a distance of 100 kilometers (60 miles) or more. The layers are not all horizontal and the reflectors are not always parallel to one another. Some of this is due to variations in surface elevation, which produce differing velocity path lengths for different reflector depths. However, some of this behavior is due to spatial variations in the deposition and removal of material in the layered deposits, a result of the recent climate history of Mars."<ref name=Lavoie2006/> "The Shallow Subsurface Radar was provided by the Italian Space Agency (ASI). Its operations are led by the University of Rome and its data are analyzed by a joint U.S.-Italian science team."<ref name=Lavoie2006/> {{clear}} ==Asteroids== {{main|Rocks/Rocky objects/Asteroids}} [[Image:Toutatis.jpg|thumb|right|250px|This is a Goldstone radar image of asteroid 4179 Toustatis. Credit: Steve Ostro, JPL.]] The image at the top of the page is of asteroid 2012 LZ1. "On Sunday, June 10, a potentially hazardous asteroid thought to have been 500 meters (0.31 miles) wide was discovered by Siding Spring Observatory in New South Wales, Australia. Fortunately for us, asteroid 2012 LZ1 drifted safely by, coming within 14 lunar distances from Earth on Thursday, June 14."<ref name=Neill>{{ cite book |author=Ian O'Neill |title=Asteroid 2012 LZ1 Just Got Supersized |publisher=Discovery Communications, LLC |location= |date=June 22, 2012 |url=http://news.discovery.com/space/asteroid-2012-lz1-just-got-supersized-120622.htm |accessdate=2013-10-24 }}</ref> "Asteroid 2012 LZ1 is actually bigger than thought… in fact, it is quite a lot bigger. 2012 LZ1 is one kilometer wide (0.62 miles), double the initial estimate."<ref name=Neill/> Asteroid "2012 LZ1′s surface is really dark, reflecting only 2-4 percent of the light that hits it — this contributed to the underestimated initial optical observations. Looking for an asteroid the shade of charcoal isn’t easy."<ref name=Neill/> “This object turned out to be quite a bit bigger than we expected, which shows how important radar observations can be, because we’re still learning a lot about the population of asteroids”.<ref name=Howell>{{ cite book |author=Ellen Howell |title=Asteroid 2012 LZ1 Just Got Supersized |publisher=Discovery Communications, LLC |location= |date=June 22, 2012 |url=http://news.discovery.com/space/asteroid-2012-lz1-just-got-supersized-120622.htm |accessdate=2013-10-24 }}</ref> “The sensitivity of our radar has permitted us to measure this asteroid’s properties and determine that it will not impact the Earth at least in the next 750 years”.<ref name=Nolan>{{ cite book |author=Mike Nolan |title=Asteroid 2012 LZ1 Just Got Supersized |publisher=Discovery Communications, LLC |location= |date=June 22, 2012 |url=http://news.discovery.com/space/asteroid-2012-lz1-just-got-supersized-120622.htm |accessdate=2013-10-24 }}</ref> The extremely accurate astrometry provided by radar is critical in long-term predictions of asteroid-Earth impacts, as illustrated by the object 99942 Apophis. At right is a Goldstone radar image of the asteroid 4179 Toutatis on November 26, 1996. The "images were recorded at NASA's Deep Space Network 70-meter and 34-meter radio/radar antennas in Goldstone, CA, and the 305-meter Arecibo Radio Telescope in Puerto Rico."<ref name=Savage>{{ cite book |author=Don Savage and Jane Platt |title=Images of Asteroid 4179 Toutatis |publisher=NASA |location=Washington, DC USA |date=November 27, 1996 |url=http://neo.jpl.nasa.gov/images/toutatis.html |accessdate=2013-10-24 }}</ref> "It's amazing that the shape of Toutatis can be determined so accurately from ground-based observations".<ref name=Jong>{{ cite book |author=Eric De Jong |title=Images of Asteroid 4179 Toutatis |publisher=NASA |location=Washington, DC USA |date=November 27, 1996 |url=http://neo.jpl.nasa.gov/images/toutatis.html |accessdate=2013-10-24 }}</ref> "This technology will provide us with startling, close-up views of thousands of asteroids that orbit near the Earth."<ref name=Jong/> "We used the computer to mathematically create a three- dimensional model of the surface and rotation of Toutatis".<ref name=Hudson>{{ cite book |author=Scott Hudson |title=Images of Asteroid 4179 Toutatis |publisher=NASA |location=Washington, DC USA |date=November 27, 1996 |url=http://neo.jpl.nasa.gov/images/toutatis.html |accessdate=2013-10-24 }}</ref> "It's as though we put a clay model in space and molded it until it matched the appearance of the actual asteroid."<ref name=Hudson/> "The video is of particular interest as Toutatis nears Earth and makes its closest approach on Friday, Nov. 29, when it will pass by at a distance of 3.3 million miles (5.3 million kilometers), or about 14 times the distance from the Earth to the Moon. In 2004, Toutatis will pass only four lunar distances from Earth, closer than any known Earth- approaching object expected to pass by in the next 60 years."<ref name=Savage/> "Toutatis poses no significant threat to Earth, at least for a few hundred years".<ref name=Ostro1996>{{ cite book |author=Steven Ostro |title=Images of Asteroid 4179 Toutatis |publisher=NASA |location=Washington, DC USA |date=November 27, 1996 |url=http://neo.jpl.nasa.gov/images/toutatis.html |accessdate=2013-10-24 }}</ref> "The discovery that we live in an asteroid swarm is important for the future of humanity".<ref name=Ostro1996/> "These leftover debris from planetary formation can teach us a good deal about the formation of our Solar System. Asteroids also contain valuable minerals and many are the cheapest possible destinations for space missions."<ref name=Ostro1996/> {{clear}} ==16 Psyche== [[Image:16Psyche (Lightcurve Inversion).png|right|thumb|250px|A three-dimensional model of 16 Psyche was computed using light curve inversion techniques. Credit: Josef Ďurech, Vojtěch Sidorin, Astronomical Institute of the Charles University.]] [[Image:AnimatedOrbitOf16Psyche.gif|left|thumb|250px|This is an animation of the revolution of 16 Psyche. Credit: [[commons:User:WilyD|WilyD]].]] [[Image:Imke radar image of 16 Psyche.jpg|right|thumb|250px|Asteroid 16 Psyche displays significant variations in radar albedo with rotation. Credit: Imke de Pater.]] A "huge, metallic asteroid named 16 Psyche [...] resides" in the asteroid belt.<ref name=Wall>{{ cite book |author=Mike Wall |title=NASA Unveils 2 New Missions to Study Truly Strange Asteroids |publisher=Space.com |location= |date=4 January 2017 |url=http://www.space.com/35221-nasa-discovery-missions-asteroids-psyche-lucy.html |accessdate=2017-01-09 }}</ref> 16 Psyche is "a 130-mile-wide (210 kilometers) metallic asteroid that may be the core of an ancient, Mars-size planet. Violent collisions billions of years ago might have stripped away the layers of rock that once lay atop this metallic object."<ref name=Tanton>{{ cite book |author=Lindy Elkins-Tanton |title=NASA Unveils 2 New Missions to Study Truly Strange Asteroids |publisher=Space.com |location= |date=4 January 2017 |url=http://www.space.com/35221-nasa-discovery-missions-asteroids-psyche-lucy.html |accessdate=2017-01-09 }}</ref> "16 Psyche is the only known object of its kind in the solar system, and this is the only way humans will ever visit a core. We learn about inner space by visiting outer space."<ref name=Tanton/> The "asteroid Psyche displays significant variations in radar and optical albedo with rotation."<ref name=Pater>{{ cite book |author=Imke de Pater |title=Prof. Imke de Pater Confirms Asteroid 16 Psyche to be the Largest Metal Asteroid in the Main Belt |publisher= |location= |date=27 October 2016 |url=http://eps.berkeley.edu/news/prof-imke-de-pater-confirms-asteroid-16-psyche-be-largest-metal-asteroid-main-belt |accessdate=2017-01-09 }}</ref> "16 Psyche [is] the largest M-class asteroid in the main belt."<ref name=Pater/> "18 radar imaging and 6 continuous wave runs in November and December 2015, [were] combined [...] with 16 continuous wave runs from 2005 and 6 recent adaptive-optics (AO) images (Drummond et al., 2016) to generate a three-dimensional shape model of Psyche."<ref name=Pater/> The "shape model has dimensions 279 × 232 × 189 km (± 10%), D_eff = 226 ± 23 km, and is 6% larger than, but within the uncertainties of, the most recently published size and shape model generated from the inversion of lightcurves (Hanus et al., 2013). Psyche is roughly ellipsoidal but displays a mass-deficit over a region spanning 90° of longitude. There is also evidence for two ∼50–70 km wide depressions near its south pole. Their size and published masses lead to an overall bulk density estimate of 4500 ± 1400 kg·m⁻³. Psyche's mean radar albedo of 0.37 ± 0.09 is consistent with a near-surface regolith composed largely of iron-nickel and ∼40% porosity. Its radar reflectivity varies by a factor of 1.6 as the asteroid rotates, suggesting global variations in metal abundance or bulk density in the near surface."<ref name=Pater/> {{clear}} ==Jupiter== {{main|Jupiter}} Between September and November 23, 1963, Jupiter is detected by radar astronomy.<ref name=Pettengill>{{ cite journal |author=Gordon H. Pettengill & Irwin I. Shapiro |title=Radar Astronomy |journal=Annual Review of Astronomy and Astrophysics |month= |year=1965 |volume=3 |issue= |pages=377-410 |url=http://adsabs.harvard.edu//abs/1965ARA&A...3..377P |arxiv= |bibcode=1965ARA&A...3..377P |doi= |pmid= |accessdate=2012-12-25 }}</ref> "The dense atmosphere makes a penetration to a hard surface (if indeed one exists at all) very unlikely. In fact, the JPL results imply a correlation of the echo with Jupiter ... which corresponds to the upper (visible) atmosphere. ... Further observations will be needed to clarify the current uncertainties surrounding radar observations of Jupiter."<ref name=Pettengill/> "Although in 1963 some claimed to have detected echoes from Jupiter, these were quite weak and have not been verified by later experiments."<ref name=Shapiro>{{ cite journal |author=Irwin I. Shapiro |title=Planetary radar astronomy |journal=Spectrum, IEEE |month=March |year=1968 |volume=5 |issue=3 |pages=70-9 |url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5214821 |arxiv= |bibcode= |doi=10.1109/MSPEC.1968.5214821 |pmid= |accessdate=2012-12-25 }}</ref> "A search for radar echoes from Jupiter at 430 MHz during the oppositions of 1964 and 1965 failed to yield positive results, despite a sensitivity several orders of magnitude better than employed by other groups in earlier (1963) attempts at higher frequencies. ... [I]t might be suspected that meteorological disturbances of a random nature were involved, and that the echoes might be returned only in exceptional circumstances. Further support for this point of view may be gleaned from the fact that JPL found positive results for only 1 (centered at 32° System I longitude) of the 8 longitude regions investigated in 1963 (Goldstein 1964) and, in fact, had no success during their observations in 1964 (see comment by Goldstein following Dyce 1965)."<ref name=Dyce>{{ cite journal |author=R. B. Dyce and G. H. Pettengill, and A. D. Sanchez |title=Radar Observations of Mars and Jupiter at 70 cm |journal=The Astronomical Journal |month=August |year=1967 |volume=72 |issue=4 |pages=771-7 |url=http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1967AJ.....72..771D&amp;data_type=PDF_HIGH&amp;whole_paper=YES&amp;type=PRINTER&amp;filetype=.pdf |arxiv= |bibcode=1967AJ.....72..771D |doi=10.1086/110307 |pmid= |accessdate=2012-12-25 }}</ref> ==Titan== [[Image:PIA10008 Seas and Lakes on Titan.jpg|thumb|right|250px|This Cassini false-color mosaic shows all synthetic-aperture radar images to date of Titan's north polar region. Credit: NASA / JPL / USGS.]] [[Image:Liquid lakes on titan.jpg|thumb|right|250px|This is a radar image of the Titan surface taken on July 22, 2006, from the Cassini probe. Credit: NASA/JPL/USGS.]] Radar detection of Titan from Arecibo Observatory, included mapping of Titan's surface. "This Cassini false-color mosaic [at right] shows all synthetic-aperture radar images to date of Titan's north polar region. Approximately 60 percent of Titan's north polar region, above 60 degrees north latitude, is now mapped with radar. About 14 percent of the mapped region is covered by what is interpreted as liquid hydrocarbon lakes."<ref name=Lavoie07>{{ cite book |author=Sue Lavoie |title=PIA10008: Titan's North Polar Region |publisher=NASA/JPL |location=Pasadena, California USA |date=October 11, 2007 |url=http://photojournal.jpl.nasa.gov/catalog/?IDNumber=PIA10008 |accessdate=2013-06-13 }}</ref> "Features thought to be liquid are shown in blue and black, and the areas likely to be solid surface are tinted brown. The terrain in the upper left of this mosaic is imaged at lower resolution than the remainder of the image".<ref name=Lavoie07/> "Most of the many lakes and seas seen so far are contained in this image, including the largest known body of liquid on Titan. These seas are most likely filled with liquid ethane, methane and dissolved nitrogen."<ref name=Lavoie07/> "Many bays, islands and presumed tributary networks are associated with the seas. The large feature in the upper right center of this image is at least 100,000 square kilometers (40,000 square miles) in area, greater in extent than Lake Superior (82,000 square kilometers or 32,000 square miles), one of Earth's largest lakes. This Titan feature covers a greater fraction of the surface, at least 0.12 percent, than the Black Sea, Earth's largest terrestrial inland sea, at 0.085 percent. Larger seas may exist, as it is probable that some of these bodies are connected, either in areas unmapped by radar or under the surface (see PIA08365)."<ref name=Lavoie07/> "Of the 400 observed lakes and seas, 70 percent of their area is taken up by large "seas" greater than 26,000 square kilometers (10,000 square miles)."<ref name=Lavoie07/> In the second image at right is another radar image of Titan's surface. "The existence of oceans or lakes of liquid methane on Saturn's moon Titan was predicted more than 20 years ago. But with a dense haze preventing a closer look it has not been possible to confirm their presence. Until the Cassini flyby of July 22, 2006, that is."<ref name=Piazza>{{ cite book |author=Enrico Piazza |title=Liquid Lakes on Titan |publisher=NASA/JPL |location=Pasadena, California USA |date=January 3, 2007 |url=http://saturn.jpl.nasa.gov/photos/imagedetails/index.cfm?imageId=2432 |accessdate=2013-06-13 }}</ref> "Radar imaging data from the flyby, published this week in the journal Nature, provide convincing evidence for large bodies of liquid. This image, used on the journal's cover, gives a taste of what Cassini saw. Intensity in this colorized image is proportional to how much radar brightness is returned, or more specifically, the logarithm of the radar backscatter cross-section. The colors are not a representation of what the human eye would see."<ref name=Piazza/> "The lakes, darker than the surrounding terrain, are emphasized here by tinting regions of low backscatter in blue. Radar-brighter regions are shown in tan. The strip of radar imagery is foreshortened to simulate an oblique view of the highest latitude region, seen from a point to its west."<ref name=Piazza/> "This radar image was acquired by the Cassini radar instrument in synthetic aperture mode on July 22, 2006. The image is centered near 80 degrees north, 35 degrees west and is about 140 kilometers (84 miles) across. Smallest details in this image are about 500 meters (1,640 feet) across."<ref name=Piazza/> {{clear}} ==Comets== {{main|Radiation astronomy/Comets}} [[Image:HMP Aug19 2011.b.gif|thumb|right|250px|Goldstone radar observations on 2011 August 19 and 20 shown detected echoes from the nucleus and coma of comet 45P/Honda-Mrkos-Pajdusakova (HMP). Credit: Goldstone.]] "Goldstone radar observations on 2011 August 19 and 20 detected echoes from the nucleus and coma of comet 45P/Honda-Mrkos-Pajdusakova (HMP). This is only the fourth Goldstone comet detection and the first since detection of comet 73P/Schwassmann-Wachmann 3 in 2006."<ref name=Goldstone>{{ cite book |author=Goldstone |title=GOLDSTONE RADAR DETECTION OF COMET HONDA-MRKOS-PAJDUSAKOVA |publisher=NASA/JPL |location=Pasasena, California USA |date=19 August 2011 |url=http://echo.jpl.nasa.gov/asteroids/45P_HMP/comet.hmp.html |accessdate=2016-09-15 }}</ref> "The [continuous wave] CW spectrum [on the right] shows the opposite-circular echo from the comet obtained on August 19. The narrow spike is the echo from the nucleus and the broad, low, asymmetric hump is the echo from coma particles. The skew of the coma echo to positive frequencies indicates that most of the coma particles were approaching Earth at the time of the observations."<ref name=Goldstone/> "The Goldstone measurements provided a range correction of 49 km for the nucleus, which significantly improved the orbit and which revealed a systematic bias in many of the optical observations."<ref name=Goldstone/> "This is only the fifteenth comet that has been detected with radar."<ref name=Goldstone/> {{clear}} ==Astrometry== "Asteroid radar astronomy began on 14 June 1968, with the detection of 1566 Icarus from Goldstone (Goldstein 1969) and Haystack (Pettengill et al. 1969)."<ref name=Ostro1991>{{ cite journal |author=Ostro, S. J.; Campbell, D. B.; Chandler, J. F.; Shapiro, I. I.; Hine, A. A.; Velez, R.; Jurgens, R. F.; Rosema, K. D.; Winkler, R.; Yeomans, D. K. |title=Asteroid radar astrometry |journal=Astronomical Journal |month=October |year=1991 |volume=102 |issue=10 |pages=1490-1502 |url=http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1991AJ....102.1490O&link_type=ARTICLE&db_key=AST&high= |arxiv= |bibcode=1991AJ....102.1490O |doi=10.1086/115975 |pmid= |accessdate=2017-07-27 }}</ref> "Radar measurements of echo Doppler frequencies and time delays permit significant refinements of orbital elements and commensurate improvements in the accuracy of prediction ephemerides because these measurements have fine fractional precision and are orthogonal to optical, angular-position measurements."<ref name=Ostro1991/> "Yeomans ''et al.'' (1987) used numerical experiments to explore the extent to which delay/ Doppler astrometry can refine orbit estimates for NEAs. They concluded that radar measurements can reduce ephemeris uncertainties dramatically for asteroids having short optical-data histories. They noted that a few radar observations of a newly discovered NEA could mean the difference between successfully recovering the object during its next close approach and losing it entirely. Even for asteroids with very long astrometric histories and secure orbits, radar measurements can significantly shrink their positional error ellipsoids for at least a decade."<ref name=Ostro1991/> "A typical transmit/receive cycle, or run, consists of signal transmission for a duration close to the roundtrip light time between the radar and the target, i.e., until the first echoes are about to come back, followed by reception of echoes for a similar duration. In continuous wave (cw) observations, one transmits a nearly monochromatic waveform and measures the distribution of echo power as a function of frequency. The resultant echo spectra can be thought of as one-dimensional images, or brightness scans across the target through a slit parallel to the asteroid’s apparent spin vector. In ranging observations, time coding of the waveform permits measurement of the distribution of echo power in time delay (range) as well."<ref name=Ostro1991/> "An asteroid’s apparent radial motion introduces a continuously changing Doppler shift into the echoes. One avoids spectral smear by tuning the receiver according to an ephemeris based on an orbit determined from astrometric asteroid observations."<ref name=Ostro1991/> "In cw experiments, voltage samples of the received signal are Fourier transformed and the results are squared to obtain an estimate of the power spectrum, with the frequency resolution equal to the reciprocal of the time series length, i.e., of the coherence time. The sampling rate is chosen to provide an unaliased bandwidth many times larger than both the ''a priori'' Doppler uncertainty and the echo bandwidth, so ''f''_est can be determined unambiguously from the received power spectrum. Normally, a number of these "single-look" spectra are averaged to improve the spectral estimates."<ref name=Ostro1991/> "In principle, range resolution can be obtained by using a coherent pulsed cw waveform—the transmitter’s carrier-frequency oscillator operates continuously but radio-frequency power is radiated only during intervals that are one delay resolution cell long and occur at intervals called the pulse repetition period (PRP). The PRP is normally much greater than the target’s intrinsic delay dispersion, thereby ensuring that the echo will consist of successive, nonoverlapping range profiles. Fourier transformation of ''N'' time samples taken at the same position (i.e., the same delay relative to τ₀) within each of N successive range profiles yields the echo power spectrum for the corresponding range cell on the target. This spectrum has an unaliased bandwidth ''B'' - l/[(PRP)(NCOH)] and a frequency resolution ''B/N'', where NCOH is the number of code cycles for which voltage samples have been coherently summed prior to Fourier transformation."<ref name=Ostro1991/> ==Locations on Earth== [[Image:France radar.jpg|thumb|right|250px|This is a Shuttle radar image of France. Credit: NASA.]] [[Image:Pakistan Khairabad digital elevation model by TanDEM-X (DLR).jpg|thumb|left|250px|Colour coded TanDEM-X digital elevation model is of Khairabad in northern Pakistan. Credit: DLR.]] The radar image of France on the right is color-coded for relative relief from darker green at or near sea level to white for mountain tops. The border of France has been shaded and given a slight elevation away from the rest of the Earth nearby. On the left is a colour coded TanDEM-X digital elevation model is of Khairabad in northern Pakistan, created on 6 August 2010. {{clear}} ==Paleogeography== [[Image:Kufra-space-radar.jpg|thumb|right|250px|A previously unknown branch of an ancient river, buried under thousands of years of windblown sand, is revealed by radar. Credit: [[commons:User:Porao|Porao]], NASA.]] "The ability of a sophisticated radar instrument to image large regions of the world from space, using different frequencies that can penetrate dry sand cover, produced the discovery in this image: a previously unknown branch of an ancient river, buried under thousands of years of windblown sand in a region of the Sahara Desert in North Africa. This area is near the Kufra Oasis in southeast Libya, centered at 23.3 degrees north latitude, 22.9 degrees east longitude. The image was acquired by the Spaceborne Imaging Radar-C/X-band Synthetic Aperture (SIR- C/X-SAR) imaging radar when it flew aboard the space shuttle Endeavour on its 60th orbit on October 4, 1994. This SIR-C image reveals a system of old, now inactive stream valleys, called "paleodrainage systems.""<ref name=Porao>{{ cite web |author=[[c:User:Porao|Porao]] |title=''File:Kufra-space-radar.jpg'' |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=4 October 1994 |url=https://commons.wikimedia.org/wiki/File:Kufra-space-radar.jpg |accessdate=2016-09-27 }}</ref> {{clear}} ==Impact craters== {{main|Radiation astronomy/Craters}} [[Image:Impact crater in northwestern Greenland.png|right|thumb|300px|The location of the 19-mile-wide (31 kilometers) impact crater in northwestern Greenland is circled in red. Credit: Kurt H. Kjær, Nicolaj K. Larsen, Tobias Binder, Anders A. Bjørk, Olaf Eisen, Mark A. Fahnestock, Svend Funder, Adam A. Garde, Henning Haack, Veit Helm, Michael Houmark-Nielsen, Kristian K. Kjeldsen1, Shfaqat A. Khan, Horst Machguth, Iain McDonald, Mathieu Morlighem, Jérémie Mouginot, John D. Paden, Tod E. Waight, Christian Weikusat, Eske Willerslev and Joseph A. MacGregor, The Natural History Museum of Denmark.{{tlx|fairuse}}]] [[Image:Bed topography based on airborne radar sounding.jpg|left|thumb|300px|Bed topography based on airborne radar sounding from 1997 to 2014 NASA data and 2016 Alfred Wegener Institute (AWI) data. Credit: Kurt H. Kjær, Nicolaj K. Larsen, Tobias Binder, Anders A. Bjørk, Olaf Eisen, Mark A. Fahnestock, Svend Funder, Adam A. Garde, Henning Haack, Veit Helm, Michael Houmark-Nielsen, Kristian K. Kjeldsen1, Shfaqat A. Khan, Horst Machguth, Iain McDonald, Mathieu Morlighem, Jérémie Mouginot, John D. Paden, Tod E. Waight, Christian Weikusat, Eske Willerslev and Joseph A. MacGregor, The Natural History Museum of Denmark.{{tlx|fairuse}}]] Per the image on the right: "In 2015 I was looking at a new map of the bedrock below the Greenland Ice Sheet and discovered a large circular feature under the Hiawatha glacier in northwest Greenland."<ref name=Larsen>{{ cite book |author=Nicolaj K. Larsen |title=A Massive Impact Crater Has Been Detected Beneath Greenland’s Ice Sheet |publisher=Gizmodo |location= |date=14 November 2018 |url=https://gizmodo.com/a-massive-impact-crater-has-been-detected-beneath-green-1830437095 |accessdate=14 November 2018 }}</ref> "You can see the rounded structure at the edge of the ice sheet, especially when flying high enough."<ref name=Paden>{{ cite book |author=John Paden |title=A Massive Impact Crater Has Been Detected Beneath Greenland’s Ice Sheet |publisher=Gizmodo |location= |date=14 November 2018 |url=https://gizmodo.com/a-massive-impact-crater-has-been-detected-beneath-green-1830437095 |accessdate=14 November 2018 }}</ref> "For the most part the crater isn’t visible out the airplane window. It’s funny that until now nobody thought, ‘Hey, what’s that semicircular feature there?’ From the airplane it is subtle and hard to see unless you already know it’s there. Using satellite imagery taken at a low sun angle that accentuates hills and valleys in the ice sheet’s terrain—you can really see the circle of the whole crater in these images."<ref name=Paden/> "It is correct that the crater is not well dated but there’s good evidence that it is geologically young, that is, it formed within the last 2 to 3 million years, and most likely it is as young as the last Ice Age [which ended around 12,000 years ago],” Larsen explained to Gizmodo. “We are currently trying to come up with ideas on how to date the impact. One idea is to drill through the ice and get bedrock samples that can be used for numerical dating."<ref name=Larsen/> On the left is an image of bed "topography based on airborne radar sounding from 1997 to 2014 NASA data and 2016 Alfred Wegener Institute (AWI) data. Black triangles represent elevated rim picks from the radargrams, and the dark purple circles represent peaks in the central uplift. Hatched red lines are field measurements of the strike of ice-marginal bedrock structures. Black circles show location of the three glaciofluvial sediment samples".<ref name=Kjær>{{ cite journal |author=Kurt H. Kjær, Nicolaj K. Larsen, Tobias Binder, Anders A. Bjørk, Olaf Eisen, Mark A. Fahnestock, Svend Funder, Adam A. Garde, Henning Haack, Veit Helm, Michael Houmark-Nielsen, Kristian K. Kjeldsen1, Shfaqat A. Khan, Horst Machguth, Iain McDonald, Mathieu Morlighem, Jérémie Mouginot, John D. Paden, Tod E. Waight, Christian Weikusat, Eske Willerslev and Joseph A. MacGregor |title=A large impact crater beneath Hiawatha Glacier in northwest Greenland |journal=Sciences Advances |date=14 November 2018 |volume=4 |issue=11 |pages=eaar8173 |url=http://advances.sciencemag.org/content/4/11/eaar8173 |arxiv= |bibcode= |doi=10.1126/sciadv.aar8173 |pmid= |accessdate=14 November 2018 }}</ref> "Glaciofluvial sediment from the largest river draining the crater contains shocked quartz and other impact-related grains. Geochemical analysis of this sediment indicates that the impactor was a fractionated iron asteroid, which must have been more than a kilometer wide to produce the identified crater. Radiostratigraphy of the ice in the crater shows that the Holocene ice is continuous and conformable, but all deeper and older ice appears to be debris rich or heavily disturbed."<ref name=Kjær/> {{clear}} ==Rainbands== [[Image:Sturmfront auf Doppler-Radar-Schirm.jpg|250px|thumb|right|Band of thunderstorms is seen on a weather radar display. Credit: NOAA.]] Rainbands are cloud and precipitation areas which are significantly elongated. Rainbands can be stratiform or convective,<ref>Glossary of Meteorology (2009). Rainband. [https://web.archive.org/web/20110606102457/http://amsglossary.allenpress.com/glossary/search?p=1&query=rainband&submit=Search] 2011-06-06 Retrieved on 2008-12-24.</ref> and are generated by differences in temperature. When noted on weather radar imagery, this precipitation elongation is referred to as banded structure.<ref>Glossary of Meteorology (2009). Banded structure. [https://web.archive.org/web/20110606102630/http://amsglossary.allenpress.com/glossary/search?id=banded-structure1] 2011-06-06 Retrieved on 2008-12-24.</ref> Rainbands in advance of warm occluded fronts and warm fronts are associated with weak upward motion,<ref>Owen Hertzman (1988). [http://adsabs.harvard.edu/abs/1988PhDT.......110H Three-Dimensional Kinematics of Rainbands in Midlatitude Cyclones.] Retrieved on 2008-12-24</ref> and tend to be wide and stratiform in nature.<ref>{{cite book |author=Yuh-Lang Lin |title=Mesoscale Dynamics |url=https://books.google.com/books?id=4KXtnQ3bDeEC&pg=PA405 |year=2007 |publisher=Cambridge University Press |isbn=978-0-521-80875-0 |page=405 }}</ref> Rainbands spawned near and ahead of cold fronts can be squall lines which are able to produce tornadoes.<ref>Glossary of Meteorology (2009). Prefrontal squall line. [url=https://web.archive.org/web/20070817224959/http://amsglossary.allenpress.com/glossary/search?id=prefrontal-squall-line1] 2007-08-17 Retrieved on 2008-12-24.</ref> Rainbands associated with cold fronts can be warped by mountain barriers perpendicular to the front's orientation due to the formation of a low-level barrier jet.<ref>J. D. Doyle (1997). The influence of mesoscale orography on a coastal jet and rainband. [https://web.archive.org/web/20120106231443/http://cat.inist.fr/?aModele=afficheN&cpsidt=2721180] 2012-01-06 Retrieved on 2008-12-25.</ref> Bands of thunderstorms can form with sea breeze and land breeze boundaries, if enough moisture is present. If sea breeze rainbands become active enough just ahead of a cold front, they can mask the location of the cold front itself.<ref>A. Rodin (1995). Interaction of a cold front with a sea-breeze front numerical simulations. [https://web.archive.org/web/20110909094659/http://cat.inist.fr/?aModele=afficheN&cpsidt=3570629] 2011-09-09 Retrieved on 2008-12-25.</ref> {{clear}} ==Extratropical cyclones== [[Image:Feb242007 blizzard.gif|thumb|350px|right|A February 24, 2007 radar image of a large extratropical cyclonic storm system at its peak over the central United States. Note the band of thunderstorms along its trailing cold front.]] Rainbands in advance of warm occluded fronts and warm fronts are associated with weak upward motion,<ref>Owen Hertzman (1988). [http://adsabs.harvard.edu/abs/1988PhDT.......110H Three-Dimensional Kinematics of Rainbands in Midlatitude Cyclones.] Retrieved on 2008-12-24</ref> and tend to be wide and stratiform in nature.<ref>Yuh-Lang Lin (2007). [https://books.google.com/books?id=4KXtnQ3bDeEC&pg=PA405&lpg=PA405&source=bl&ots=CL5KD0HLAJ&sig=Gz5bwKi9yu8j25EbXLD3TVNNQ68&hl=en&sa=X&oi=book_result&resnum=10&ct=result Mesoscale Dynamics.] Retrieved on 2008-12-25.</ref> In an atmosphere with rich low level moisture and vertical wind shear,<ref>Richard H. Grumm (2006). 16 November Narrow Frontal Rain band Floods and severe weather [https://web.archive.org/web/20110720032822/http://nws.met.psu.edu/severe/2006/16Nov2006.pdf] 2011-07-20 Retrieved on 2008-12-26.</ref> narrow, convective rainbands known as squall lines generally in the cyclone's warm sector, ahead of strong cold fronts associated with extratropical cyclones.<ref>Glossary of Meteorology (2009). Prefrontal squall line. [https://web.archive.org/web/20070817224959/http://amsglossary.allenpress.com/glossary/search?id=prefrontal-squall-line1] 2007-08-17 Retrieved on 2008-12-24.</ref> Wider rain bands can occur behind cold fronts, which tend to have more stratiform, and less convective, precipitation.<ref>K. A. Browning and Robert J. Gurney (1999). [https://books.google.com/books?id=DO5K1NK_ZewC&pg=PA116&lpg=PA116&source=web&ots=j2xQaps00_&sig=SSYUXEW1H4QDe4UI0K3T0WIW6Tk&hl=en&sa=X&oi=book_result&resnum=9&ct=result#PPA117,M1 Global Energy and Water Cycles.] Retrieved on 2008-12-26.</ref> Within the cold sector north to northwest of a cyclone center, in colder cyclones, small scale, or mesoscale, bands of heavy snow can occur within a cyclone's comma head precipitation pattern with a width of {{convert|20|mi|km|order=flip}} to {{convert|50|mi|km|order=flip}}.<ref>KELLY HEIDBREDER (2007). [http://www.theweatherprediction.com/weatherpapers/023/index.html Mesoscale snow banding.] Retrieved on 2008-12-24.</ref> These bands in the comma head are associated with areas of frontogensis, or zones of strengthening temperature contrast.<ref>David R. Novak, Lance F. Bosart, Daniel Keyser, and Jeff S. Waldstreicher (2002). [http://cstar.cestm.albany.edu/CAP_Projects/Project4/Banded%20Precip/novakWAF.pdf A CLIMATOLOGICAL AND COMPOSITE STUDY OF COLD SEASON BANDED PRECIPITATION IN THE NORTHEAST UNITED STATES.] Retrieved on 2008-12-26.</ref> Southwest of extratropical cyclones, curved flow bringing cold air across the relatively warm Great Lakes can lead to narrow lake effect snow bands which bring significant localized snowfall.<ref>{{cite web |author=B. Geerts |year=1998 |url=http://www-das.uwyo.edu/~geerts/cwx/notes/chap10/lake_effect_snow.html |title=Lake Effect Snow |accessdate=2008-12-24 |publisher=University of Wyoming }}</ref> {{clear}} ==Recent history== [[Image:ADU-1000-3.jpg|thumb|250px|right|This image shows the early planetary radar at Pluton, USSR, 1960. Credit: [[c:User:Rumlin|Rumlin]].{{tlx|free media}}]] The '''recent history''' period dates from around 1,000 b2k to present. The "Arecibo telescope was completed in 1963 at the initiative of Cornell electrical engineering professor William E. Gordon".<ref name=Brand/> At right is an image of the Pluton radar complex used for radar astronomy since 1960. {{clear}} ==Arecibo Observatory== [[Image:Arecibo Observatory Aerial View2.jpg|thumb|right|250px|The Arecibo Radio Telescope, Arecibo, Puerto Rico, at 1000 feet (305 m) across, is the largest dish antenna in the world. Credit: Michael D. Bicay.]] The "Arecibo Observatory in Puerto Rico [is] the world's largest, and most sensitive, single-dish radio telescope."<ref name=Brand>{{ cite book | author=David Brand | title=Astrophysicist Robert Brown, leader in telescope development, named to head NAIC and its main facility, Arecibo Observatory | publisher=Cornell University | date=21 January 2003 | url=http://www.news.cornell.edu/releases/Jan03/NAIC.director.deb.html | accessdate=2008-09-02 }}</ref> "The 1,000-foot-diameter (305 meters) Arecibo telescope [... provides] access to state-of-the-art observing for scientists in [[Radiation astronomy/Radios|radio astronomy]], solar system radar and [[Radiation astronomy/Atmospheres|atmospheric studies]], and the observatory has the unique capability for solar system and ionosphere (the atmosphere's ionized upper layers) radar remote sensing."<ref name=Brand/> It contains the largest curved focusing dish on Earth, giving Arecibo the largest electromagnetic-wave-gathering capacity.<ref name=Castel>{{ cite book | title=Arecibo: Celestial Eavesdropper | author=Frederic Castel | publisher=Space.com | date=8 May 2000 | accessdate=2008-09-02 |url=http://web.archive.org/web/20000619110005/http://www.space.com/scienceastronomy/astronomy/arecibo_profile_000508.html }}</ref> The dish surface is made of 38,778 perforated aluminum panels, each measuring about 3 by 6 feet (1 by 2 m), supported by a mesh of steel cables. The telescope has three radar transmitters, with effective isotropic radiated powers [EIRP] of 20 TW at 2380&nbsp;MHz, 2.5 TW (pulse peak) at 430&nbsp;MHz, and 300 MW at 47&nbsp;MHz. The telescope is a spherical reflector, not a parabolic reflector. To aim the telescope, the receiver is moved to intercept signals reflected from different directions by the spherical dish surface. A parabolic mirror would induce a varying astigmatism when the receiver is in different positions off the focal point, but the error of a spherical mirror is the same in every direction. The receiver is located on a 900-ton platform which is suspended 150&nbsp;m (500&nbsp;ft) in the air above the dish by 18 cables running from three reinforced concrete towers, one of which is 110&nbsp;m (365&nbsp;ft) high and the other two of which are 80&nbsp;m (265&nbsp;ft) high (the tops of the three towers are at the same elevation). The platform has a 93-meter-long rotating bow-shaped track called the azimuth arm on which receiving antennas, secondary and tertiary reflectors are mounted. This allows the telescope to observe any region of the sky within a forty-degree cone of visibility about the local zenith (between −1 and 38 degrees of declination). Puerto Rico's location near the equator allows Arecibo to view all of the planets in the Solar System, though the round trip light time to objects beyond Saturn is longer than the time the telescope can track it, preventing radar observations of more distant objects. {{clear}} ==Goldstone Deep Space Communication Complex== [[Image:Goldstone Deep Space Communication Complex - GPN-2000-000506.jpg|thumb|right|250px|This is an image of the Goldstone Deep Space Communication Complex. Credit: NASA.]] Shown at right are the three "34m (110 ft.) diameter Beam Waveguide antennas located at the Goldstone Deep Space Communications Complex, situated in the Mojave Desert in California. This is one of three complexes which comprise NASA's Deep Space Network (DSN). The DSN provides radio communications for all of NASA's interplanetary spacecraft and is also utilized for radio astronomy and radar observations of the solar system and the universe."<ref name=Hahn>{{ cite book |author=Michael Hahn |title=Goldstone Deep Space Communication Complex |publisher=NASA HQ |location=Washington, DC USA |date=January 1, 1990 |url=http://grin.hq.nasa.gov/ABSTRACTS/GPN-2000-000506.html |accessdate=2013-10-24 }}</ref> {{clear}} ==Synthetic Aperture Radar Satellite== [[Image:NISAR artist concept.jpg|thumb|right|250px|Artist concept is of the Nasa-Isro synthetic aperture radar (NISAR) satellite in orbit. Credit: NASA.]] "Using advanced radar imaging that will provide an unprecedented, detailed view of Earth, the NASA-ISRO Synthetic Aperture Radar, or NISAR, satellite is designed to observe and take measurements of some of the planet's most complex processes, including ecosystem disturbances, ice-sheet collapse, and natural hazards such as earthquakes, tsunamis, volcanoes and landslides."<ref name=Nelson>{{ cite book |author=Jon Nelson |title=NASA-ISRO Synthetic Aperture Radar |publisher=NASA/JPL |location=Pasadena, California USA |date=2 July 2014 |url=http://www.jpl.nasa.gov/missions/nasa-isro-synthetic-aperture-radar-nisar/ |accessdate=2015-03-10 }}</ref> "Data collected from NISAR will reveal information about the evolution and state of Earth's crust, help scientists better understand our planet's processes and changing climate, and aid future resource and hazard management. The mission is a partnership between NASA and the Indian Space Research Organization [ISRO]."<ref name=Nelson/> "Scientific Instrument(s) - L-band (24-centimeter wavelength) Polarimetric Synthetic Aperture Radar [JPL] - S-band (12-centimeter wavelength) Polarimetric Synthetic Aperture Radar [ISRO]"<ref name=Nelson/> {{clear}} ==Interferograms== [[Image:Izmit interferogram.jpg|thumb|right|250px|This image is an interferogram created using pairs of images taken by Synthetic Aperture Radar (SAR). Credit: NASA/JPL-Caltech.]] [[Image:L C band topo interferograms.jpg|thumb|left|250px|This image of Fort Irwin in California's Mojave Desert compares interferometric radar signatures topography. Credit: NASA/JPL-Caltech STS-65 (STS-68) crew.{{tlx|free media}}]] "This image is an interferogram that was created using pairs of images taken by Synthetic Aperture Radar (SAR). The images, acquired at two different times, have been combined to measure surface deformation or changes that may have occurred during the time between data acquisition. The images were collected by the European Space Agency's Remote Sensing satellite (ERS-2) on 13 August 1999 and 17 September 1999 and were combined to produce these image maps of the apparent surface deformation, or changes, during and after the 17 August 1999 Izmit, Turkey earthquake. This magnitude 7.6 earthquake was the largest in 60 years in Turkey and caused extensive damage and loss of life. Each of the color contours of the interferogram represents 28 mm (1.1 inches) of motion towards the satellite, or about 70 mm (2.8 inches) of horizontal motion. White areas are outside the SAR image or water of seas and lakes. The North Anatolian Fault that broke during the Izmit earthquake moved more than 2.5 meters (8.1 feet) to produce the pattern measured by the interferogram. Thin red lines show the locations of fault breaks mapped on the surface. The SAR interferogram shows that the deformation and fault slip extended west of the surface faults, underneath the Gulf of Izmit. Thick black lines mark the fault rupture inferred from the SAR data. Scientists are using the SAR interferometry along with other data collected on the ground to estimate the pattern of slip that occurred during the Izmit earthquake. This then used to improve computer models that predict how this deformation transferred stress to other faults and to the continuation of the North Anatolian Fault, which extends to the west past the large city of Istanbul. These models show that the Izmit earthquake further increased the already high probability of a major earthquake near Istanbul."<ref name=Lavoie1999>{{ cite book |author=Sue Lavoie |title=PIA00557: Izmit, Turkey 1999 Earthquake Interferogram |publisher=NASA/JPL |location=Pasadena, California USA |date=17 September 1999 |url=http://photojournal.jpl.nasa.gov/catalog/PIA00557 |accessdate=2016-09-27 }}</ref> "This image [on the left] of Fort Irwin in California's Mojave Desert compares interferometric radar signatures topography -- data that were obtained by multiple imaging of the same region to produce three-dimensional elevation maps -- as it was obtained on October 7-8, 1994 by the Spaceborne Imaging Radar-C/X-band Synthetic Aperture Radar aboard the space shuttle Endeavour. Data were acquired using the L-band (24 centimeter wavelength) and C-band (6 centimeter wavelength). The image covers an area about 25 kilometers by 70 kilometers (15.5 miles by 43 miles). North is to the lower right of the image. The color contours shown are proportional to the topographic elevation. With a wavelength one-fourth that of the L-band, the results from the C-band cycle through the color contours four times faster for a given elevation change. Detailed comparisons of these multiple frequency data over different terrain types will provide insights in the future into wavelength-dependent effects of penetration and scattering on the topography measurement accuracy. Fort Irwin is an ideal site for such detailed digital elevation model comparisons because a number of high precision digital models of the area already exist from conventional measurements as well as from airborne interferometric SAR data."<ref name=Lavoie1994>{{ cite book |author=Sue Lavoie |title=PIA01759: SPace Radar Image of Fort Irwin, California |publisher=NASA/JPL |location=Pasadena, California USA |date=7 October 1994 |url=http://photojournal.jpl.nasa.gov/catalog/PIA01759 |accessdate=2016-09-27 }}</ref> "Spaceborne Imaging Radar-C and X-band Synthetic Aperture Radar (SIR-C/X-SAR) is part of NASA's Mission to Planet Earth. The radars illuminate Earth with microwaves, allowing detailed observations at any time, regardless of weather or sunlight conditions. SIR-C/X-SAR uses three microwave wavelengths: L-band (24 cm), C-band (6 cm) and X-band (3 cm). The multi-frequency data will be used by the international scientific community to better understand the global environment and how it is changing. The SIR-C/X-SAR data, complemented by aircraft and ground studies, will give scientists clearer insights into those environmental changes which are caused by nature and those changes which are induced by human activity."<ref name=Lavoie1994/> "SIR-C was developed by NASA's Jet Propulsion Laboratory. X-SAR was developed by the Dornier and Alenia Spazio companies for the German space agency, Deutsche Agentur fuer Raumfahrtangelegenheiten (DARA), and the Italian space agency, Agenzia Spaziale Italiana (ASI), with the Deutsche Forschungsanstalt fuer Luft und Raumfahrt e.V.(DLR), the major partner in science, operations and data processing of X-SAR."<ref name=Lavoie1994/> {{clear}} ==Gyrotrons== "Other important applications of gyrotrons are high-power microwave sources include high resolution radar ranging and imaging in atmospheric and planetary science as well as deep-space and specialized satellite communications and RF drivers for next-generation high-gradient linear accelerators".<ref name=Ying>{{ cite journal |author=Liu Ying-hui; Li Hong-fu ; Li Hao ; Wang E-feng ; Xu Yong ; Sun Yu |title=Analysis of RF Field in Open Cavity by Mode-Matching Technique, In: ''Infrared Millimeter Waves and 14th International Conference on Teraherz Electronics'' |publisher=IEEE |location=Shanghai |month=September 18-22, |year=2006 |editor= |pages=78 |url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4222020 |arxiv= |bibcode= |doi=10.1109/ICIMW.2006.368286 |pmid= |isbn=1-4244-0400-2 |accessdate=2013-10-20 }}</ref> ==Aerial gravity gradiometry== [[Image:Tpdc-on-cgg.jpg|right|thumb|300px|Aerial high-resolution gravity gradiometry system is in combination with LIDAR digital terrain mapping, electromagnetics, digital video, and gamma-ray spectrometry. Credit: Mozambique Resources Post.{{tlx|fairuse}}]] The aircraft imaged on the right carried-out aerial high-resolution gravity gradiometry system in combination with LIDAR digital terrain mapping, electromagnetics, digital video, and gamma-ray spectrometry over "onshore areas along the South-Eastern Tanzanian Coastal Basin and the eastern arm of the East African Rift."<ref name=Post>{{ cite web |author=Mozambique Resources Post |title=Africa Oil & Gas: TPDC awards CGG airborne gravity gradiometer surveys onshore Tanzania |publisher=Mozambique Resources Post |location=Tanzania |date=23 July 2015 |url=https://mozambiqueminingpost.wordpress.com/2015/07/23/africa-oil-gas-tpdc-awards-cgg-airborne-gravity-gradiometer-surveys-onshore-tanzania/ |accessdate=2017-10-02 }}</ref> {{clear}} ==Hypotheses== {{main|Hypotheses}} # A radar frequency exists that is reflected back by the Sun. # A control group for radar astronomy would likely contain a standard target that allows maximum radar facility resolution and imaging. ==See also== {{div col|colwidth=20em}} * [[Radiation astronomy/Gravitationals|Gravitational astronomy]] * [[Radiation astronomy/Infrareds|Infrared astronomy]] * [[Radiation astronomy/Microwaves|Microwave astronomy]] * [[Radiation astronomy/Radios|Radio astronomy]] * [[Radiation astronomy/Reds|Red astronomy]] * [[Radiation astronomy/Submillimeters|Submillimeter astronomy]] {{Div col end}} ==References== {{reflist|2}} ==External links== * [http://www.iau.org/ International Astronomical Union] * [http://nedwww.ipac.caltech.edu/ NASA/IPAC Extragalactic Database - NED] * [http://nssdc.gsfc.nasa.gov/ NASA's National Space Science Data Center] * [http://www.osti.gov/ Office of Scientific & Technical Information] * [http://www.questia.com/ Questia - The Online Library of Books and Journals] * [http://online.sagepub.com/ SAGE journals online] * [http://www.adsabs.harvard.edu/ The SAO/NASA Astrophysics Data System] * [http://www.scirus.com/srsapp/advanced/index.jsp?q1= Scirus for scientific information only advanced search] * [http://cas.sdss.org/astrodr6/en/tools/quicklook/quickobj.asp SDSS Quick Look tool: SkyServer] * [http://simbad.u-strasbg.fr/simbad/ SIMBAD Astronomical Database] * [http://simbad.harvard.edu/simbad/ SIMBAD Web interface, Harvard alternate] * [http://nssdc.gsfc.nasa.gov/nmc/SpacecraftQuery.jsp Spacecraft Query at NASA] * [http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/convcoord/convcoord.pl Universal coordinate converter] <!-- footer templates --> {{tlx|Flight resouces}}{{tlx|Geology resources}}{{tlx|Materials science resources}}{{Principles of radiation astronomy}}{{tlx|Technology resources}}{{Sisterlinks|Radar astronomy}} <!-- categories --> [[Category:Astrophysics/Lectures]] [[Category:Astronomy/Lectures]] [[Category:Radiation/Lectures]] [[Category:Radiation astronomy/Lectures]] [[Category:Technology/Lectures]] d7awukeqtrjteeduqfivwpmxo1xq36j 103 160200 2410795 2176374 2022-08-01T15:25:30Z Isivile Nkungu Khenkebe 2947451 wikitext text/x-wiki == Cultures == i'd like to talk about all cultures i have been conflicted to since three days of age,my music is British and i would not hide it, i accossed 91200 kilometres for it, i find myself old enough to change it and i wouldn't to evoluate more than that stage, i have enough to many propositions i'd to retire, and i could never across more than 10 000 km aroud. the adventure was very expensives i have known lot of artists and worked with, collaborate with, and we 'd never come back to reform it because we have passed the age, because today we got married and our wives devided us, and we worked a lot.our wives learned us to become ourself and better way to liberate from drugs and alcool. we say that life of artists is like life of dogs, i the chinese sign of dog is very artist but it would become another subject or another problem when it's nervous or anxious so that we will not undertand them. {{unsigned|41.103.139.187|18 August 2010}} Isivile Nkungu Khenkebe Is the one of the Drama class [[User:Isivile Nkungu Khenkebe|Isivile Nkungu Khenkebe]] ([[User talk:Isivile Nkungu Khenkebe|discuss]] • [[Special:Contributions/Isivile Nkungu Khenkebe|contribs]]) 15:25, 1 August 2022 (UTC) == "Portal:Arts" vs. "School:Arts and design" == Don't we already have an "Arts and design" school under Humanities? And isn't this redundant? --[[User:RayneVanDunem|RayneVanDunem]] ([[User talk:RayneVanDunem|talk]]) 05:32, 11 August 2012 (UTC) == Car Painting == i am a calligrapher artist .I want to write letters in a way that it give embossing effect. I have an idea to write with potene ,material used in car painting. Is there any person who knows about this kind of materials --[[User:Imranrafik|Imranrafik]] ([[User talk:Imranrafik|discuss]] • [[Special:Contributions/Imranrafik|contribs]]) 09:25, 26 April 2014 (UTC) k97vjqwda0khsedinhoe2sg9j22nad7 2410796 2410795 2022-08-01T15:26:29Z Isivile Nkungu Khenkebe 2947451 /* Isivile Nkungu Khenkebe */ new section wikitext text/x-wiki == Cultures == i'd like to talk about all cultures i have been conflicted to since three days of age,my music is British and i would not hide it, i accossed 91200 kilometres for it, i find myself old enough to change it and i wouldn't to evoluate more than that stage, i have enough to many propositions i'd to retire, and i could never across more than 10 000 km aroud. the adventure was very expensives i have known lot of artists and worked with, collaborate with, and we 'd never come back to reform it because we have passed the age, because today we got married and our wives devided us, and we worked a lot.our wives learned us to become ourself and better way to liberate from drugs and alcool. we say that life of artists is like life of dogs, i the chinese sign of dog is very artist but it would become another subject or another problem when it's nervous or anxious so that we will not undertand them. {{unsigned|41.103.139.187|18 August 2010}} Isivile Nkungu Khenkebe Is the one of the Drama class [[User:Isivile Nkungu Khenkebe|Isivile Nkungu Khenkebe]] ([[User talk:Isivile Nkungu Khenkebe|discuss]] • [[Special:Contributions/Isivile Nkungu Khenkebe|contribs]]) 15:25, 1 August 2022 (UTC) == "Portal:Arts" vs. "School:Arts and design" == Don't we already have an "Arts and design" school under Humanities? And isn't this redundant? --[[User:RayneVanDunem|RayneVanDunem]] ([[User talk:RayneVanDunem|talk]]) 05:32, 11 August 2012 (UTC) == Car Painting == i am a calligrapher artist .I want to write letters in a way that it give embossing effect. I have an idea to write with potene ,material used in car painting. Is there any person who knows about this kind of materials --[[User:Imranrafik|Imranrafik]] ([[User talk:Imranrafik|discuss]] • [[Special:Contributions/Imranrafik|contribs]]) 09:25, 26 April 2014 (UTC) == Isivile Nkungu Khenkebe == I am 15 years old [[User:Isivile Nkungu Khenkebe|Isivile Nkungu Khenkebe]] ([[User talk:Isivile Nkungu Khenkebe|discuss]] • [[Special:Contributions/Isivile Nkungu Khenkebe|contribs]]) 15:26, 1 August 2022 (UTC) ch9stre94povhtpkkf2qwzngjzyxv80 4 162205 2410827 2409289 2022-08-01T21:21:53Z MediaWiki message delivery 983498 /* Tech News: 2022-31 */ new section wikitext text/x-wiki {{Archive box|[[/2014/]] · [[/2015/]] · [[/2016/]] · [[/2017/]] · [[/2018/]] · [[/2019/]] · [[/2020/]] · [[/2021/]]}} __TOC__ {{Clear}} == [[m:Special:MyLanguage/Tech/News/2022/02|Tech News: 2022-02]] == <section begin="technews-2022-W02"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/02|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] A <bdi lang="zxx" dir="ltr"><code>oauth_consumer</code></bdi> variable has been added to the [[mw:Special:MyLanguage/AbuseFilter|AbuseFilter]] to enable identifying changes made by specific tools. [https://phabricator.wikimedia.org/T298281] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets are [[mw:Special:MyLanguage/ResourceLoader/Migration_guide_(users)#Package_Gadgets|now able to directly include JSON pages]]. This means some gadgets can now be configured by administrators without needing the interface administrator permission, such as with the Geonotice gadget. [https://phabricator.wikimedia.org/T198758] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets [[mw:Extension:Gadgets#Options|can now specify page actions]] on which they are available. For example, <bdi lang="zxx" dir="ltr"><code>|actions=edit,history</code></bdi> will load a gadget only while editing and on history pages. [https://phabricator.wikimedia.org/T63007] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets can now be loaded on demand with the <bdi lang="zxx" dir="ltr"><code>withgadget</code></bdi> URL parameter. This can be used to replace [[mw:Special:MyLanguage/Snippets/Load JS and CSS by URL|an earlier snippet]] that typically looks like <bdi lang="zxx" dir="ltr"><code>withJS</code></bdi> or <bdi lang="zxx" dir="ltr"><code>withCSS</code></bdi>. [https://phabricator.wikimedia.org/T29766] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] At wikis where [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list|the Mentorship system is configured]], you can now use the Action API to get a list of a [[mw:Special:MyLanguage/Growth/Mentor_dashboard|mentor's]] mentees. [https://phabricator.wikimedia.org/T291966] * The heading on the main page can now be configured using <span class="mw-content-ltr" lang="en" dir="ltr">[[MediaWiki:Mainpage-title-loggedin]]</span> for logged-in users and <span class="mw-content-ltr" lang="en" dir="ltr">[[MediaWiki:Mainpage-title]]</span> for logged-out users. Any CSS that was previously used to hide the heading should be removed. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Small_wiki_toolkits/Starter_kit/Main_page_customization#hide-heading] [https://phabricator.wikimedia.org/T298715] * Four special pages (and their API counterparts) now have a maximum database query execution time of 30 seconds. These special pages are: RecentChanges, Watchlist, Contributions, and Log. This change will help with site performance and stability. You can read [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IPJNO75HYAQWIGTHI5LJHTDVLVOC4LJP/ more details about this change] including some possible solutions if this affects your workflows. [https://phabricator.wikimedia.org/T297708] * The [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Sticky Header|sticky header]] has been deployed for 50% of logged-in users on [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Frequently asked questions#pilot-wikis|more than 10 wikis]]. This is part of the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Desktop Improvements]]. See [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Participate|how to take part in the project]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-12|en}}. It will be on all wikis from {{#time:j xg|2022-01-13|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Events''' * [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey 2022]] begins. All contributors to the Wikimedia projects can propose for tools and platform improvements. The proposal phase takes place from {{#time:j xg|2022-01-10|en}} 18:00 UTC to {{#time:j xg|2022-01-23|en}} 18:00 UTC. [[m:Special:MyLanguage/Community_Wishlist_Survey/FAQ|Learn more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/02|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W02"/> 01:23, 11 January 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22562156 --> == [[m:Special:MyLanguage/Tech/News/2022/03|Tech News: 2022-03]] == <section begin="technews-2022-W03"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/03|Translations]] are available. '''Recent changes''' * When using [[mw:Special:MyLanguage/Extension:WikiEditor|WikiEditor]] (also known as the 2010 wikitext editor), people will now see a warning if they link to disambiguation pages. If you click "{{int:Disambiguator-review-link}}" in the warning, it will ask you to correct the link to a more specific term. You can [[m:Community Wishlist Survey 2021/Warn when linking to disambiguation pages#Jan 12, 2021: Turning on the changes for all Wikis|read more information]] about this completed 2021 Community Wishlist item. * You can [[mw:Special:MyLanguage/Help:DiscussionTools#subscribe|automatically subscribe to all of the talk page discussions]] that you start or comment in using [[mw:Special:MyLanguage/Talk pages project/Feature summary|DiscussionTools]]. You will receive [[mw:Special:MyLanguage/Notifications|notifications]] when another editor replies. This is available at most wikis. Go to your [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]] and turn on "{{int:discussiontools-preference-autotopicsub}}". [https://phabricator.wikimedia.org/T263819] * When asked to create a new page or talk page section, input fields can be [[mw:Special:MyLanguage/Manual:Creating_pages_with_preloaded_text|"preloaded" with some text]]. This feature is now limited to wikitext pages. This is so users can't be tricked into making malicious edits. There is a discussion about [[phab:T297725|if this feature should be re-enabled]] for some content types. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-19|en}}. It will be on all wikis from {{#time:j xg|2022-01-20|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Events''' * [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey 2022]] continues. All contributors to the Wikimedia projects can propose for tools and platform improvements. The proposal phase takes place from {{#time:j xg|2022-01-10|en}} 18:00 UTC to {{#time:j xg|2022-01-23|en}} 18:00 UTC. [[m:Special:MyLanguage/Community_Wishlist_Survey/FAQ|Learn more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/03|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W03"/> 19:55, 17 January 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22620285 --> == [[m:Special:MyLanguage/Tech/News/2022/04|Tech News: 2022-04]] == <section begin="technews-2022-W04"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/04|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-26|en}}. It will be on all wikis from {{#time:j xg|2022-01-27|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * The following languages can now be used with [[mw:Special:MyLanguage/Extension:SyntaxHighlight|syntax highlighting]]: BDD, Elpi, LilyPond, Maxima, Rita, Savi, Sed, Sophia, Spice, .SRCINFO. * You can now access your watchlist from outside of the user menu in the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|new Vector skin]]. The watchlist link appears next to the notification icons if you are at the top of the page. [https://phabricator.wikimedia.org/T289619] '''Events''' * You can see the results of the [[m:Special:MyLanguage/Coolest Tool Award|Coolest Tool Award 2021]] and learn more about 14 tools which were selected this year. * You can [[m:Special:MyLanguage/Community_Wishlist_Survey/Help_us|translate, promote]], or comment on [[m:Special:MyLanguage/Community Wishlist Survey 2022/Proposals|the proposals]] in the Community Wishlist Survey. Voting will begin on {{#time:j xg|2022-01-28|en}}. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/04|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W04"/> 21:38, 24 January 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22644148 --> == [[m:Special:MyLanguage/Tech/News/2022/05|Tech News: 2022-05]] == <section begin="technews-2022-W05"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/05|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] If a gadget should support the new <bdi lang="zxx" dir="ltr"><code>?withgadget</code></bdi> URL parameter that was [[m:Special:MyLanguage/Tech/News/2022/02|announced]] 3 weeks ago, then it must now also specify <bdi lang="zxx" dir="ltr"><code>supportsUrlLoad</code></bdi> in the gadget definition ([[mw:Special:MyLanguage/Extension:Gadgets#supportsUrlLoad|documentation]]). [https://phabricator.wikimedia.org/T29766] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-02|en}}. It will be on all wikis from {{#time:j xg|2022-02-03|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Future changes''' * A change that was [[m:Special:MyLanguage/Tech/News/2021/16|announced]] last year was delayed. It is now ready to move ahead: ** The user group <code>oversight</code> will be renamed <code>suppress</code>. This is for [[phab:T109327|technical reasons]]. This is the technical name. It doesn't affect what you call the editors with this user right on your wiki. This is planned to happen in three weeks. You can comment [[phab:T112147|in Phabricator]] if you have objections. As usual, these labels can be translated on translatewiki ([[phab:T112147|direct links are available]]) or by administrators on your wiki. '''Events''' * You can vote on proposals in the [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey]] between 28 January and 11 February. The survey decides what the [[m:Special:MyLanguage/Community Tech|Community Tech team]] will work on. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/05|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W05"/> 17:42, 31 January 2022 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22721804 --> == [[m:Special:MyLanguage/Tech/News/2022/06|Tech News: 2022-06]] == <section begin="technews-2022-W06"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/06|Translations]] are available. '''Recent changes''' * English Wikipedia recently set up a gadget for dark mode. You can enable it there, or request help from an [[m:Special:MyLanguage/Interface administrators|interface administrator]] to set it up on your wiki ([[w:en:Wikipedia:Dark mode (gadget)|instructions and screenshot]]). * Category counts are sometimes wrong. They will now be completely recounted at the beginning of every month. [https://phabricator.wikimedia.org/T299823] '''Problems''' * A code-change last week to fix a bug with [[mw:Special:MyLanguage/Manual:Live preview|Live Preview]] may have caused problems with some local gadgets and user-scripts. Any code with skin-specific behaviour for <bdi lang="zxx" dir="ltr"><code>vector</code></bdi> should be updated to also check for <bdi lang="zxx" dir="ltr"><code>vector-2022</code></bdi>. [[phab:T300987|A code-snippet, global search, and example are available]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-09|en}}. It will be on all wikis from {{#time:j xg|2022-02-10|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/06|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W06"/> 21:15, 7 February 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22765948 --> == [[m:Special:MyLanguage/Tech/News/2022/07|Tech News: 2022-07]] == <section begin="technews-2022-W07"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/07|Translations]] are available. '''Recent changes''' * [[mw:Special:MyLanguage/Manual:Purge|Purging]] a category page with fewer than 5,000 members will now recount it completely. This will allow editors to fix incorrect counts when it is wrong. [https://phabricator.wikimedia.org/T85696] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-16|en}}. It will be on all wikis from {{#time:j xg|2022-02-17|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * [[File:Octicons-tools.svg|15px|link=|Advanced item]] In the [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] extension, the <code dir=ltr>rmspecials()</code> function has been updated so that it does not remove the "space" character. Wikis are advised to wrap all the uses of <code dir=ltr>rmspecials()</code> with <code dir=ltr>rmwhitespace()</code> wherever necessary to keep filters' behavior unchanged. You can use the search function on [[Special:AbuseFilter]] to locate its usage. [https://phabricator.wikimedia.org/T263024] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/07|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W07"/> 19:18, 14 February 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22821788 --> == [[m:Special:MyLanguage/Tech/News/2022/08|Tech News: 2022-08]] == <section begin="technews-2022-W08"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/08|Translations]] are available. '''Recent changes''' * [[Special:Nuke|Special:Nuke]] will now provide the standard deletion reasons (editable at <bdi lang="en" dir="ltr">[[MediaWiki:Deletereason-dropdown]]</bdi>) to use when mass-deleting pages. This was [[m:Community Wishlist Survey 2022/Admins and patrollers/Mass-delete to offer drop-down of standard reasons, or templated reasons.|a request in the 2022 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T25020] * At Wikipedias, all new accounts now get the [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] by default when creating an account. Communities are encouraged to [[mw:Special:MyLanguage/Help:Growth/Tools/Account_creation|update their help resources]]. Previously, only 80% of new accounts would get the Growth features. A few Wikipedias remain unaffected by this change. [https://phabricator.wikimedia.org/T301820] * You can now prevent specific images that are used in a page from appearing in other locations, such as within PagePreviews or Search results. This is done with the markup <bdi lang="zxx" dir="ltr"><code><nowiki>class=notpageimage</nowiki></code></bdi>. For example, <code><nowiki>[[File:Example.png|class=notpageimage]]</nowiki></code>. [https://phabricator.wikimedia.org/T301588] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] There has been a change to the HTML of Special:Contributions, Special:MergeHistory, and History pages, to support the grouping of changes by date in [[mw:Special:MyLanguage/Skin:Minerva_Neue|the mobile skin]]. While unlikely, this may affect gadgets and user scripts. A [[phab:T298638|list of all the HTML changes]] is on Phabricator. '''Events''' * [[m:Special:MyLanguage/Community Wishlist Survey 2022/Results|Community Wishlist Survey results]] have been published. The [[m:Special:MyLanguage/Community Wishlist Survey/Updates/2022 results#leaderboard|ranking of prioritized proposals]] is also available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-22|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-23|en}}. It will be on all wikis from {{#time:j xg|2022-02-24|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Future changes''' * The software to play videos and audio files on pages will change soon on all wikis. The old player will be removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Toolforge's underlying operating system is being updated. If you maintain any tools there, there are two options for migrating your tools into the new system. There are [[wikitech:News/Toolforge Stretch deprecation|details, deadlines, and instructions]] on Wikitech. [https://lists.wikimedia.org/hyperkitty/list/cloud-announce@lists.wikimedia.org/thread/EPJFISC52T7OOEFH5YYMZNL57O4VGSPR/] * Administrators will soon have [[m:Special:MyLanguage/Community Wishlist Survey 2021/(Un)delete associated talk page|the option to delete/undelete]] the associated "talk" page when they are deleting a given page. An API endpoint with this option will also be available. This was [[m:Community Wishlist Survey 2021/Admins and patrollers/(Un)delete associated talk page|a request from the 2021 Wishlist Survey]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/08|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W08"/> 19:12, 21 February 2022 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22847768 --> == [[m:Special:MyLanguage/Tech/News/2022/09|Tech News: 2022-09]] == <section begin="technews-2022-W09"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/09|Translations]] are available. '''Recent changes''' * When searching for edits by [[mw:Special:MyLanguage/Help:Tags|change tags]], e.g. in page history or user contributions, there is now a dropdown list of possible tags. This was [[m:Community Wishlist Survey 2022/Miscellaneous/Improve plain-text change tag selector|a request in the 2022 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T27909] * Mentors using the [[mw:Special:MyLanguage/Growth/Mentor_dashboard|Growth Mentor dashboard]] will now see newcomers assigned to them who have made at least one edit, up to 200 edits. Previously, all newcomers assigned to the mentor were visible on the dashboard, even ones without any edit or ones who made hundred of edits. Mentors can still change these values using the filters on their dashboard. Also, the last choice of filters will now be saved. [https://phabricator.wikimedia.org/T301268][https://phabricator.wikimedia.org/T294460] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The user group <code>oversight</code> was renamed <code>suppress</code>. This is for [[phab:T109327|technical reasons]]. You may need to update any local references to the old name, e.g. gadgets, links to Special:Listusers, or uses of [[mw:Special:MyLanguage/Help:Magic_words|NUMBERINGROUP]]. '''Problems''' * The recent change to the HTML of [[mw:Special:MyLanguage/Help:Tracking changes|tracking changes]] pages caused some problems for screenreaders. This is being fixed. [https://phabricator.wikimedia.org/T298638] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.24|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-02|en}}. It will be on all wikis from {{#time:j xg|2022-03-03|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Future changes''' * Working with templates will become easier. [[m:WMDE_Technical_Wishes/Templates|Several improvements]] are planned for March 9 on most wikis and on March 16 on English Wikipedia. The improvements include: Bracket matching, syntax highlighting colors, finding and inserting templates, and related visual editor features. * If you are a template developer or an interface administrator, and you are intentionally overriding or using the default CSS styles of user feedback boxes (the classes: <code dir=ltr>successbox, messagebox, errorbox, warningbox</code>), please note that these classes and associated CSS will soon be removed from MediaWiki core. This is to prevent problems when the same class-names are also used on a wiki. Please let us know by commenting at [[phab:T300314]] if you think you might be affected. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/09|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W09"/> 22:59, 28 February 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22902593 --> == [[m:Special:MyLanguage/Tech/News/2022/10|Tech News: 2022-10]] == <section begin="technews-2022-W10"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/10|Translations]] are available. '''Problems''' * There was a problem with some interface labels last week. It will be fixed this week. This change was part of ongoing work to simplify the support for skins which do not have active maintainers. [https://phabricator.wikimedia.org/T301203] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-09|en}}. It will be on all wikis from {{#time:j xg|2022-03-10|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/10|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W10"/> 21:16, 7 March 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22958074 --> == [[m:Special:MyLanguage/Tech/News/2022/11|Tech News: 2022-11]] == <section begin="technews-2022-W11"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/11|Translations]] are available. '''Recent changes''' * In the Wikipedia Android app [[mw:Special:MyLanguage/Wikimedia_Apps/Team/Android/Communication#Updates|it is now possible]] to change the toolbar at the bottom so the tools you use more often are easier to click on. The app now also has a focused reading mode. [https://phabricator.wikimedia.org/T296753][https://phabricator.wikimedia.org/T254771] '''Problems''' * There was a problem with the collection of some page-view data from June 2021 to January 2022 on all wikis. This means the statistics are incomplete. To help calculate which projects and regions were most affected, relevant datasets are being retained for 30 extra days. You can [[m:Talk:Data_retention_guidelines#Added_exception_for_page_views_investigation|read more on Meta-wiki]]. * There was a problem with the databases on March 10. All wikis were unreachable for logged-in users for 12 minutes. Logged-out users could read pages but could not edit or access uncached content then. [https://wikitech.wikimedia.org/wiki/Incident_documentation/2022-03-10_MediaWiki_availability] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-16|en}}. It will be on all wikis from {{#time:j xg|2022-03-17|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * When [[mw:Special:MyLanguage/Help:System_message#Finding_messages_and_documentation|using <bdi lang="zxx" dir="ltr"><code>uselang=qqx</code></bdi> to find localisation messages]], it will now show all possible message keys for navigation tabs such as "{{int:vector-view-history}}". [https://phabricator.wikimedia.org/T300069] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Access to [[{{#special:RevisionDelete}}]] has been expanded to include users who have <code dir=ltr>deletelogentry</code> and <code dir=ltr>deletedhistory</code> rights through their group memberships. Before, only those with the <code dir=ltr>deleterevision</code> right could access this special page. [https://phabricator.wikimedia.org/T301928] * On the [[{{#special:Undelete}}]] pages for diffs and revisions, there will be a link back to the main Undelete page with the list of revisions. [https://phabricator.wikimedia.org/T284114] '''Future changes''' * The Wikimedia Foundation has announced the IP Masking implementation strategy and next steps. The [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation#feb25|announcement can be read here]]. * The [[mw:Special:MyLanguage/Wikimedia Apps/Android FAQ|Wikipedia Android app]] developers are working on [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Communication|new functions]] for user talk pages and article talk pages. [https://phabricator.wikimedia.org/T297617] '''Events''' * The [[mw:Wikimedia Hackathon 2022|Wikimedia Hackathon 2022]] will take place as a hybrid event on 20-22 May 2022. The Hackathon will be held online and there are grants available to support local in-person meetups around the world. Grants can be requested until 20 March. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/11|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W11"/> 22:07, 14 March 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22993074 --> == [[m:Special:MyLanguage/Tech/News/2022/12|Tech News: 2022-12]] == <section begin="technews-2022-W12"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/12|Translations]] are available. '''New code release schedule for this week''' * There will be four MediaWiki releases this week, instead of just one. This is an experiment which should lead to fewer problems and to faster feature updates. The releases will be on all wikis, at different times, on Monday, Tuesday, and Wednesday. You can [[mw:Special:MyLanguage/Wikimedia Release Engineering Team/Trainsperiment week|read more about this project]]. '''Recent changes''' * You can now set how many search results to show by default in [[Special:Preferences#mw-prefsection-searchoptions|your Preferences]]. This was the 12th most popular wish in the [[m:Special:MyLanguage/Community Wishlist Survey 2022/Results|Community Wishlist Survey 2022]]. [https://phabricator.wikimedia.org/T215716] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The Jupyter notebooks tool [[wikitech:PAWS|PAWS]] has been updated to a new interface. [https://phabricator.wikimedia.org/T295043] '''Future changes''' * Interactive maps via [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] will soon work on wikis using the [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevisions]] extension. [https://wikimedia.sslsurvey.de/Kartographer-Workflows-EN/ Please tell us] which improvements you want to see in Kartographer. You can take this survey in simple English. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/12|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W12"/> 16:01, 21 March 2022 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23034693 --> == [[m:Special:MyLanguage/Tech/News/2022/13|Tech News: 2022-13]] == <section begin="technews-2022-W13"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/13|Translations]] are available. '''Recent changes''' * There is a simple new Wikimedia Commons upload tool available for macOS users, [[c:Commons:Sunflower|Sunflower]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-30|en}}. It will be on all wikis from {{#time:j xg|2022-03-31|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * Some wikis will be in read-only for a few minutes because of regular database maintenance. It will be performed on {{#time:j xg|2022-03-29|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]) and on {{#time:j xg|2022-03-31|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]). [https://phabricator.wikimedia.org/T301850][https://phabricator.wikimedia.org/T303798] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/13|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W13"/> 19:54, 28 March 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23073711 --> == [[m:Special:MyLanguage/Tech/News/2022/14|Tech News: 2022-14]] == <section begin="technews-2022-W14"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/14|Translations]] are available. '''Problems''' * For a few days last week, edits that were suggested to newcomers were not tagged in the [[{{#special:recentchanges}}]] feed. This bug has been fixed. [https://phabricator.wikimedia.org/T304747] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-06|en}}. It will be on all wikis from {{#time:j xg|2022-04-07|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-07|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]). '''Future changes''' * Starting next week, Tech News' title will be translatable. When the newsletter is distributed, its title may not be <code dir=ltr>Tech News: 2022-14</code> anymore. It may affect some filters that have been set up by some communities. [https://phabricator.wikimedia.org/T302920] * Over the next few months, the "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" Growth feature [[phab:T304110|will become available to more Wikipedias]]. Each week, a few wikis will get the feature. You can test this tool at [[mw:Special:MyLanguage/Growth#deploymentstable|a few wikis where "Link recommendation" is already available]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/14|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W14"/> 21:01, 4 April 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23097604 --> == Tech News: 2022-15 == <section begin="technews-2022-W15"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/15|Translations]] are available. '''Recent changes''' * There is a new public status page at <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wikimediastatus.net/ www.wikimediastatus.net]</span>. This site shows five automated high-level metrics where you can see the overall health and performance of our wikis' technical environment. It also contains manually-written updates for widespread incidents, which are written as quickly as the engineers are able to do so while also fixing the actual problem. The site is separated from our production infrastructure and hosted by an external service, so that it can be accessed even if the wikis are briefly unavailable. You can [https://diff.wikimedia.org/2022/03/31/announcing-www-wikimediastatus-net/ read more about this project]. * On Wiktionary wikis, the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-13|en}}. It will be on all wikis from {{#time:j xg|2022-04-14|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/15|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W15"/> 19:44, 11 April 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23124108 --> == Tech News: 2022-16 == <section begin="technews-2022-W16"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/16|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-20|en}}. It will be on all wikis from {{#time:j xg|2022-04-21|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-19|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]) and on {{#time:j xg|2022-04-21|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s8.dblist targeted wikis]). * Administrators will now have [[m:Community Wishlist Survey 2021/(Un)delete associated talk page|the option to delete/undelete the associated "Talk" page]] when they are deleting a given page. An API endpoint with this option is also available. This concludes the [[m:Community Wishlist Survey 2021/Admins and patrollers/(Un)delete associated talk page|11th wish of the 2021 Community Wishlist Survey]]. * On [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements#test-wikis|selected wikis]], 50% of logged-in users will see the new [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Table of contents|table of contents]]. When scrolling up and down the page, the table of contents will stay in the same place on the screen. This is part of the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Desktop Improvements]] project. [https://phabricator.wikimedia.org/T304169] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Message boxes produced by MediaWiki code will no longer have these CSS classes: <code dir=ltr>successbox</code>, <code dir=ltr>errorbox</code>, <code dir=ltr>warningbox</code>. The styles for those classes and <code dir=ltr>messagebox</code> will be removed from MediaWiki core. This only affects wikis that use these classes in wikitext, or change their appearance within site-wide CSS. Please review any local usage and definitions for these classes you may have. This was previously announced in the [[m:Special:MyLanguage/Tech/News/2022/09|28 February issue of Tech News]]. '''Future changes''' * [[mw:Special:MyLanguage/Extension:Kartographer|Kartographer]] will become compatible with [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevisions page stabilization]]. Kartographer maps will also work on pages with [[mw:Special:MyLanguage/Help:Pending changes|pending changes]]. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation#Project_descriptions] The Kartographer documentation has been thoroughly updated. [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Extension:Kartographer/Getting_started] [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:VisualEditor/Maps] [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Extension:Kartographer] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/16|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W16"/> 23:11, 18 April 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23167004 --> == Tech News: 2022-17 == <section begin="technews-2022-W17"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/17|Translations]] are available. '''Recent changes''' * On [https://noc.wikimedia.org/conf/dblists/group1.dblist many wikis] (group 1), the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-27|en}}. It will be on all wikis from {{#time:j xg|2022-04-28|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-26|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s2.dblist targeted wikis]). * Some very old browsers and operating systems are no longer supported. Some things on the wikis might look weird or not work in very old browsers like Internet Explorer 9 or 10, Android 4, or Firefox 38 or older. [https://phabricator.wikimedia.org/T306486] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/17|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W17"/> 22:56, 25 April 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23187115 --> == Tech News: 2022-18 == <section begin="technews-2022-W18"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/18|Translations]] are available. '''Recent changes''' * On [https://noc.wikimedia.org/conf/dblists/group2.dblist all remaining wikis] (group 2), the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-03|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-04|en}}. It will be on all wikis from {{#time:j xg|2022-05-05|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''Future changes''' * The developers are working on talk pages in the [[mw:Wikimedia Apps/Team/iOS|Wikipedia app for iOS]]. You can [https://wikimedia.qualtrics.com/jfe/form/SV_9GBcHczQGLbQWTY give feedback]. You can take the survey in English, German, Hebrew or Chinese. * [[m:WMDE_Technical_Wishes/VisualEditor_template_dialog_improvements#Status_and_next_steps|Most wikis]] will receive an [[m:WMDE_Technical_Wishes/VisualEditor_template_dialog_improvements|improved template dialog]] in VisualEditor and New Wikitext mode. [https://phabricator.wikimedia.org/T296759] [https://phabricator.wikimedia.org/T306967] * If you use syntax highlighting while editing wikitext, you can soon activate a [[m:WMDE_Technical_Wishes/Improved_Color_Scheme_of_Syntax_Highlighting#Color-blind_mode|colorblind-friendly color scheme]]. [https://phabricator.wikimedia.org/T306867] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Several CSS IDs related to MediaWiki interface messages will be removed. Technical editors should please [[phab:T304363|review the list of IDs and links to their existing uses]]. These include <code dir=ltr>#mw-anon-edit-warning</code>, <code dir=ltr>#mw-undelete-revision</code> and 3 others. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/18|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W18"/> 19:33, 2 May 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23232924 --> == Tech News: 2022-19 == <section begin="technews-2022-W19"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/19|Translations]] are available. '''Recent changes''' * You can now see categories in the [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android|Wikipedia app for Android]]. [https://phabricator.wikimedia.org/T73966] '''Problems''' * Last week, there was a problem with Wikidata's search autocomplete. This has now been fixed. [https://phabricator.wikimedia.org/T307586] * Last week, all wikis had slow access or no access for 20 minutes, for logged-in users and non-cached pages. This was caused by a problem with a database change. [https://phabricator.wikimedia.org/T307647] '''Changes later this week''' * There is no new MediaWiki version this week. [https://phabricator.wikimedia.org/T305217#7894966] * [[m:WMDE Technical Wishes/Geoinformation#Current issues|Incompatibility issues]] with [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] and the [[mw:Special:MyLanguage/Help:Extension:FlaggedRevs|FlaggedRevs extension]] will be fixed: Deployment is planned for May 10 on all wikis. Kartographer will then be enabled on the [[phab:T307348|five wikis which have not yet enabled the extension]] on May 24. * The [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector (2022)]] skin will be set as the default on several more wikis, including Arabic and Catalan Wikipedias. Logged-in users will be able to switch back to the old Vector (2010). See the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/2022-04 for the largest wikis|latest update]] about Vector (2022). '''Future meetings''' * The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place on 17 May. The following meetings are currently planned for: 7 June, 21 June, 5 July, 19 July. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/19|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W19"/> 15:22, 9 May 2022 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23256717 --> == Tech News: 2022-20 == <section begin="technews-2022-W20"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/20|Translations]] are available. '''Changes later this week''' * Some wikis can soon use the [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|add a link]] feature. This will start on Wednesday. The wikis are {{int:project-localized-name-cawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ptwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-simplewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-svwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ukwiki/en}}. This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304542] * The [[mw:Special:MyLanguage/Wikimedia Hackathon 2022|Wikimedia Hackathon 2022]] will take place online on May 20–22. It will be in English. There are also local [[mw:Special:MyLanguage/Wikimedia Hackathon 2022/Meetups|hackathon meetups]] in Germany, Ghana, Greece, India, Nigeria and the United States. Technically interested Wikimedians can work on software projects and learn new skills. You can also host a session or post a project you want to work on. * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-18|en}}. It will be on all wikis from {{#time:j xg|2022-05-19|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''Future changes''' * You can soon edit translatable pages in the visual editor. Translatable pages exist on for examples Meta and Commons. [https://diff.wikimedia.org/2022/05/12/mediawiki-1-38-brings-support-for-editing-translatable-pages-with-the-visual-editor/] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/20|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W20"/> 18:58, 16 May 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23291515 --> == Tech News: 2022-21 == <section begin="technews-2022-W21"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/21|Translations]] are available. '''Recent changes''' * Administrators using the mobile web interface can now access Special:Block directly from user pages. [https://phabricator.wikimedia.org/T307341] * The <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wiktionary.org/ www.wiktionary.org]</span> portal page now uses an automated update system. Other [[m:Project_portals|project portals]] will be updated over the next few months. [https://phabricator.wikimedia.org/T304629] '''Problems''' * The Growth team maintains a mentorship program for newcomers. Previously, newcomers weren't able to opt out from the program. Starting May 19, 2022, newcomers are able to fully opt out from Growth mentorship, in case they do not wish to have any mentor at all. [https://phabricator.wikimedia.org/T287915] * Some editors cannot access the content translation tool if they load it by clicking from the contributions menu. This problem is being worked on. It should still work properly if accessed directly via Special:ContentTranslation. [https://phabricator.wikimedia.org/T308802] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-25|en}}. It will be on all wikis from {{#time:j xg|2022-05-26|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Gadget and user scripts developers are invited to give feedback on a [[mw:User:Jdlrobson/Extension:Gadget/Policy|proposed technical policy]] aiming to improve support from MediaWiki developers. [https://phabricator.wikimedia.org/T308686] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/21|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W21"/> 00:21, 24 May 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23317250 --> == Tech News: 2022-22 == <section begin="technews-2022-W22"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/22|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] In the [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] extension, an <code dir=ltr>ip_in_ranges()</code> function has been introduced to check if an IP is in any of the ranges. Wikis are advised to combine multiple <code dir=ltr>ip_in_range()</code> expressions joined by <code>|</code> into a single expression for better performance. You can use the search function on [[Special:AbuseFilter|Special:AbuseFilter]] to locate its usage. [https://phabricator.wikimedia.org/T305017] * The [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature|IP Info feature]] which helps abuse fighters access information about IPs, [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#May 24, 2022|has been deployed]] to all wikis as a beta feature. This comes after weeks of beta testing on test.wikipedia.org. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.14|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-01|en}}. It will be on all wikis from {{#time:j xg|2022-06-02|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-05-31|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]). * The [[mw:Special:MyLanguage/Help:DiscussionTools#New topic tool|New Topic Tool]] will be deployed for all editors at most wikis soon. You will be able to opt out from within the tool and in [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Talk_pages_project/New_discussion][https://phabricator.wikimedia.org/T287804] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[:mw:Special:ApiHelp/query+usercontribs|list=usercontribs API]] will support fetching contributions from an [[mw:Special:MyLanguage/Help:Range blocks#Non-technical explanation|IP range]] soon. API users can set the <code>uciprange</code> parameter to get contributions from any IP range within [[:mw:Manual:$wgRangeContributionsCIDRLimit|the limit]]. [https://phabricator.wikimedia.org/T177150] * A new parser function will be introduced: <bdi lang="zxx" dir="ltr"><code><nowiki>{{=}}</nowiki></code></bdi>. It will replace existing templates named "=". It will insert an [[w:en:Equals sign|equal sign]]. This can be used to escape the equal sign in the parameter values of templates. [https://phabricator.wikimedia.org/T91154] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/22|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W22"/> 20:28, 30 May 2022 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23340178 --> == Tech News: 2022-23 == <section begin="technews-2022-W23"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/23|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-08|en}}. It will be on all wikis from {{#time:j xg|2022-06-09|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] A new <bdi lang="zxx" dir="ltr"><code>str_replace_regexp()</code></bdi> function can be used in [[Special:AbuseFilter|abuse filters]] to replace parts of text using a [[w:en:Regular expression|regular expression]]. [https://phabricator.wikimedia.org/T285468] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/23|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W23"/> 02:46, 7 June 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23366979 --> == Tech News: 2022-24 == <section begin="technews-2022-W24"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/24|Translations]] are available. '''Recent changes''' * All wikis can now use [[mw:Special:MyLanguage/Extension:Kartographer|Kartographer]] maps. Kartographer maps now also work on pages with [[mw:Special:MyLanguage/Help:Pending changes|pending changes]]. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation#Project_descriptions][https://phabricator.wikimedia.org/T307348] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-15|en}}. It will be on all wikis from {{#time:j xg|2022-06-16|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-14|en}} at 06:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]). [https://phabricator.wikimedia.org/T300471] * Starting on Wednesday, a new set of Wikipedias will get "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" ({{int:project-localized-name-abwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-acewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-adywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-afwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-akwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-alswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-amwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-anwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-angwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-arcwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-arzwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-astwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-atjwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-avwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-aywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-azwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-azbwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304548] * The [[mw:Special:MyLanguage/Help:DiscussionTools#New topic tool|New Topic Tool]] will be deployed for all editors at Commons, Wikidata, and some other wikis soon. You will be able to opt out from within the tool and in [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Talk_pages_project/New_discussion][https://phabricator.wikimedia.org/T287804] '''Future meetings''' * The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place today (13 June). The following meetings will take place on: 28 June, 12 July, 26 July. '''Future changes''' * By the end of July, the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector 2022]] skin should be ready to become the default across all wikis. Discussions on how to adjust it to the communities' needs will begin in the next weeks. It will always be possible to revert to the previous version on an individual basis. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/2022-04 for the largest wikis|Learn more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/24|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W24"/> 16:58, 13 June 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23389956 --> == Tech News: 2022-25 == <section begin="technews-2022-W25"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/25|Translations]] are available. '''Recent changes''' * The [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android|Wikipedia App for Android]] now has an option for editing the whole page at once, located in the overflow menu (three-dots menu [[File:Ic more vert 36px.svg|15px|link=|alt=]]). [https://phabricator.wikimedia.org/T103622] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Some recent database changes may affect queries using the [[m:Research:Quarry|Quarry tool]]. Queries for <bdi lang="zxx" dir="ltr"><code>site_stats</code></bdi> at English Wikipedia, Commons, and Wikidata will need to be updated. [[phab:T306589|Read more]]. * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] A new <bdi lang="zxx" dir="ltr"><code>user_global_editcount</code></bdi> variable can be used in [[Special:AbuseFilter|abuse filters]] to avoid affecting globally active users. [https://phabricator.wikimedia.org/T130439] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-22|en}}. It will be on all wikis from {{#time:j xg|2022-06-23|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * Users of non-responsive skins (e.g. MonoBook or Vector) on mobile devices may notice a slight change in the default zoom level. This is intended to optimize zooming and ensure all interface elements are present on the page (for example the table of contents on Vector 2022). In the unlikely event this causes any problems with how you use the site, we'd love to understand better, please ping <span class="mw-content-ltr" lang="en" dir="ltr">[[m:User:Jon (WMF)|Jon (WMF)]]</span> to any on-wiki conversations. [https://phabricator.wikimedia.org/T306910] '''Future changes''' * The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July. Discussions will look different. You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Parsoid's HTML output will soon stop annotating file links with different <bdi lang="zxx" dir="ltr"><code>typeof</code></bdi> attribute values, and instead use <bdi lang="zxx" dir="ltr"><code>mw:File</code></bdi> for all types. Tool authors should adjust any code that expects: <bdi lang="zxx" dir="ltr"><code>mw:Image</code></bdi>, <bdi lang="zxx" dir="ltr"><code>mw:Audio</code></bdi>, or <bdi lang="zxx" dir="ltr"><code>mw:Video</code></bdi>. [https://phabricator.wikimedia.org/T273505] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/25|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W25"/> 20:18, 20 June 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23425855 --> == Tech News: 2022-26 == <section begin="technews-2022-W26"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/26|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] API service now has self-service accounts with free on-demand requests and monthly snapshots ([https://enterprise.wikimedia.com/docs/ API documentation]). Community access [[m:Special:MyLanguage/Wikimedia Enterprise/FAQ#community-access|via database dumps & Wikimedia Cloud Services]] continues. * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] [[d:Special:MyLanguage/Wikidata:Wiktionary#lua|All Wikimedia wikis can now use Wikidata Lexemes in Lua]] after creating local modules and templates. Discussions are welcome [[d:Wikidata_talk:Lexicographical_data#You_can_now_reuse_Wikidata_Lexemes_on_all_wikis|on the project talk page]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-29|en}}. It will be on all wikis from {{#time:j xg|2022-06-30|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-28|en}} at 06:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]). [https://phabricator.wikimedia.org/T311033] * Some global and cross-wiki services will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-30|en}} at 06:00 UTC. This will impact ContentTranslation, Echo, StructuredDiscussions, Growth experiments and a few more services. [https://phabricator.wikimedia.org/T300472] * Users will be able to sort columns within sortable tables in the mobile skin. [https://phabricator.wikimedia.org/T233340] '''Future meetings''' * The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place tomorrow (28 June). The following meetings will take place on 12 July and 26 July. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/26|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W26"/> 20:02, 27 June 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23453785 --> == Tech News: 2022-27 == <section begin="technews-2022-W27"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/27|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-06|en}}. It will be on all wikis from {{#time:j xg|2022-07-07|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-07-05|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]) and on {{#time:j xg|2022-07-07|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]). * The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July. Discussions will look different. You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. * [[File:Octicons-tools.svg|15px|link=|alt=| Advanced item]] This change only affects pages in the main namespace in Wikisource. The Javascript config variable <bdi lang="zxx" dir="ltr"><code>proofreadpage_source_href</code></bdi> will be removed from <bdi lang="zxx" dir="ltr"><code>[[mw:Special:MyLanguage/Manual:Interface/JavaScript#mw.config|mw.config]]</code></bdi> and be replaced with the variable <bdi lang="zxx" dir="ltr"><code>prpSourceIndexPage</code></bdi>. [https://phabricator.wikimedia.org/T309490] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/27|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W27"/> 19:32, 4 July 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23466250 --> == Tech News: 2022-28 == <section begin="technews-2022-W28"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/28|Translations]] are available. '''Recent changes''' * In the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector 2022 skin]], the page title is now displayed above the tabs such as Discussion, Read, Edit, View history, or More. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates#Page title/tabs switch|Learn more]]. [https://phabricator.wikimedia.org/T303549] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] It is now possible to easily view most of the configuration settings that apply to just one wiki, and to compare settings between two wikis if those settings are different. For example: [https://noc.wikimedia.org/wiki.php?wiki=jawiktionary Japanese Wiktionary settings], or [https://noc.wikimedia.org/wiki.php?wiki=eswiki&compare=eowiki settings that are different between the Spanish and Esperanto Wikipedias]. Local communities may want to [[m:Special:MyLanguage/Requesting_wiki_configuration_changes|discuss and propose changes]] to their local settings. Details about each of the named settings can be found by [[mw:Special:Search|searching MediaWiki.org]]. [https://phabricator.wikimedia.org/T308932] *The Anti-Harassment Tools team [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#May|recently deployed]] the IP Info Feature as a [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature at all wikis]]. This feature allows abuse fighters to access information about IP addresses. Please check our update on [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#April|how to find and use the tool]]. Please share your feedback using a link you will be given within the tool itself. '''Changes later this week''' * There is no new MediaWiki version this week. * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-07-12|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]). '''Future changes''' * The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July. Discussions will look different. You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/28|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W28"/> 19:24, 11 July 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23502519 --> == Tech News: 2022-29 == <section begin="technews-2022-W29"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/29|Translations]] are available. '''Problems''' * The feature on mobile web for [[mw:Special:MyLanguage/Extension:NearbyPages|Nearby Pages]] was missing last week. It will be fixed this week. [https://phabricator.wikimedia.org/T312864] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-20|en}}. It will be on all wikis from {{#time:j xg|2022-07-21|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''Future changes''' * The [[mw:Technical_decision_making/Forum|Technical Decision Forum]] is seeking [[mw:Technical_decision_making/Community_representation|community representatives]]. You can apply on wiki or by emailing <span class="mw-content-ltr" lang="en" dir="ltr">TDFSupport@wikimedia.org</span> before 12 August. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/29|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W29"/> 22:59, 18 July 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23517957 --> == Tech News: 2022-30 == <section begin="technews-2022-W30"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/30|Translations]] are available. '''Recent changes''' * The <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wikibooks.org/ www.wikibooks.org]</span> and <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wikiquote.org/ www.wikiquote.org]</span> portal pages now use an automated update system. Other [[m:Project_portals|project portals]] will be updated over the next few months. [https://phabricator.wikimedia.org/T273179] '''Problems''' * Last week, some wikis were in read-only mode for a few minutes because of an emergency switch of their main database ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]). [https://phabricator.wikimedia.org/T313383] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-27|en}}. It will be on all wikis from {{#time:j xg|2022-07-28|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * The external link icon will change slightly in the skins Vector legacy and Vector 2022. The new icon uses simpler shapes to be more recognizable on low-fidelity screens. [https://phabricator.wikimedia.org/T261391] * Administrators will now see buttons on user pages for "{{int:changeblockip}}" and "{{int:unblockip}}" instead of just "{{int:blockip}}" if the user is already blocked. [https://phabricator.wikimedia.org/T308570] '''Future meetings''' * The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place tomorrow (26 July). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/30|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W30"/> 19:27, 25 July 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23545370 --> == Tech News: 2022-31 == <section begin="technews-2022-W31"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/31|Translations]] are available. '''Recent changes''' * Improved [[m:Special:MyLanguage/Help:Displaying_a_formula#Phantom|LaTeX capabilities for math rendering]] are now available in the wikis thanks to supporting <bdi lang="zxx" dir="ltr"><code>Phantom</code></bdi> tags. This completes part of [[m:Community_Wishlist_Survey_2022/Editing/Missing_LaTeX_capabilities_for_math_rendering|the #59 wish]] of the 2022 Community Wishlist Survey. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-08-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-08-03|en}}. It will be on all wikis from {{#time:j xg|2022-08-04|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * The [[mw:Special:MyLanguage/Help:Extension:WikiEditor/Realtime_Preview|Realtime Preview]] will be available as a Beta Feature on wikis in [https://noc.wikimedia.org/conf/highlight.php?file=dblists%2Fgroup0.dblist Group 0]. This feature was built in order to fulfill [[m:Special:MyLanguage/Community_Wishlist_Survey_2021/Real_Time_Preview_for_Wikitext|one of the Community Wishlist Survey proposals]]. '''Future changes''' * The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout August. Discussions will look different. You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. '''Future meetings''' * This week, three meetings about [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector (2022)]] with live interpretation will take place. On Tuesday, interpretation in Russian will be provided. On Thursday, meetings for Arabic and Spanish speakers will take place. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|See how to join]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/31|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W31"/> 21:21, 1 August 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23615613 --> l03lw6utl9su0jyq6rile70wfv1z45u 1 170731 2410877 2409126 2022-08-02T03:12:11Z Mikael Häggström 12130 /* Editorial board application of Luis Rafael Moscote-Salazar */ Support wikitext text/x-wiki <noinclude> {{WikiJournal editorial application top |archive box = {{Archive box|[[/Archive 2015-2017]] <br>[[/Archive 2018]] <br>[[/Archive 2019]] <br>[[/Archive 2020]] <br>[[/Archive 2021]] <br>[[/Archive 2022]] }} }} </noinclude> ==Associate editor application of James Bibey== {{WikiJournal editor application submitted | position =Associate editor | name =James Bibey | qualifications =2nd Year Medical Student | link =N/A | areas_of_expertise =General medicine (basic anatomy, physiology, biochemistry, statistics, ethics) | professional_experience =Maths lecturing, anatomical prosection preparation. | publishing_experience =N/A | open_experience =Significant editing history on English Wikipedia (primarily medicine and anatomy), Wikimedia Commons, and Wikidata under username "Bibeyjj". | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Medicine. [[User:Bibeyjj|Bibeyjj]] ([[User talk:Bibeyjj|discuss]] • [[Special:Contributions/Bibeyjj|contribs]]) 19:23, 24 September 2021 (UTC) }} *{{Support}}. The applicant is [https://en.wikipedia.org/wiki/Special:Contributions/Bibeyjj active in related topics in Wikipedia], and I think we can really need the help for our journal too. [[User:Bibeyjj|Bibeyjj]], I hope you are still interested in this position. I'm sorry for the late response to your application, as you see we are quite busy with the everyday matters of the project. If elected, would you be willing to help out for instance in [[WikiJournal_of_Medicine/Editorial_guidelines#Arranging_peer_review|finding peer reviewers]] for article submissions to the journal? [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 19:10, 22 May 2022 (UTC) *{{Support}}. Mikael summarises the reasons well above and I agree [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 06:56, 23 May 2022 (UTC) *They seem a good candidate for assoc editor status, and it would be useful experience for them as well as helpful skills for us. We've been a bit stalled on applications over the last year, so it will be good to get organised again. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 11:59, 11 June 2022 (UTC) :I agree. I made [https://en.wikipedia.org/wiki/User_talk:Bibeyjj#Associate_editor_application an entry on the user's talk page] whether he's still interested. If so, I think we can go ahead and approve this application. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 15:52, 16 June 2022 (UTC) ==Editorial board application of Daniel Barton Oerther== {{WikiJournal editor application submitted | position =Editorial board | Q = Q18001544 | name =Daniel Barton Oerther | qualifications =PHD | link =https://people.mst.edu/faculty/oertherd/ | areas_of_expertise =environmental health | professional_experience =Professor of environmental health engineering; Certified environmental health specialist and leader of a number of national and international organizations specializing in environmental health | publishing_experience =Editorial boards of Perspectives in Public Health, Water Environment Research, and Journal of Environmental Engineering ASCE | open_experience =Wikipedia and regular reviewer for open access Nursing Open | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Medicine. [[User:Oertherdb|Oertherdb]] ([[User talk:Oertherdb|discuss]] • [[Special:Contributions/Oertherdb|contribs]]) 15:23, 25 February 2022 (UTC) }} *{{Support}}. The applicant has substantial credentials including health and leadership. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 19:16, 22 May 2022 (UTC) *{{Support}}. I was the person who approached this candidate on the basis of his interesting background, enthusiasm for nursing and health and his Wikipedia experience [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 06:57, 23 May 2022 (UTC) *{{Support}}. [[User:Editor WikiJMed|Editor WikiJMed]] ([[User talk:Editor WikiJMed|discuss]] • [[Special:Contributions/Editor WikiJMed|contribs]]) 21:07, 23 May 2022 (UTC) *{{support}} - I also suppoprt this application (apologies for not noting earlier). I particular, they have a usefully interdisciplinary knowledgebase connecting medicine to engineering and environmental research. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 11:56, 11 June 2022 (UTC) '''Result: Accepted into the editorial board.''' :[[WikiJournal_User_Group/Editorial_guidelines#Adding_editorial_board_members|Next steps]] (add <code>DONE</code> or <code><nowiki>{{Done}}</nowiki></code> after someone has performed the task): # {{Done}} [[{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member|Send a welcome message and confirm their preferred email address]] (usually in their provided website link, else via [[Special:EmailUser]]) # {{Done}} {{clickable button 2 | url=https://en.wikiversity.org/wiki/{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member|Onboarding email template}} # {{Done}} Copy their information over to [[{{ROOTPAGENAME}}/Editorial board|editorial board page]] using the {{tlx|WikiJournal editor summary}} template # __ Add their name and start data to the [[d:{{WJQboard|default=Q75674277}}|relevant editorial board]] on wikidata # {{Done}} Direct-add them to the {{WJX}}board mailing list ([https://groups.google.com/forum/?utm_medium=email&utm_source=footer#!managemembers/{{WJX}}board/add via this link]) which will grant them access to the private page only visible to board members # {{Done}} Welcome them at the {{WJX}}board mailing list so that they are informed # __ Finally, move the application to [[Talk:{{ROOTPAGENAME}}/Editors/Archive_{{CURRENTYEAR}}|this year's archive page]] ([[{{ROOTPAGENAME}}/Editorial guidelines/Message_templates#Onboarding_a_new_board_member|Suggested email template]])<br> [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 12:36, 13 June 2022 (UTC) ==Editorial board application of Luis Rafael Moscote-Salazar== {{WikiJournal editor application submitted | journal =WikiJournal of Medicine | position =Editorial board member | name =Luis Rafael Moscote-Salazar | qualifications =MD | link =https://neuroclani.org/ | areas_of_expertise =Neurosurgery, Neurotrauma, Stroke, Neurointervention, Neurocritical Care, Neurointervention, Evidence Based Medicine | professional_experience =Neurosurgeon graduated from the University of Cartagena, Founder of the Colombian Clinical Research Group in Neurocritical Care and Co-founder of the Latinamerican Council of Neurocritical Care (CLaNi). Research communication, research leadership, mentorship. | publishing_experience = Experience with publishing in peer-reviewed journals (see Google Scholar profile. Peer-reviewer for several international journals in Neurosurgery and Medicine (see Publons profile). | open_experience = | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Medicine. [[User:NeuroQuimbaya|NeuroQuimbaya]] ([[User talk:NeuroQuimbaya|discuss]]) 2022-07-12 }} :No details were provided, so I have contacted the applicant to request that they add more information. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 01:27, 7 July 2022 (UTC) ::I've updated the application above with the new replacement information that they sent through. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 03:55, 13 July 2022 (UTC) :::'''Support'''. This person seems active in research and publishing according to orcid ([https://orcid.org/0000-0002-4180-6962]). I think it is appropriate to let him join us. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 03:12, 2 August 2022 (UTC) i0t0glba0j6ormi8lfxl7q6i1hpf7zl 2410880 2410877 2022-08-02T03:13:30Z Mikael Häggström 12130 /* Associate editor application of James Bibey */ Emailed too wikitext text/x-wiki <noinclude> {{WikiJournal editorial application top |archive box = {{Archive box|[[/Archive 2015-2017]] <br>[[/Archive 2018]] <br>[[/Archive 2019]] <br>[[/Archive 2020]] <br>[[/Archive 2021]] <br>[[/Archive 2022]] }} }} </noinclude> ==Associate editor application of James Bibey== {{WikiJournal editor application submitted | position =Associate editor | name =James Bibey | qualifications =2nd Year Medical Student | link =N/A | areas_of_expertise =General medicine (basic anatomy, physiology, biochemistry, statistics, ethics) | professional_experience =Maths lecturing, anatomical prosection preparation. | publishing_experience =N/A | open_experience =Significant editing history on English Wikipedia (primarily medicine and anatomy), Wikimedia Commons, and Wikidata under username "Bibeyjj". | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Medicine. [[User:Bibeyjj|Bibeyjj]] ([[User talk:Bibeyjj|discuss]] • [[Special:Contributions/Bibeyjj|contribs]]) 19:23, 24 September 2021 (UTC) }} *{{Support}}. The applicant is [https://en.wikipedia.org/wiki/Special:Contributions/Bibeyjj active in related topics in Wikipedia], and I think we can really need the help for our journal too. [[User:Bibeyjj|Bibeyjj]], I hope you are still interested in this position. I'm sorry for the late response to your application, as you see we are quite busy with the everyday matters of the project. If elected, would you be willing to help out for instance in [[WikiJournal_of_Medicine/Editorial_guidelines#Arranging_peer_review|finding peer reviewers]] for article submissions to the journal? [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 19:10, 22 May 2022 (UTC) *{{Support}}. Mikael summarises the reasons well above and I agree [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 06:56, 23 May 2022 (UTC) *They seem a good candidate for assoc editor status, and it would be useful experience for them as well as helpful skills for us. We've been a bit stalled on applications over the last year, so it will be good to get organised again. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 11:59, 11 June 2022 (UTC) :I agree. I made [https://en.wikipedia.org/wiki/User_talk:Bibeyjj#Associate_editor_application an entry on the user's talk page] whether he's still interested. If so, I think we can go ahead and approve this application. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 15:52, 16 June 2022 (UTC) ::I haven't heard back from the wiki talk page, so I sent an email through the wiki system as well. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 03:13, 2 August 2022 (UTC) ==Editorial board application of Daniel Barton Oerther== {{WikiJournal editor application submitted | position =Editorial board | Q = Q18001544 | name =Daniel Barton Oerther | qualifications =PHD | link =https://people.mst.edu/faculty/oertherd/ | areas_of_expertise =environmental health | professional_experience =Professor of environmental health engineering; Certified environmental health specialist and leader of a number of national and international organizations specializing in environmental health | publishing_experience =Editorial boards of Perspectives in Public Health, Water Environment Research, and Journal of Environmental Engineering ASCE | open_experience =Wikipedia and regular reviewer for open access Nursing Open | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Medicine. [[User:Oertherdb|Oertherdb]] ([[User talk:Oertherdb|discuss]] • [[Special:Contributions/Oertherdb|contribs]]) 15:23, 25 February 2022 (UTC) }} *{{Support}}. The applicant has substantial credentials including health and leadership. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 19:16, 22 May 2022 (UTC) *{{Support}}. I was the person who approached this candidate on the basis of his interesting background, enthusiasm for nursing and health and his Wikipedia experience [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 06:57, 23 May 2022 (UTC) *{{Support}}. [[User:Editor WikiJMed|Editor WikiJMed]] ([[User talk:Editor WikiJMed|discuss]] • [[Special:Contributions/Editor WikiJMed|contribs]]) 21:07, 23 May 2022 (UTC) *{{support}} - I also suppoprt this application (apologies for not noting earlier). I particular, they have a usefully interdisciplinary knowledgebase connecting medicine to engineering and environmental research. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 11:56, 11 June 2022 (UTC) '''Result: Accepted into the editorial board.''' :[[WikiJournal_User_Group/Editorial_guidelines#Adding_editorial_board_members|Next steps]] (add <code>DONE</code> or <code><nowiki>{{Done}}</nowiki></code> after someone has performed the task): # {{Done}} [[{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member|Send a welcome message and confirm their preferred email address]] (usually in their provided website link, else via [[Special:EmailUser]]) # {{Done}} {{clickable button 2 | url=https://en.wikiversity.org/wiki/{{ROOTPAGENAMEE}}/Editorial_guidelines/Message_templates#Onboarding_a_new_board_member|Onboarding email template}} # {{Done}} Copy their information over to [[{{ROOTPAGENAME}}/Editorial board|editorial board page]] using the {{tlx|WikiJournal editor summary}} template # __ Add their name and start data to the [[d:{{WJQboard|default=Q75674277}}|relevant editorial board]] on wikidata # {{Done}} Direct-add them to the {{WJX}}board mailing list ([https://groups.google.com/forum/?utm_medium=email&utm_source=footer#!managemembers/{{WJX}}board/add via this link]) which will grant them access to the private page only visible to board members # {{Done}} Welcome them at the {{WJX}}board mailing list so that they are informed # __ Finally, move the application to [[Talk:{{ROOTPAGENAME}}/Editors/Archive_{{CURRENTYEAR}}|this year's archive page]] ([[{{ROOTPAGENAME}}/Editorial guidelines/Message_templates#Onboarding_a_new_board_member|Suggested email template]])<br> [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 12:36, 13 June 2022 (UTC) ==Editorial board application of Luis Rafael Moscote-Salazar== {{WikiJournal editor application submitted | journal =WikiJournal of Medicine | position =Editorial board member | name =Luis Rafael Moscote-Salazar | qualifications =MD | link =https://neuroclani.org/ | areas_of_expertise =Neurosurgery, Neurotrauma, Stroke, Neurointervention, Neurocritical Care, Neurointervention, Evidence Based Medicine | professional_experience =Neurosurgeon graduated from the University of Cartagena, Founder of the Colombian Clinical Research Group in Neurocritical Care and Co-founder of the Latinamerican Council of Neurocritical Care (CLaNi). Research communication, research leadership, mentorship. | publishing_experience = Experience with publishing in peer-reviewed journals (see Google Scholar profile. Peer-reviewer for several international journals in Neurosurgery and Medicine (see Publons profile). | open_experience = | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Medicine. [[User:NeuroQuimbaya|NeuroQuimbaya]] ([[User talk:NeuroQuimbaya|discuss]]) 2022-07-12 }} :No details were provided, so I have contacted the applicant to request that they add more information. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 01:27, 7 July 2022 (UTC) ::I've updated the application above with the new replacement information that they sent through. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 03:55, 13 July 2022 (UTC) :::'''Support'''. This person seems active in research and publishing according to orcid ([https://orcid.org/0000-0002-4180-6962]). I think it is appropriate to let him join us. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 03:12, 2 August 2022 (UTC) bvzh1olana4bxlneqwb9lnj9a0jaq7d 0 171005 2410781 2410394 2022-08-01T13:26:14Z Young1lim 21186 /* Geometric Series Examples */ wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Applications ([[Media:Laurent.6.Application.6A.20220801.pdf|A.pdf]], [[Media:Laurent.5.Application.6B.20220723.pdf|B.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] 4r01vyx7ccuo2jxw1muioxt6movjdfz 14 199765 2410910 1413398 2022-08-02T05:47:59Z Jtneill 10242 [[Category:Addiction studies]] wikitext text/x-wiki [[Category:Addiction studies]] [[Category:Motivation and emotion/Book]] ksiddvu75lkcmlfuynv2b51lqx1z4vu 0 206087 2410812 2344464 2022-08-01T19:27:59Z Sophivorus 85690 Remove myself per lack of involvement wikitext text/x-wiki <noinclude>{{WikiJSci top menu}}{{WikiJSci right menu}} __NOTOC__ The '''editorial board''' handles any confidential article submissions, and has final responsibility for ensuring that robust academic peer review is performed in a timely manner. Information about editorial board responsibilities, and how to apply to join the editorial board, are found below. {{wjs_h2|Editorial board members}} </noinclude><div style = "column-width: 40em;"> {{Editor info | Q = Q38382414 | description = Dr Thomas Shafee is an evolutionary biochemist and data scientist at [[Wikipedia:La Trobe Institute for Molecular Science|La Trobe University]], Melbourne, Australia. He is particularly interested in how protein superfamilies evolve and how that information can be used to engineer proteins for human uses. }} {{Editor_info | Q = Q99632752 | description = My original research efforts have been performed through participation per numerous grants from the USA Department of Energy at Argonne National Laboratory and Department of Defense at the Naval Research Laboratory. }} {{Editor_info | Q = Q75491774 | description = }} {{Editor_info | Q = Q30078576 | description = Markus Pössel is Managing Scientist at Haus der Astronomie, Heidelberg's Center for Astronomy Education and Outreach. He has helped to bring [[w:General relativity|general relativity]] and [[w:Introduction to general relativity|introduction to general relativity]], among other articles, to [[w:WP:FA|Featured Article]] status. }} {{Editor_info | Q = Q75493029 | description = Ian Alexander is a retired systems engineer who has helped to bring over 100 [[w:User:Chiswick_Chap/Articles|Wikipedia articles]] on biology, and its history and philosophy, to '[[w:Good articles|Good Article]]' status. }} {{Editor_info | Q = Q75494408 | description = Dr Florian Weller is an ecologist working with population models of endangered species, currently at the [[w:University of Cape Town|University of Cape Town]]. }} {{Editor_info | Q = Q75496213 | description = Daniele Pugliesi is a multipotentialite Lecturer and Technical Consultant providing training and assistance in many fields related to industrial activities, management and sustainability. }} {{Editor_info | Q = Q75497001 | description = Sylvain Ribault is a researcher at the Institute for Theoretical Physics in Saclay. He sometimes [http://researchpracticesandtools.blogspot.fr/ blogs] on the subjects of scientific publishing and conformal field theory. }} {{Editor_info | Q = Q37604025 | description = Melanie Stefan is an Edinburgh-Zhejiang lecturer at the University of Edinburgh, whose research group focuses on using computational tools to understand learning and memory. }} {{Editor_info | Q = Q61657548 | description = Jack Nunn is a researcher in the Department of Public Health at [[La Trobe University]]. His research is in creating and evaluating equitable and ethical ways for people to participate in all aspects of research and science. Jack is currently a PhD candidate exploring genomics research and ways of involving people. Jack is the founder and Director of the not-for-profit education organisation [https://scienceforall.world/ 'Science for All']. }} {{Editor_info | Q = Q75502264 | description = <!-- 1-2 sentence description --> }} {{Editor_info | Q = Q55120687 | description = Guy Vandegrift teaches physics and astronomy at the Lake Campus of [[w:Wright State University|Wright State University]], and is currently working to develop [[QB|Quizbank]]. }} {{Editor_info | Q = Q57628851 | description = Dr. Thijs van Vlijmen completed his Ph.D. at [[Wikipedia:Utrecht University|Utrecht University]] in the Department of Cell Biology in the [[wikipedia:University Medical Center Utrecht|University Medical Center Utrecht]] in 2008. After that he started working at [[Wikipedia:Springer Science+Business Media|Springer Science+Business Media]] as a publishing editor. He is currently Associate Editorial Director for Australia and New Zealand at [[wikipedia:Taylor & Francis|Taylor & Francis]]. }} {{Editor_info | Q = Q75788989 | description = Kelee Pacion is the Biology and Environmental Sciences Librarian at Princeton University. Her interests are science communication, information literacy, and education. }} {{Editor_info | Q = Q50551463 | description = Ed Palermo is an Assistant Professor of Materials Science and Engineering at Rensselaer Polytechnic Institute (Troy, NY). His interests are in polymer science and biomaterials. }} {{Editor_info | Q = Q75789306 | description = Tina Qin is the science and engineering librarian and lecturer at Department of Chemistry at Vanderbilt University. She is particular interested in e-Science, scholarly communication and chemical informatics. }} {{Editor_info | Q = Q57907525 | description = <!-- 1-2 sentence description --> }} {{Editor_info | Q = Q59874238 | description = Paula Diaconescu is a Professor of Chemistry and Biochemistry at the University of California, Los Angeles. Her research group focuses on switchable catalysis, a new area of chemistry inspired by nature’s sensory processes. Switchable catalysts use external triggers in order to toggle between multiple species with different reactivity. }} {{Editor_info | Q = Q75446119 | description = Andrew Leung is a climatologist working at [[:w:Environment and Climate Change Canada|Environment and Climate Change Canada]]. He is a contributor to Wikipedia and many of its sister projects. }} {{Editor_info | Q = Q57204503 | description = José Lages is an Associate Professor of Theoretical Physics at the University of Burgundy Franche-Comté, Besançon, France. His recent researches focus on Chaos in Quantum Physics, Chaos in Astrophysics, and Complex Networks. He co-invented the [http://perso.utinam.cnrs.fr/~lages/datasets/WRWU13/ Wikipedia Ranking of World Universities] obtained from the statistical analysis of Wikipedia articles network for different language editions. }} {{Editor_info | Q = Q37376938 | description = Thais Morata is a Research Audiologist at the US [[w:National_Institute_for_Occupational_Safety_and_Health|NIOSH]] and coordinator of the [[w:National_Occupational_Research_Agenda#NORA's_Research_Organization:_Sectors_and_Cross-Sectors|NORA Manufacturing Sector Council]]. She created the Safe-in-Sound Excellence in Hearing Loss Prevention Awards™, is a founding associate editor for the [[w:International_Journal_of_Audiology|International Journal of Audiology]], and a founding editor of the [[w:Cochrane_(organisation)|Cochrane]] Work Review Group. She co-instructs several graduate classes and promotes the Wiki Edu platform. }} {{Editor_info | Q = Q75790178 | description = <!-- 1-2 sentence description --> }} {{Editor_info | Q = Q7932748 | description = <!-- 1-2 sentence description --> }} {{Editor_info | Q = Q30513172 | description = Ed Baker is an orthopterist and bioacoustician at the University of York, and Scientific Associate of the Natural History Museum, London. }} {{Editor_info | Q = Q19842825 | description = Herpetologist, taxonomist and paleontologist with a major interest in nomenclature and the development of good science through best practices, good publishing methods and teaching as well as a multidisciplinary approach. }} {{Editor_info | Q = Q28649100 | image = Mad-portrait-photo-201910.jpg | description = Trained in biotech, now highly interdisciplinary focus on empowering individuals & community to take an active role in leading and contributing to research about themselves. }} {{Editor_info | Q = Q53510778 | description = Jeff Lundeen is the Canada Research Chair in Quantum Photonics. He heads an experimental laboratory at the University of Ottawa that uses photons to study the unique characteristics of quantum physics, to carry information for quantum computing, and to build quantum-enhanced sensing and measurement devices. }} {{Editor_info | Q = Q27452378 | description = 30 years as faculty at University of British Columbia (research, teaching) - [https://scholar.google.ca/citations?user=Ax7V79gAAAAJ&hl=en Publications]. }} {{Editor_info | Q = Q103880232 | description = Research career in chemistry: coordination chemistry, small molecule crystallography, medicinal, and boron chemistry. As faculty at TCU, I was leading a research group in the the development of dopamine targeting drugs. Currently, at the University of Michigan, my research interests are in data science and scientometrics as related to chemistry and biomedical research. }} {{Editor_info | Q = Q42883470 | description = Moritz Schubotz and his group research on Mathematical Information Retrieval at the Department of Mathematics at FIZ Karlsruhe – Leibniz Institute for Information Infrastructure. His main interests are: research, teaching, building research infrastructure, establish a FAIR scholary practice. }} </div><noinclude> {{wjs_h2|Previous board members}} {{Editor_info | Q = Q38381954}} {{Editor_info | Q = Q28914646}} {{:WikiJournal User Group/Editorial board}} {{wjs_h2|Duties of editorial board members}} {{#section-h:WikiJournal_User_Group/Ethics_statement|Duties of the editorial board}} {{wjs_h2|Duties of the editor-in-chief}} {{#section-h:WikiJournal_User_Group/Ethics_statement|Duties of the editor-in-chief}} *''[[WikiJournal User Group/Ethics statement|Full ethics statement]]'', by the WikiJournal User Group </noinclude> [[Category:WikiJournal of Science]] 8rg8c2v4ttbsv7misrqncqjfz5xbxvk 2 211310 2410808 2410774 2022-08-01T18:34:29Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! algebra |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> |- | 300 | -1 | 3 | <math>2^{3/12} \approx1.18921</math> |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> |- | 400 | -1 | 4 | <math>2^{4/12} \approx1.25992</math> |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> |} eq3xoki0gu6openo70v3jvm9dgwbi7t 2410814 2410808 2022-08-01T19:49:50Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! algebra ! error |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | TBA |- | 300 | -1 | 3 | <math>2^{3/12} \approx1.18921</math> | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | TBA |- | 400 | -1 | 4 | <math>2^{4/12} \approx1.25992</math> | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | TBA |} onwua5scq92d9anhnx00g5msd4z2g7r 2410815 2410814 2022-08-01T19:52:50Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! algebra ! error ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | TBA | TBA |- | 300 | -1 | 3 | <math>2^{3/12} \approx1.18921</math> | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | TBA | TBA |- | 400 | -1 | 4 | <math>2^{4/12} \approx1.25992</math> | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | TBA | TBA |} 30xxzizb2u316437r89oicb1xug5dhy 2410817 2410815 2022-08-01T20:02:11Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! error ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | TBA | TBA |- | 300 | -1 | 3 | <math>2^{3/12} \approx1.18921</math> | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | TBA | TBA |- | 400 | -1 | 4 | <math>2^{4/12} \approx1.25992</math> | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | TBA | TBA |} e79hnzhcc5p3ub7m5x5zqpyb4mr76yk 2410820 2410817 2022-08-01T20:37:28Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! error ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | 0.13 | TBA |- | 300 | -1 | 3 | <math>2^{3/12} \approx1.18921</math> | 0.00 | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | -0.15 | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | -0.36 | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | 0.31 | TBA |- | 400 | -1 | 4 | <math>2^{4/12} \approx1.25992</math> | 0.00 | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | -0.18 | TBA |} 4yanj40ibuu579u4akiogxrwa58isug 2410822 2410820 2022-08-01T20:46:00Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! error ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>+0.13</math> | TBA |- | 300 | -1 | 3 | <math>2^{3/12} \approx1.18921</math> | <math>\;0.00</math> | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>-0.15</math> | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>-0.36</math> | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>+0.31</math> | TBA |- | 400 | -1 | 4 | <math>2^{4/12} \approx1.25992</math> | <math>\;0.00</math> | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>-0.18</math> | TBA |} 0ulvuybb32ipquht36yk3tviyf7lod5 2410823 2410822 2022-08-01T20:53:33Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! error ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>+0.13</math> | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0.00</math> | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>-0.15</math> | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>-0.36</math> | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>+0.31</math> | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0.00</math> | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>-0.18</math> | TBA |} nkat7tyo5scvoslnbscce41yb47v6ro 2410824 2410823 2022-08-01T21:15:57Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! error ! pitch<br>(cents) ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | TBA | TBA | TBA |} 0vo424jmleiwqcjchomcsm1ulvl9ee2 2410825 2410824 2022-08-01T21:17:33Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! error ! pitch<br>(cents) ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>+0.13</math> | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0.00</math> | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>-0.15</math> | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>-0.36</math> | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>+0.31</math> | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0.00</math> | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>-0.18</math> | TBA | TBA | TBA |} 6adm4omy899iwikg7c8bfwmx8z4mmi4 2410826 2410825 2022-08-01T21:19:13Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! error ! pitch<br>(cents) ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>+0.13</math> | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0.00</math> | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>-0.15</math> | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>-0.36</math> | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>+0.31</math> | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0.00</math> | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>-0.18</math> | TBA | TBA |} 9xzw5ovpdqed1cmyffnllwzod87u3fq 2410829 2410826 2022-08-01T21:28:31Z Guy vandegrift 813252 Undo all revisions. Resource is empty, but not [[Wikiversity:Deletions|deleted]]. wikitext text/x-wiki phoiac9h4m842xq45sp7s6u21eteeq1 2410830 2410829 2022-08-01T21:45:53Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! error ! pitch<br>(cents) ! interval ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>+0.13</math> | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0.00</math> | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>-0.15</math> | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>-0.36</math> | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>+0.31</math> | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0.00</math> | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>-0.18</math> | TBA | TBA | TBA |} cug0tgjcpbrickbrlms16t6moqtooe2 2410848 2410830 2022-08-01T22:59:51Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta; n ! pitch<br>(cents) ! interval ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>+0.13</math> | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0.00</math> | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>-0.15</math> | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>-0.36</math> | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>+0.31</math> | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0.00</math> | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>-0.18</math> | TBA | TBA | TBA |} g8by0iflwactu0ku6wy38no58ady7aw 2410849 2410848 2022-08-01T23:02:47Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n ! n<br>(cents) ! interval ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>+0.13</math> | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0.00</math> | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>-0.15</math> | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>-0.36</math> | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>+0.31</math> | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0.00</math> | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>-0.18</math> | TBA | TBA | TBA |} 8porj83opn3pta0cn38wkptre1cj1rv 2410850 2410849 2022-08-01T23:15:08Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>+0.13</math> | 294.13 | TBA | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0.00</math> | 300.00 | TBA | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>-0.15</math> | 301.85 | TBA | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>-0.36</math> | 315.64 | TBA | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>+0.31</math> | 386.31 | TBA | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0.00</math> | 400.00 | TBA | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>-0.18</math> | 407.82 | TBA | TBA | TBA | TBA | TBA |} k0609iiwjcs5so7w4uiv1iilkliptf7 2410875 2410850 2022-08-02T03:10:15Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>+0.13</math> | 294.13 | TBA | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0.00</math> | 300.00 | TBA | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>-0.15</math> | 301.85 | TBA | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>-0.36</math> | 315.64 | TBA | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>+0.31</math> | 386.31 | TBA | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0.00</math> | 400.00 | TBA | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>-0.18</math> | 407.82 | TBA | TBA | TBA | TBA | TBA |- | 427.3725723 | 32 | 25 | <math>\frac{32}{25}= \frac{2^{5}}{5^{2}} \approx 1.28</math> | <math>+0.37</math> | 427.37 | TBA | TBA | TBA | TBA | TBA |- | 435.0840953 | 9 | 7 | <math>\frac{9}{7}= \frac{3^{2}}{7} \approx 1.28571</math> | <math>+0.08</math> | 435.08 | TBA | TBA | TBA | TBA | TBA |- | 498.0449991 | 4 | 3 | <math>\frac{4}{3}= \frac{2^{2}}{3} \approx 1.33333</math> | <math>+0.04</math> | 498.04 | TBA | TBA | TBA | TBA | TBA |- | 500 | -1 | 5 | <math> \quad 2^{5/12} \approx1.33484</math> | <math>\;0.00</math> | 500.00 | TBA | TBA | TBA | TBA | TBA |- | 519.5512887 | 27 | 20 | <math>\frac{27}{20}= \frac{3^{3}}{2^{2}\cdot 5} \approx 1.35</math> | <math>-0.45</math> | 519.55 | TBA | TBA | TBA | TBA | TBA |- | 568.717426 | 25 | 18 | <math>\frac{25}{18}= \frac{5^{2}}{2\cdot 3^{2}} \approx 1.38889</math> | <math>-0.28</math> | 568.72 | TBA | TBA | TBA | TBA | TBA |- | 582.5121926 | 7 | 5 | <math>\frac{7}{5}= \frac{7}{5} \approx 1.4</math> | <math>-0.49</math> | 582.51 | TBA | TBA | TBA | TBA | TBA |- | 590.2237156 | 45 | 32 | <math>\frac{45}{32}= \frac{3^{2}\cdot 5}{2^{5}} \approx 1.40625</math> | <math>+0.22</math> | 590.22 | TBA | TBA | TBA | TBA | TBA |- | 600 | -1 | 6 | <math> \quad 2^{6/12} \approx1.41421</math> | <math>\;0.00</math> | 600.00 | TBA | TBA | TBA | TBA | TBA |- | 609.7762844 | 64 | 45 | <math>\frac{64}{45}= \frac{2^{6}}{3^{2}\cdot 5} \approx 1.42222</math> | <math>-0.22</math> | 609.78 | TBA | TBA | TBA | TBA | TBA |- | 617.4878074 | 10 | 7 | <math>\frac{10}{7}= \frac{2\cdot 5}{7} \approx 1.42857</math> | <math>+0.49</math> | 617.49 | TBA | TBA | TBA | TBA | TBA |- | 631.282574 | 36 | 25 | <math>\frac{36}{25}= \frac{2^{2}\cdot 3^{2}}{5^{2}} \approx 1.44</math> | <math>+0.28</math> | 631.28 | TBA | TBA | TBA | TBA | TBA |- | 680.4487113 | 40 | 27 | <math>\frac{40}{27}= \frac{2^{3}\cdot 5}{3^{3}} \approx 1.48148</math> | <math>+0.45</math> | 680.45 | TBA | TBA | TBA | TBA | TBA |- | 700 | -1 | 7 | <math> \quad 2^{7/12} \approx1.49831</math> | <math>\;0.00</math> | 700.00 | TBA | TBA | TBA | TBA | TBA |- | 701.9550009 | 3 | 2 | <math>\frac{3}{2}= \frac{3}{2} \approx 1.5</math> | <math>-0.04</math> | 701.96 | TBA | TBA | TBA | TBA | TBA |- | 764.9159047 | 14 | 9 | <math>\frac{14}{9}= \frac{2\cdot 7}{3^{2}} \approx 1.55556</math> | <math>-0.08</math> | 764.92 | TBA | TBA | TBA | TBA | TBA |- | 772.6274277 | 25 | 16 | <math>\frac{25}{16}= \frac{5^{2}}{2^{4}} \approx 1.5625</math> | <math>-0.37</math> | 772.63 | TBA | TBA | TBA | TBA | TBA |- | 800 | -1 | 8 | <math> \quad 2^{8/12} \approx1.5874</math> | <math>\;0.00</math> | 800.00 | TBA | TBA | TBA | TBA | TBA |- | 813.6862861 | 8 | 5 | <math>\frac{8}{5}= \frac{2^{3}}{5} \approx 1.6</math> | <math>-0.31</math> | 813.69 | TBA | TBA | TBA | TBA | TBA |- | 835.1925757 | 81 | 50 | <math>\frac{81}{50}= \frac{3^{4}}{2\cdot 5^{2}} \approx 1.62</math> | <math>+0.19</math> | 835.19 | TBA | TBA | TBA | TBA | TBA |- | 884.358713 | 5 | 3 | <math>\frac{5}{3}= \frac{5}{3} \approx 1.66667</math> | <math>+0.36</math> | 884.36 | TBA | TBA | TBA | TBA | TBA |- | 900 | -1 | 9 | <math> \quad 2^{9/12} \approx1.68179</math> | <math>\;0.00</math> | 900.00 | TBA | TBA | TBA | TBA | TBA |- | 905.8650026 | 27 | 16 | <math>\frac{27}{16}= \frac{3^{3}}{2^{4}} \approx 1.6875</math> | <math>-0.13</math> | 905.87 | TBA | TBA | TBA | TBA | TBA |- | 933.1290944 | 12 | 7 | <math>\frac{12}{7}= \frac{2^{2}\cdot 3}{7} \approx 1.71429</math> | <math>+0.13</math> | 933.13 | TBA | TBA | TBA | TBA | TBA |- | 968.8259065 | 7 | 4 | <math>\frac{7}{4}= \frac{7}{2^{2}} \approx 1.75</math> | <math>-0.17</math> | 968.83 | TBA | TBA | TBA | TBA | TBA |- | 996.0899983 | 16 | 9 | <math>\frac{16}{9}= \frac{2^{4}}{3^{2}} \approx 1.77778</math> | <math>+0.09</math> | 996.09 | TBA | TBA | TBA | TBA | TBA |- | 1000 | -1 | 10 | <math> \quad 2^{10/12} \approx1.7818</math> | <math>\;0.00</math> | 1000.00 | TBA | TBA | TBA | TBA | TBA |- | 1017.596288 | 9 | 5 | <math>\frac{9}{5}= \frac{3^{2}}{5} \approx 1.8</math> | <math>-0.40</math> | 1017.60 | TBA | TBA | TBA | TBA | TBA |} izoswkxk0bnbzrndir6s6k58koykq4k 2410876 2410875 2022-08-02T03:10:39Z Guy vandegrift 813252 wikitext text/x-wiki ==s== {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>+0.13</math> | 294.13 | TBA | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0.00</math> | 300.00 | TBA | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>-0.15</math> | 301.85 | TBA | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>-0.36</math> | 315.64 | TBA | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>+0.31</math> | 386.31 | TBA | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0.00</math> | 400.00 | TBA | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>-0.18</math> | 407.82 | TBA | TBA | TBA | TBA | TBA |- | 427.3725723 | 32 | 25 | <math>\frac{32}{25}= \frac{2^{5}}{5^{2}} \approx 1.28</math> | <math>+0.37</math> | 427.37 | TBA | TBA | TBA | TBA | TBA |- | 435.0840953 | 9 | 7 | <math>\frac{9}{7}= \frac{3^{2}}{7} \approx 1.28571</math> | <math>+0.08</math> | 435.08 | TBA | TBA | TBA | TBA | TBA |- | 498.0449991 | 4 | 3 | <math>\frac{4}{3}= \frac{2^{2}}{3} \approx 1.33333</math> | <math>+0.04</math> | 498.04 | TBA | TBA | TBA | TBA | TBA |- | 500 | -1 | 5 | <math> \quad 2^{5/12} \approx1.33484</math> | <math>\;0.00</math> | 500.00 | TBA | TBA | TBA | TBA | TBA |- | 519.5512887 | 27 | 20 | <math>\frac{27}{20}= \frac{3^{3}}{2^{2}\cdot 5} \approx 1.35</math> | <math>-0.45</math> | 519.55 | TBA | TBA | TBA | TBA | TBA |- | 568.717426 | 25 | 18 | <math>\frac{25}{18}= \frac{5^{2}}{2\cdot 3^{2}} \approx 1.38889</math> | <math>-0.28</math> | 568.72 | TBA | TBA | TBA | TBA | TBA |- | 582.5121926 | 7 | 5 | <math>\frac{7}{5}= \frac{7}{5} \approx 1.4</math> | <math>-0.49</math> | 582.51 | TBA | TBA | TBA | TBA | TBA |- | 590.2237156 | 45 | 32 | <math>\frac{45}{32}= \frac{3^{2}\cdot 5}{2^{5}} \approx 1.40625</math> | <math>+0.22</math> | 590.22 | TBA | TBA | TBA | TBA | TBA |- | 600 | -1 | 6 | <math> \quad 2^{6/12} \approx1.41421</math> | <math>\;0.00</math> | 600.00 | TBA | TBA | TBA | TBA | TBA |- | 609.7762844 | 64 | 45 | <math>\frac{64}{45}= \frac{2^{6}}{3^{2}\cdot 5} \approx 1.42222</math> | <math>-0.22</math> | 609.78 | TBA | TBA | TBA | TBA | TBA |- | 617.4878074 | 10 | 7 | <math>\frac{10}{7}= \frac{2\cdot 5}{7} \approx 1.42857</math> | <math>+0.49</math> | 617.49 | TBA | TBA | TBA | TBA | TBA |- | 631.282574 | 36 | 25 | <math>\frac{36}{25}= \frac{2^{2}\cdot 3^{2}}{5^{2}} \approx 1.44</math> | <math>+0.28</math> | 631.28 | TBA | TBA | TBA | TBA | TBA |- | 680.4487113 | 40 | 27 | <math>\frac{40}{27}= \frac{2^{3}\cdot 5}{3^{3}} \approx 1.48148</math> | <math>+0.45</math> | 680.45 | TBA | TBA | TBA | TBA | TBA |- | 700 | -1 | 7 | <math> \quad 2^{7/12} \approx1.49831</math> | <math>\;0.00</math> | 700.00 | TBA | TBA | TBA | TBA | TBA |- | 701.9550009 | 3 | 2 | <math>\frac{3}{2}= \frac{3}{2} \approx 1.5</math> | <math>-0.04</math> | 701.96 | TBA | TBA | TBA | TBA | TBA |- | 764.9159047 | 14 | 9 | <math>\frac{14}{9}= \frac{2\cdot 7}{3^{2}} \approx 1.55556</math> | <math>-0.08</math> | 764.92 | TBA | TBA | TBA | TBA | TBA |- | 772.6274277 | 25 | 16 | <math>\frac{25}{16}= \frac{5^{2}}{2^{4}} \approx 1.5625</math> | <math>-0.37</math> | 772.63 | TBA | TBA | TBA | TBA | TBA |- | 800 | -1 | 8 | <math> \quad 2^{8/12} \approx1.5874</math> | <math>\;0.00</math> | 800.00 | TBA | TBA | TBA | TBA | TBA |- | 813.6862861 | 8 | 5 | <math>\frac{8}{5}= \frac{2^{3}}{5} \approx 1.6</math> | <math>-0.31</math> | 813.69 | TBA | TBA | TBA | TBA | TBA |- | 835.1925757 | 81 | 50 | <math>\frac{81}{50}= \frac{3^{4}}{2\cdot 5^{2}} \approx 1.62</math> | <math>+0.19</math> | 835.19 | TBA | TBA | TBA | TBA | TBA |- | 884.358713 | 5 | 3 | <math>\frac{5}{3}= \frac{5}{3} \approx 1.66667</math> | <math>+0.36</math> | 884.36 | TBA | TBA | TBA | TBA | TBA |- | 900 | -1 | 9 | <math> \quad 2^{9/12} \approx1.68179</math> | <math>\;0.00</math> | 900.00 | TBA | TBA | TBA | TBA | TBA |- | 905.8650026 | 27 | 16 | <math>\frac{27}{16}= \frac{3^{3}}{2^{4}} \approx 1.6875</math> | <math>-0.13</math> | 905.87 | TBA | TBA | TBA | TBA | TBA |- | 933.1290944 | 12 | 7 | <math>\frac{12}{7}= \frac{2^{2}\cdot 3}{7} \approx 1.71429</math> | <math>+0.13</math> | 933.13 | TBA | TBA | TBA | TBA | TBA |- | 968.8259065 | 7 | 4 | <math>\frac{7}{4}= \frac{7}{2^{2}} \approx 1.75</math> | <math>-0.17</math> | 968.83 | TBA | TBA | TBA | TBA | TBA |- | 996.0899983 | 16 | 9 | <math>\frac{16}{9}= \frac{2^{4}}{3^{2}} \approx 1.77778</math> | <math>+0.09</math> | 996.09 | TBA | TBA | TBA | TBA | TBA |- | 1000 | -1 | 10 | <math> \quad 2^{10/12} \approx1.7818</math> | <math>\;0.00</math> | 1000.00 | TBA | TBA | TBA | TBA | TBA |- | 1017.596288 | 9 | 5 | <math>\frac{9}{5}= \frac{3^{2}}{5} \approx 1.8</math> | <math>-0.40</math> | 1017.60 | TBA | TBA | TBA | TBA | TBA |} c10iqgfst5mraffabbyon6c9ro9d4k7 2410878 2410876 2022-08-02T03:12:56Z Guy vandegrift 813252 Undo all revisions. Resource is empty, but not [[Wikiversity:Deletions|deleted]]. wikitext text/x-wiki phoiac9h4m842xq45sp7s6u21eteeq1 2410881 2410878 2022-08-02T03:14:07Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>+0.13</math> | 294.13 | TBA | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0.00</math> | 300.00 | TBA | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>-0.15</math> | 301.85 | TBA | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>-0.36</math> | 315.64 | TBA | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>+0.31</math> | 386.31 | TBA | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0.00</math> | 400.00 | TBA | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>-0.18</math> | 407.82 | TBA | TBA | TBA | TBA | TBA |} k0609iiwjcs5so7w4uiv1iilkliptf7 2410883 2410881 2022-08-02T03:41:26Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>-5.9</math> | 294.13 | TBA | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0.0</math> | 300.00 | TBA | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>+1.8</math> | 301.85 | TBA | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>+15.6</math> | 315.64 | TBA | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>-13.7</math> | 386.31 | TBA | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0.0</math> | 400.00 | TBA | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>+7.8</math> | 407.82 | TBA | TBA | TBA | TBA | TBA |- | 427.3725723 | 32 | 25 | <math>\frac{32}{25}= \frac{2^{5}}{5^{2}} \approx 1.28</math> | <math>+27.4</math> | 427.37 | TBA | TBA | TBA | TBA | TBA |- | 435.0840953 | 9 | 7 | <math>\frac{9}{7}= \frac{3^{2}}{7} \approx 1.28571</math> | <math>+35.1</math> | 435.08 | TBA | TBA | TBA | TBA | TBA |- | 498.0449991 | 4 | 3 | <math>\frac{4}{3}= \frac{2^{2}}{3} \approx 1.33333</math> | <math>-2.0</math> | 498.04 | TBA | TBA | TBA | TBA | TBA |- | 500 | -1 | 5 | <math> \quad 2^{5/12} \approx1.33484</math> | <math>\;0.0</math> | 500.00 | TBA | TBA | TBA | TBA | TBA |- | 519.5512887 | 27 | 20 | <math>\frac{27}{20}= \frac{3^{3}}{2^{2}\cdot 5} \approx 1.35</math> | <math>+19.6</math> | 519.55 | TBA | TBA | TBA | TBA | TBA |- | 568.717426 | 25 | 18 | <math>\frac{25}{18}= \frac{5^{2}}{2\cdot 3^{2}} \approx 1.38889</math> | <math>-31.3</math> | 568.72 | TBA | TBA | TBA | TBA | TBA |- | 582.5121926 | 7 | 5 | <math>\frac{7}{5}= \frac{7}{5} \approx 1.4</math> | <math>-17.5</math> | 582.51 | TBA | TBA | TBA | TBA | TBA |- | 590.2237156 | 45 | 32 | <math>\frac{45}{32}= \frac{3^{2}\cdot 5}{2^{5}} \approx 1.40625</math> | <math>-9.8</math> | 590.22 | TBA | TBA | TBA | TBA | TBA |- | 600 | -1 | 6 | <math> \quad 2^{6/12} \approx1.41421</math> | <math>\;0.0</math> | 600.00 | TBA | TBA | TBA | TBA | TBA |- | 609.7762844 | 64 | 45 | <math>\frac{64}{45}= \frac{2^{6}}{3^{2}\cdot 5} \approx 1.42222</math> | <math>+9.8</math> | 609.78 | TBA | TBA | TBA | TBA | TBA |- | 617.4878074 | 10 | 7 | <math>\frac{10}{7}= \frac{2\cdot 5}{7} \approx 1.42857</math> | <math>+17.5</math> | 617.49 | TBA | TBA | TBA | TBA | TBA |- | 631.282574 | 36 | 25 | <math>\frac{36}{25}= \frac{2^{2}\cdot 3^{2}}{5^{2}} \approx 1.44</math> | <math>+31.3</math> | 631.28 | TBA | TBA | TBA | TBA | TBA |- | 680.4487113 | 40 | 27 | <math>\frac{40}{27}= \frac{2^{3}\cdot 5}{3^{3}} \approx 1.48148</math> | <math>-19.6</math> | 680.45 | TBA | TBA | TBA | TBA | TBA |- | 700 | -1 | 7 | <math> \quad 2^{7/12} \approx1.49831</math> | <math>\;0.0</math> | 700.00 | TBA | TBA | TBA | TBA | TBA |- | 701.9550009 | 3 | 2 | <math>\frac{3}{2}= \frac{3}{2} \approx 1.5</math> | <math>+2.0</math> | 701.96 | TBA | TBA | TBA | TBA | TBA |- | 764.9159047 | 14 | 9 | <math>\frac{14}{9}= \frac{2\cdot 7}{3^{2}} \approx 1.55556</math> | <math>-35.1</math> | 764.92 | TBA | TBA | TBA | TBA | TBA |- | 772.6274277 | 25 | 16 | <math>\frac{25}{16}= \frac{5^{2}}{2^{4}} \approx 1.5625</math> | <math>-27.4</math> | 772.63 | TBA | TBA | TBA | TBA | TBA |- | 800 | -1 | 8 | <math> \quad 2^{8/12} \approx1.5874</math> | <math>\;0.0</math> | 800.00 | TBA | TBA | TBA | TBA | TBA |- | 813.6862861 | 8 | 5 | <math>\frac{8}{5}= \frac{2^{3}}{5} \approx 1.6</math> | <math>+13.7</math> | 813.69 | TBA | TBA | TBA | TBA | TBA |- | 835.1925757 | 81 | 50 | <math>\frac{81}{50}= \frac{3^{4}}{2\cdot 5^{2}} \approx 1.62</math> | <math>+35.2</math> | 835.19 | TBA | TBA | TBA | TBA | TBA |- | 884.358713 | 5 | 3 | <math>\frac{5}{3}= \frac{5}{3} \approx 1.66667</math> | <math>-15.6</math> | 884.36 | TBA | TBA | TBA | TBA | TBA |- | 900 | -1 | 9 | <math> \quad 2^{9/12} \approx1.68179</math> | <math>\;0.0</math> | 900.00 | TBA | TBA | TBA | TBA | TBA |- | 905.8650026 | 27 | 16 | <math>\frac{27}{16}= \frac{3^{3}}{2^{4}} \approx 1.6875</math> | <math>+5.9</math> | 905.87 | TBA | TBA | TBA | TBA | TBA |- | 933.1290944 | 12 | 7 | <math>\frac{12}{7}= \frac{2^{2}\cdot 3}{7} \approx 1.71429</math> | <math>+33.1</math> | 933.13 | TBA | TBA | TBA | TBA | TBA |- | 968.8259065 | 7 | 4 | <math>\frac{7}{4}= \frac{7}{2^{2}} \approx 1.75</math> | <math>-31.2</math> | 968.83 | TBA | TBA | TBA | TBA | TBA |- | 996.0899983 | 16 | 9 | <math>\frac{16}{9}= \frac{2^{4}}{3^{2}} \approx 1.77778</math> | <math>-3.9</math> | 996.09 | TBA | TBA | TBA | TBA | TBA |- | 1000 | -1 | 10 | <math> \quad 2^{10/12} \approx1.7818</math> | <math>\;0.0</math> | 1000.00 | TBA | TBA | TBA | TBA | TBA |- | 1017.596288 | 9 | 5 | <math>\frac{9}{5}= \frac{3^{2}}{5} \approx 1.8</math> | <math>+17.6</math> | 1017.60 | TBA | TBA | TBA | TBA | TBA |} cmk37llb8ya4bm739ekza598501rrx1 2410884 2410883 2022-08-02T03:41:47Z Guy vandegrift 813252 wikitext text/x-wiki 222 {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>-5.9</math> | 294.13 | TBA | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0.0</math> | 300.00 | TBA | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>+1.8</math> | 301.85 | TBA | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>+15.6</math> | 315.64 | TBA | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>-13.7</math> | 386.31 | TBA | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0.0</math> | 400.00 | TBA | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>+7.8</math> | 407.82 | TBA | TBA | TBA | TBA | TBA |- | 427.3725723 | 32 | 25 | <math>\frac{32}{25}= \frac{2^{5}}{5^{2}} \approx 1.28</math> | <math>+27.4</math> | 427.37 | TBA | TBA | TBA | TBA | TBA |- | 435.0840953 | 9 | 7 | <math>\frac{9}{7}= \frac{3^{2}}{7} \approx 1.28571</math> | <math>+35.1</math> | 435.08 | TBA | TBA | TBA | TBA | TBA |- | 498.0449991 | 4 | 3 | <math>\frac{4}{3}= \frac{2^{2}}{3} \approx 1.33333</math> | <math>-2.0</math> | 498.04 | TBA | TBA | TBA | TBA | TBA |- | 500 | -1 | 5 | <math> \quad 2^{5/12} \approx1.33484</math> | <math>\;0.0</math> | 500.00 | TBA | TBA | TBA | TBA | TBA |- | 519.5512887 | 27 | 20 | <math>\frac{27}{20}= \frac{3^{3}}{2^{2}\cdot 5} \approx 1.35</math> | <math>+19.6</math> | 519.55 | TBA | TBA | TBA | TBA | TBA |- | 568.717426 | 25 | 18 | <math>\frac{25}{18}= \frac{5^{2}}{2\cdot 3^{2}} \approx 1.38889</math> | <math>-31.3</math> | 568.72 | TBA | TBA | TBA | TBA | TBA |- | 582.5121926 | 7 | 5 | <math>\frac{7}{5}= \frac{7}{5} \approx 1.4</math> | <math>-17.5</math> | 582.51 | TBA | TBA | TBA | TBA | TBA |- | 590.2237156 | 45 | 32 | <math>\frac{45}{32}= \frac{3^{2}\cdot 5}{2^{5}} \approx 1.40625</math> | <math>-9.8</math> | 590.22 | TBA | TBA | TBA | TBA | TBA |- | 600 | -1 | 6 | <math> \quad 2^{6/12} \approx1.41421</math> | <math>\;0.0</math> | 600.00 | TBA | TBA | TBA | TBA | TBA |- | 609.7762844 | 64 | 45 | <math>\frac{64}{45}= \frac{2^{6}}{3^{2}\cdot 5} \approx 1.42222</math> | <math>+9.8</math> | 609.78 | TBA | TBA | TBA | TBA | TBA |- | 617.4878074 | 10 | 7 | <math>\frac{10}{7}= \frac{2\cdot 5}{7} \approx 1.42857</math> | <math>+17.5</math> | 617.49 | TBA | TBA | TBA | TBA | TBA |- | 631.282574 | 36 | 25 | <math>\frac{36}{25}= \frac{2^{2}\cdot 3^{2}}{5^{2}} \approx 1.44</math> | <math>+31.3</math> | 631.28 | TBA | TBA | TBA | TBA | TBA |- | 680.4487113 | 40 | 27 | <math>\frac{40}{27}= \frac{2^{3}\cdot 5}{3^{3}} \approx 1.48148</math> | <math>-19.6</math> | 680.45 | TBA | TBA | TBA | TBA | TBA |- | 700 | -1 | 7 | <math> \quad 2^{7/12} \approx1.49831</math> | <math>\;0.0</math> | 700.00 | TBA | TBA | TBA | TBA | TBA |- | 701.9550009 | 3 | 2 | <math>\frac{3}{2}= \frac{3}{2} \approx 1.5</math> | <math>+2.0</math> | 701.96 | TBA | TBA | TBA | TBA | TBA |- | 764.9159047 | 14 | 9 | <math>\frac{14}{9}= \frac{2\cdot 7}{3^{2}} \approx 1.55556</math> | <math>-35.1</math> | 764.92 | TBA | TBA | TBA | TBA | TBA |- | 772.6274277 | 25 | 16 | <math>\frac{25}{16}= \frac{5^{2}}{2^{4}} \approx 1.5625</math> | <math>-27.4</math> | 772.63 | TBA | TBA | TBA | TBA | TBA |- | 800 | -1 | 8 | <math> \quad 2^{8/12} \approx1.5874</math> | <math>\;0.0</math> | 800.00 | TBA | TBA | TBA | TBA | TBA |- | 813.6862861 | 8 | 5 | <math>\frac{8}{5}= \frac{2^{3}}{5} \approx 1.6</math> | <math>+13.7</math> | 813.69 | TBA | TBA | TBA | TBA | TBA |- | 835.1925757 | 81 | 50 | <math>\frac{81}{50}= \frac{3^{4}}{2\cdot 5^{2}} \approx 1.62</math> | <math>+35.2</math> | 835.19 | TBA | TBA | TBA | TBA | TBA |- | 884.358713 | 5 | 3 | <math>\frac{5}{3}= \frac{5}{3} \approx 1.66667</math> | <math>-15.6</math> | 884.36 | TBA | TBA | TBA | TBA | TBA |- | 900 | -1 | 9 | <math> \quad 2^{9/12} \approx1.68179</math> | <math>\;0.0</math> | 900.00 | TBA | TBA | TBA | TBA | TBA |- | 905.8650026 | 27 | 16 | <math>\frac{27}{16}= \frac{3^{3}}{2^{4}} \approx 1.6875</math> | <math>+5.9</math> | 905.87 | TBA | TBA | TBA | TBA | TBA |- | 933.1290944 | 12 | 7 | <math>\frac{12}{7}= \frac{2^{2}\cdot 3}{7} \approx 1.71429</math> | <math>+33.1</math> | 933.13 | TBA | TBA | TBA | TBA | TBA |- | 968.8259065 | 7 | 4 | <math>\frac{7}{4}= \frac{7}{2^{2}} \approx 1.75</math> | <math>-31.2</math> | 968.83 | TBA | TBA | TBA | TBA | TBA |- | 996.0899983 | 16 | 9 | <math>\frac{16}{9}= \frac{2^{4}}{3^{2}} \approx 1.77778</math> | <math>-3.9</math> | 996.09 | TBA | TBA | TBA | TBA | TBA |- | 1000 | -1 | 10 | <math> \quad 2^{10/12} \approx1.7818</math> | <math>\;0.0</math> | 1000.00 | TBA | TBA | TBA | TBA | TBA |- | 1017.596288 | 9 | 5 | <math>\frac{9}{5}= \frac{3^{2}}{5} \approx 1.8</math> | <math>+17.6</math> | 1017.60 | TBA | TBA | TBA | TBA | TBA |} h3eu3zuatjwx01rs3swxblqa159spk6 2410885 2410884 2022-08-02T03:44:18Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>-5.865003</math> | 294.13 | TBA | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0.000000</math> | 300.00 | TBA | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>+1.846520</math> | 301.85 | TBA | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>+15.641287</math> | 315.64 | TBA | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>-13.686286</math> | 386.31 | TBA | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0.000000</math> | 400.00 | TBA | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>+7.820003</math> | 407.82 | TBA | TBA | TBA | TBA | TBA |- | 427.3725723 | 32 | 25 | <math>\frac{32}{25}= \frac{2^{5}}{5^{2}} \approx 1.28</math> | <math>+27.372572</math> | 427.37 | TBA | TBA | TBA | TBA | TBA |- | 435.0840953 | 9 | 7 | <math>\frac{9}{7}= \frac{3^{2}}{7} \approx 1.28571</math> | <math>+35.084095</math> | 435.08 | TBA | TBA | TBA | TBA | TBA |- | 498.0449991 | 4 | 3 | <math>\frac{4}{3}= \frac{2^{2}}{3} \approx 1.33333</math> | <math>-1.955001</math> | 498.04 | TBA | TBA | TBA | TBA | TBA |- | 500 | -1 | 5 | <math> \quad 2^{5/12} \approx1.33484</math> | <math>\;0.000000</math> | 500.00 | TBA | TBA | TBA | TBA | TBA |- | 519.5512887 | 27 | 20 | <math>\frac{27}{20}= \frac{3^{3}}{2^{2}\cdot 5} \approx 1.35</math> | <math>+19.551289</math> | 519.55 | TBA | TBA | TBA | TBA | TBA |- | 568.717426 | 25 | 18 | <math>\frac{25}{18}= \frac{5^{2}}{2\cdot 3^{2}} \approx 1.38889</math> | <math>-31.282574</math> | 568.72 | TBA | TBA | TBA | TBA | TBA |- | 582.5121926 | 7 | 5 | <math>\frac{7}{5}= \frac{7}{5} \approx 1.4</math> | <math>-17.487807</math> | 582.51 | TBA | TBA | TBA | TBA | TBA |- | 590.2237156 | 45 | 32 | <math>\frac{45}{32}= \frac{3^{2}\cdot 5}{2^{5}} \approx 1.40625</math> | <math>-9.776284</math> | 590.22 | TBA | TBA | TBA | TBA | TBA |- | 600 | -1 | 6 | <math> \quad 2^{6/12} \approx1.41421</math> | <math>\;0.000000</math> | 600.00 | TBA | TBA | TBA | TBA | TBA |- | 609.7762844 | 64 | 45 | <math>\frac{64}{45}= \frac{2^{6}}{3^{2}\cdot 5} \approx 1.42222</math> | <math>+9.776284</math> | 609.78 | TBA | TBA | TBA | TBA | TBA |- | 617.4878074 | 10 | 7 | <math>\frac{10}{7}= \frac{2\cdot 5}{7} \approx 1.42857</math> | <math>+17.487807</math> | 617.49 | TBA | TBA | TBA | TBA | TBA |- | 631.282574 | 36 | 25 | <math>\frac{36}{25}= \frac{2^{2}\cdot 3^{2}}{5^{2}} \approx 1.44</math> | <math>+31.282574</math> | 631.28 | TBA | TBA | TBA | TBA | TBA |- | 680.4487113 | 40 | 27 | <math>\frac{40}{27}= \frac{2^{3}\cdot 5}{3^{3}} \approx 1.48148</math> | <math>-19.551289</math> | 680.45 | TBA | TBA | TBA | TBA | TBA |- | 700 | -1 | 7 | <math> \quad 2^{7/12} \approx1.49831</math> | <math>\;0.000000</math> | 700.00 | TBA | TBA | TBA | TBA | TBA |- | 701.9550009 | 3 | 2 | <math>\frac{3}{2}= \frac{3}{2} \approx 1.5</math> | <math>+1.955001</math> | 701.96 | TBA | TBA | TBA | TBA | TBA |- | 764.9159047 | 14 | 9 | <math>\frac{14}{9}= \frac{2\cdot 7}{3^{2}} \approx 1.55556</math> | <math>-35.084095</math> | 764.92 | TBA | TBA | TBA | TBA | TBA |- | 772.6274277 | 25 | 16 | <math>\frac{25}{16}= \frac{5^{2}}{2^{4}} \approx 1.5625</math> | <math>-27.372572</math> | 772.63 | TBA | TBA | TBA | TBA | TBA |- | 800 | -1 | 8 | <math> \quad 2^{8/12} \approx1.5874</math> | <math>\;0.000000</math> | 800.00 | TBA | TBA | TBA | TBA | TBA |- | 813.6862861 | 8 | 5 | <math>\frac{8}{5}= \frac{2^{3}}{5} \approx 1.6</math> | <math>+13.686286</math> | 813.69 | TBA | TBA | TBA | TBA | TBA |- | 835.1925757 | 81 | 50 | <math>\frac{81}{50}= \frac{3^{4}}{2\cdot 5^{2}} \approx 1.62</math> | <math>+35.192576</math> | 835.19 | TBA | TBA | TBA | TBA | TBA |- | 884.358713 | 5 | 3 | <math>\frac{5}{3}= \frac{5}{3} \approx 1.66667</math> | <math>-15.641287</math> | 884.36 | TBA | TBA | TBA | TBA | TBA |- | 900 | -1 | 9 | <math> \quad 2^{9/12} \approx1.68179</math> | <math>\;0.000000</math> | 900.00 | TBA | TBA | TBA | TBA | TBA |- | 905.8650026 | 27 | 16 | <math>\frac{27}{16}= \frac{3^{3}}{2^{4}} \approx 1.6875</math> | <math>+5.865003</math> | 905.87 | TBA | TBA | TBA | TBA | TBA |- | 933.1290944 | 12 | 7 | <math>\frac{12}{7}= \frac{2^{2}\cdot 3}{7} \approx 1.71429</math> | <math>+33.129094</math> | 933.13 | TBA | TBA | TBA | TBA | TBA |- | 968.8259065 | 7 | 4 | <math>\frac{7}{4}= \frac{7}{2^{2}} \approx 1.75</math> | <math>-31.174094</math> | 968.83 | TBA | TBA | TBA | TBA | TBA |- | 996.0899983 | 16 | 9 | <math>\frac{16}{9}= \frac{2^{4}}{3^{2}} \approx 1.77778</math> | <math>-3.910002</math> | 996.09 | TBA | TBA | TBA | TBA | TBA |- | 1000 | -1 | 10 | <math> \quad 2^{10/12} \approx1.7818</math> | <math>\;0.000000</math> | 1000.00 | TBA | TBA | TBA | TBA | TBA |- | 1017.596288 | 9 | 5 | <math>\frac{9}{5}= \frac{3^{2}}{5} \approx 1.8</math> | <math>+17.596288</math> | 1017.60 | TBA | TBA | TBA | TBA | TBA |} nw6k43k8dy4kenf63re6b3sg7fgc6tj 2410886 2410885 2022-08-02T03:44:31Z Guy vandegrift 813252 wikitext text/x-wiki 666 {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>-5.865003</math> | 294.13 | TBA | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0.000000</math> | 300.00 | TBA | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>+1.846520</math> | 301.85 | TBA | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>+15.641287</math> | 315.64 | TBA | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>-13.686286</math> | 386.31 | TBA | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0.000000</math> | 400.00 | TBA | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>+7.820003</math> | 407.82 | TBA | TBA | TBA | TBA | TBA |- | 427.3725723 | 32 | 25 | <math>\frac{32}{25}= \frac{2^{5}}{5^{2}} \approx 1.28</math> | <math>+27.372572</math> | 427.37 | TBA | TBA | TBA | TBA | TBA |- | 435.0840953 | 9 | 7 | <math>\frac{9}{7}= \frac{3^{2}}{7} \approx 1.28571</math> | <math>+35.084095</math> | 435.08 | TBA | TBA | TBA | TBA | TBA |- | 498.0449991 | 4 | 3 | <math>\frac{4}{3}= \frac{2^{2}}{3} \approx 1.33333</math> | <math>-1.955001</math> | 498.04 | TBA | TBA | TBA | TBA | TBA |- | 500 | -1 | 5 | <math> \quad 2^{5/12} \approx1.33484</math> | <math>\;0.000000</math> | 500.00 | TBA | TBA | TBA | TBA | TBA |- | 519.5512887 | 27 | 20 | <math>\frac{27}{20}= \frac{3^{3}}{2^{2}\cdot 5} \approx 1.35</math> | <math>+19.551289</math> | 519.55 | TBA | TBA | TBA | TBA | TBA |- | 568.717426 | 25 | 18 | <math>\frac{25}{18}= \frac{5^{2}}{2\cdot 3^{2}} \approx 1.38889</math> | <math>-31.282574</math> | 568.72 | TBA | TBA | TBA | TBA | TBA |- | 582.5121926 | 7 | 5 | <math>\frac{7}{5}= \frac{7}{5} \approx 1.4</math> | <math>-17.487807</math> | 582.51 | TBA | TBA | TBA | TBA | TBA |- | 590.2237156 | 45 | 32 | <math>\frac{45}{32}= \frac{3^{2}\cdot 5}{2^{5}} \approx 1.40625</math> | <math>-9.776284</math> | 590.22 | TBA | TBA | TBA | TBA | TBA |- | 600 | -1 | 6 | <math> \quad 2^{6/12} \approx1.41421</math> | <math>\;0.000000</math> | 600.00 | TBA | TBA | TBA | TBA | TBA |- | 609.7762844 | 64 | 45 | <math>\frac{64}{45}= \frac{2^{6}}{3^{2}\cdot 5} \approx 1.42222</math> | <math>+9.776284</math> | 609.78 | TBA | TBA | TBA | TBA | TBA |- | 617.4878074 | 10 | 7 | <math>\frac{10}{7}= \frac{2\cdot 5}{7} \approx 1.42857</math> | <math>+17.487807</math> | 617.49 | TBA | TBA | TBA | TBA | TBA |- | 631.282574 | 36 | 25 | <math>\frac{36}{25}= \frac{2^{2}\cdot 3^{2}}{5^{2}} \approx 1.44</math> | <math>+31.282574</math> | 631.28 | TBA | TBA | TBA | TBA | TBA |- | 680.4487113 | 40 | 27 | <math>\frac{40}{27}= \frac{2^{3}\cdot 5}{3^{3}} \approx 1.48148</math> | <math>-19.551289</math> | 680.45 | TBA | TBA | TBA | TBA | TBA |- | 700 | -1 | 7 | <math> \quad 2^{7/12} \approx1.49831</math> | <math>\;0.000000</math> | 700.00 | TBA | TBA | TBA | TBA | TBA |- | 701.9550009 | 3 | 2 | <math>\frac{3}{2}= \frac{3}{2} \approx 1.5</math> | <math>+1.955001</math> | 701.96 | TBA | TBA | TBA | TBA | TBA |- | 764.9159047 | 14 | 9 | <math>\frac{14}{9}= \frac{2\cdot 7}{3^{2}} \approx 1.55556</math> | <math>-35.084095</math> | 764.92 | TBA | TBA | TBA | TBA | TBA |- | 772.6274277 | 25 | 16 | <math>\frac{25}{16}= \frac{5^{2}}{2^{4}} \approx 1.5625</math> | <math>-27.372572</math> | 772.63 | TBA | TBA | TBA | TBA | TBA |- | 800 | -1 | 8 | <math> \quad 2^{8/12} \approx1.5874</math> | <math>\;0.000000</math> | 800.00 | TBA | TBA | TBA | TBA | TBA |- | 813.6862861 | 8 | 5 | <math>\frac{8}{5}= \frac{2^{3}}{5} \approx 1.6</math> | <math>+13.686286</math> | 813.69 | TBA | TBA | TBA | TBA | TBA |- | 835.1925757 | 81 | 50 | <math>\frac{81}{50}= \frac{3^{4}}{2\cdot 5^{2}} \approx 1.62</math> | <math>+35.192576</math> | 835.19 | TBA | TBA | TBA | TBA | TBA |- | 884.358713 | 5 | 3 | <math>\frac{5}{3}= \frac{5}{3} \approx 1.66667</math> | <math>-15.641287</math> | 884.36 | TBA | TBA | TBA | TBA | TBA |- | 900 | -1 | 9 | <math> \quad 2^{9/12} \approx1.68179</math> | <math>\;0.000000</math> | 900.00 | TBA | TBA | TBA | TBA | TBA |- | 905.8650026 | 27 | 16 | <math>\frac{27}{16}= \frac{3^{3}}{2^{4}} \approx 1.6875</math> | <math>+5.865003</math> | 905.87 | TBA | TBA | TBA | TBA | TBA |- | 933.1290944 | 12 | 7 | <math>\frac{12}{7}= \frac{2^{2}\cdot 3}{7} \approx 1.71429</math> | <math>+33.129094</math> | 933.13 | TBA | TBA | TBA | TBA | TBA |- | 968.8259065 | 7 | 4 | <math>\frac{7}{4}= \frac{7}{2^{2}} \approx 1.75</math> | <math>-31.174094</math> | 968.83 | TBA | TBA | TBA | TBA | TBA |- | 996.0899983 | 16 | 9 | <math>\frac{16}{9}= \frac{2^{4}}{3^{2}} \approx 1.77778</math> | <math>-3.910002</math> | 996.09 | TBA | TBA | TBA | TBA | TBA |- | 1000 | -1 | 10 | <math> \quad 2^{10/12} \approx1.7818</math> | <math>\;0.000000</math> | 1000.00 | TBA | TBA | TBA | TBA | TBA |- | 1017.596288 | 9 | 5 | <math>\frac{9}{5}= \frac{3^{2}}{5} \approx 1.8</math> | <math>+17.596288</math> | 1017.60 | TBA | TBA | TBA | TBA | TBA |} 71ywdayn2v7unsdirdtd1wsr6qooast 2410888 2410886 2022-08-02T03:46:57Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>-6</math> | 294.13 | TBA | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0</math> | 300.00 | TBA | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>+2</math> | 301.85 | TBA | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>+16</math> | 315.64 | TBA | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>-14</math> | 386.31 | TBA | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0</math> | 400.00 | TBA | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>+8</math> | 407.82 | TBA | TBA | TBA | TBA | TBA |- | 427.3725723 | 32 | 25 | <math>\frac{32}{25}= \frac{2^{5}}{5^{2}} \approx 1.28</math> | <math>+27</math> | 427.37 | TBA | TBA | TBA | TBA | TBA |- | 435.0840953 | 9 | 7 | <math>\frac{9}{7}= \frac{3^{2}}{7} \approx 1.28571</math> | <math>+35</math> | 435.08 | TBA | TBA | TBA | TBA | TBA |- | 498.0449991 | 4 | 3 | <math>\frac{4}{3}= \frac{2^{2}}{3} \approx 1.33333</math> | <math>-2</math> | 498.04 | TBA | TBA | TBA | TBA | TBA |- | 500 | -1 | 5 | <math> \quad 2^{5/12} \approx1.33484</math> | <math>\;0</math> | 500.00 | TBA | TBA | TBA | TBA | TBA |- | 519.5512887 | 27 | 20 | <math>\frac{27}{20}= \frac{3^{3}}{2^{2}\cdot 5} \approx 1.35</math> | <math>+20</math> | 519.55 | TBA | TBA | TBA | TBA | TBA |- | 568.717426 | 25 | 18 | <math>\frac{25}{18}= \frac{5^{2}}{2\cdot 3^{2}} \approx 1.38889</math> | <math>-31</math> | 568.72 | TBA | TBA | TBA | TBA | TBA |- | 582.5121926 | 7 | 5 | <math>\frac{7}{5}= \frac{7}{5} \approx 1.4</math> | <math>-17</math> | 582.51 | TBA | TBA | TBA | TBA | TBA |- | 590.2237156 | 45 | 32 | <math>\frac{45}{32}= \frac{3^{2}\cdot 5}{2^{5}} \approx 1.40625</math> | <math>-10</math> | 590.22 | TBA | TBA | TBA | TBA | TBA |- | 600 | -1 | 6 | <math> \quad 2^{6/12} \approx1.41421</math> | <math>\;0</math> | 600.00 | TBA | TBA | TBA | TBA | TBA |- | 609.7762844 | 64 | 45 | <math>\frac{64}{45}= \frac{2^{6}}{3^{2}\cdot 5} \approx 1.42222</math> | <math>+10</math> | 609.78 | TBA | TBA | TBA | TBA | TBA |- | 617.4878074 | 10 | 7 | <math>\frac{10}{7}= \frac{2\cdot 5}{7} \approx 1.42857</math> | <math>+17</math> | 617.49 | TBA | TBA | TBA | TBA | TBA |- | 631.282574 | 36 | 25 | <math>\frac{36}{25}= \frac{2^{2}\cdot 3^{2}}{5^{2}} \approx 1.44</math> | <math>+31</math> | 631.28 | TBA | TBA | TBA | TBA | TBA |- | 680.4487113 | 40 | 27 | <math>\frac{40}{27}= \frac{2^{3}\cdot 5}{3^{3}} \approx 1.48148</math> | <math>-20</math> | 680.45 | TBA | TBA | TBA | TBA | TBA |- | 700 | -1 | 7 | <math> \quad 2^{7/12} \approx1.49831</math> | <math>\;0</math> | 700.00 | TBA | TBA | TBA | TBA | TBA |- | 701.9550009 | 3 | 2 | <math>\frac{3}{2}= \frac{3}{2} \approx 1.5</math> | <math>+2</math> | 701.96 | TBA | TBA | TBA | TBA | TBA |- | 764.9159047 | 14 | 9 | <math>\frac{14}{9}= \frac{2\cdot 7}{3^{2}} \approx 1.55556</math> | <math>-35</math> | 764.92 | TBA | TBA | TBA | TBA | TBA |- | 772.6274277 | 25 | 16 | <math>\frac{25}{16}= \frac{5^{2}}{2^{4}} \approx 1.5625</math> | <math>-27</math> | 772.63 | TBA | TBA | TBA | TBA | TBA |- | 800 | -1 | 8 | <math> \quad 2^{8/12} \approx1.5874</math> | <math>\;0</math> | 800.00 | TBA | TBA | TBA | TBA | TBA |- | 813.6862861 | 8 | 5 | <math>\frac{8}{5}= \frac{2^{3}}{5} \approx 1.6</math> | <math>+14</math> | 813.69 | TBA | TBA | TBA | TBA | TBA |- | 835.1925757 | 81 | 50 | <math>\frac{81}{50}= \frac{3^{4}}{2\cdot 5^{2}} \approx 1.62</math> | <math>+35</math> | 835.19 | TBA | TBA | TBA | TBA | TBA |- | 884.358713 | 5 | 3 | <math>\frac{5}{3}= \frac{5}{3} \approx 1.66667</math> | <math>-16</math> | 884.36 | TBA | TBA | TBA | TBA | TBA |- | 900 | -1 | 9 | <math> \quad 2^{9/12} \approx1.68179</math> | <math>\;0</math> | 900.00 | TBA | TBA | TBA | TBA | TBA |- | 905.8650026 | 27 | 16 | <math>\frac{27}{16}= \frac{3^{3}}{2^{4}} \approx 1.6875</math> | <math>+6</math> | 905.87 | TBA | TBA | TBA | TBA | TBA |- | 933.1290944 | 12 | 7 | <math>\frac{12}{7}= \frac{2^{2}\cdot 3}{7} \approx 1.71429</math> | <math>+33</math> | 933.13 | TBA | TBA | TBA | TBA | TBA |- | 968.8259065 | 7 | 4 | <math>\frac{7}{4}= \frac{7}{2^{2}} \approx 1.75</math> | <math>-31</math> | 968.83 | TBA | TBA | TBA | TBA | TBA |- | 996.0899983 | 16 | 9 | <math>\frac{16}{9}= \frac{2^{4}}{3^{2}} \approx 1.77778</math> | <math>-4</math> | 996.09 | TBA | TBA | TBA | TBA | TBA |- | 1000 | -1 | 10 | <math> \quad 2^{10/12} \approx1.7818</math> | <math>\;0</math> | 1000.00 | TBA | TBA | TBA | TBA | TBA |- | 1017.596288 | 9 | 5 | <math>\frac{9}{5}= \frac{3^{2}}{5} \approx 1.8</math> | <math>+18</math> | 1017.60 | TBA | TBA | TBA | TBA | TBA |} kpo79gjlcmpvy8mu1ilyewban2p5nhq 2410889 2410888 2022-08-02T03:49:40Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n<br>(cents) ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>-6</math> | 294.13 | TBA | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0</math> | 300.00 | TBA | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>+2</math> | 301.85 | TBA | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>+16</math> | 315.64 | TBA | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>-14</math> | 386.31 | TBA | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0</math> | 400.00 | TBA | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>+8</math> | 407.82 | TBA | TBA | TBA | TBA | TBA |- | 427.3725723 | 32 | 25 | <math>\frac{32}{25}= \frac{2^{5}}{5^{2}} \approx 1.28</math> | <math>+27</math> | 427.37 | TBA | TBA | TBA | TBA | TBA |- | 435.0840953 | 9 | 7 | <math>\frac{9}{7}= \frac{3^{2}}{7} \approx 1.28571</math> | <math>+35</math> | 435.08 | TBA | TBA | TBA | TBA | TBA |- | 498.0449991 | 4 | 3 | <math>\frac{4}{3}= \frac{2^{2}}{3} \approx 1.33333</math> | <math>-2</math> | 498.04 | TBA | TBA | TBA | TBA | TBA |- | 500 | -1 | 5 | <math> \quad 2^{5/12} \approx1.33484</math> | <math>\;0</math> | 500.00 | TBA | TBA | TBA | TBA | TBA |- | 519.5512887 | 27 | 20 | <math>\frac{27}{20}= \frac{3^{3}}{2^{2}\cdot 5} \approx 1.35</math> | <math>+20</math> | 519.55 | TBA | TBA | TBA | TBA | TBA |- | 568.717426 | 25 | 18 | <math>\frac{25}{18}= \frac{5^{2}}{2\cdot 3^{2}} \approx 1.38889</math> | <math>-31</math> | 568.72 | TBA | TBA | TBA | TBA | TBA |- | 582.5121926 | 7 | 5 | <math>\frac{7}{5}= \frac{7}{5} \approx 1.4</math> | <math>-17</math> | 582.51 | TBA | TBA | TBA | TBA | TBA |- | 590.2237156 | 45 | 32 | <math>\frac{45}{32}= \frac{3^{2}\cdot 5}{2^{5}} \approx 1.40625</math> | <math>-10</math> | 590.22 | TBA | TBA | TBA | TBA | TBA |- | 600 | -1 | 6 | <math> \quad 2^{6/12} \approx1.41421</math> | <math>\;0</math> | 600.00 | TBA | TBA | TBA | TBA | TBA |- | 609.7762844 | 64 | 45 | <math>\frac{64}{45}= \frac{2^{6}}{3^{2}\cdot 5} \approx 1.42222</math> | <math>+10</math> | 609.78 | TBA | TBA | TBA | TBA | TBA |- | 617.4878074 | 10 | 7 | <math>\frac{10}{7}= \frac{2\cdot 5}{7} \approx 1.42857</math> | <math>+17</math> | 617.49 | TBA | TBA | TBA | TBA | TBA |- | 631.282574 | 36 | 25 | <math>\frac{36}{25}= \frac{2^{2}\cdot 3^{2}}{5^{2}} \approx 1.44</math> | <math>+31</math> | 631.28 | TBA | TBA | TBA | TBA | TBA |- | 680.4487113 | 40 | 27 | <math>\frac{40}{27}= \frac{2^{3}\cdot 5}{3^{3}} \approx 1.48148</math> | <math>-20</math> | 680.45 | TBA | TBA | TBA | TBA | TBA |- | 700 | -1 | 7 | <math> \quad 2^{7/12} \approx1.49831</math> | <math>\;0</math> | 700.00 | TBA | TBA | TBA | TBA | TBA |- | 701.9550009 | 3 | 2 | <math>\frac{3}{2}= \frac{3}{2} \approx 1.5</math> | <math>+2</math> | 701.96 | TBA | TBA | TBA | TBA | TBA |- | 764.9159047 | 14 | 9 | <math>\frac{14}{9}= \frac{2\cdot 7}{3^{2}} \approx 1.55556</math> | <math>-35</math> | 764.92 | TBA | TBA | TBA | TBA | TBA |- | 772.6274277 | 25 | 16 | <math>\frac{25}{16}= \frac{5^{2}}{2^{4}} \approx 1.5625</math> | <math>-27</math> | 772.63 | TBA | TBA | TBA | TBA | TBA |- | 800 | -1 | 8 | <math> \quad 2^{8/12} \approx1.5874</math> | <math>\;0</math> | 800.00 | TBA | TBA | TBA | TBA | TBA |- | 813.6862861 | 8 | 5 | <math>\frac{8}{5}= \frac{2^{3}}{5} \approx 1.6</math> | <math>+14</math> | 813.69 | TBA | TBA | TBA | TBA | TBA |- | 835.1925757 | 81 | 50 | <math>\frac{81}{50}= \frac{3^{4}}{2\cdot 5^{2}} \approx 1.62</math> | <math>+35</math> | 835.19 | TBA | TBA | TBA | TBA | TBA |- | 884.358713 | 5 | 3 | <math>\frac{5}{3}= \frac{5}{3} \approx 1.66667</math> | <math>-16</math> | 884.36 | TBA | TBA | TBA | TBA | TBA |- | 900 | -1 | 9 | <math> \quad 2^{9/12} \approx1.68179</math> | <math>\;0</math> | 900.00 | TBA | TBA | TBA | TBA | TBA |- | 905.8650026 | 27 | 16 | <math>\frac{27}{16}= \frac{3^{3}}{2^{4}} \approx 1.6875</math> | <math>+6</math> | 905.87 | TBA | TBA | TBA | TBA | TBA |- | 933.1290944 | 12 | 7 | <math>\frac{12}{7}= \frac{2^{2}\cdot 3}{7} \approx 1.71429</math> | <math>+33</math> | 933.13 | TBA | TBA | TBA | TBA | TBA |- | 968.8259065 | 7 | 4 | <math>\frac{7}{4}= \frac{7}{2^{2}} \approx 1.75</math> | <math>-31</math> | 968.83 | TBA | TBA | TBA | TBA | TBA |- | 996.0899983 | 16 | 9 | <math>\frac{16}{9}= \frac{2^{4}}{3^{2}} \approx 1.77778</math> | <math>-4</math> | 996.09 | TBA | TBA | TBA | TBA | TBA |- | 1000 | -1 | 10 | <math> \quad 2^{10/12} \approx1.7818</math> | <math>\;0</math> | 1000.00 | TBA | TBA | TBA | TBA | TBA |- | 1017.596288 | 9 | 5 | <math>\frac{9}{5}= \frac{3^{2}}{5} \approx 1.8</math> | <math>+18</math> | 1017.60 | TBA | TBA | TBA | TBA | TBA |} cfvzh14yg3n7q4z57frmlqtns4o59z6 2410894 2410889 2022-08-02T04:07:21Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n<br>(cents) ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>-6</math> | 294.13 | 294 | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0</math> | 300.00 | 300 | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>+2</math> | 301.85 | 302 | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>+16</math> | 315.64 | 316 | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>-14</math> | 386.31 | 386 | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0</math> | 400.00 | 400 | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>+8</math> | 407.82 | 408 | TBA | TBA | TBA | TBA |- | 427.3725723 | 32 | 25 | <math>\frac{32}{25}= \frac{2^{5}}{5^{2}} \approx 1.28</math> | <math>+27</math> | 427.37 | 427 | TBA | TBA | TBA | TBA |- | 435.0840953 | 9 | 7 | <math>\frac{9}{7}= \frac{3^{2}}{7} \approx 1.28571</math> | <math>+35</math> | 435.08 | 435 | TBA | TBA | TBA | TBA |- | 498.0449991 | 4 | 3 | <math>\frac{4}{3}= \frac{2^{2}}{3} \approx 1.33333</math> | <math>-2</math> | 498.04 | 498 | TBA | TBA | TBA | TBA |- | 500 | -1 | 5 | <math> \quad 2^{5/12} \approx1.33484</math> | <math>\;0</math> | 500.00 | 500 | TBA | TBA | TBA | TBA |- | 519.5512887 | 27 | 20 | <math>\frac{27}{20}= \frac{3^{3}}{2^{2}\cdot 5} \approx 1.35</math> | <math>+20</math> | 519.55 | 520 | TBA | TBA | TBA | TBA |- | 568.717426 | 25 | 18 | <math>\frac{25}{18}= \frac{5^{2}}{2\cdot 3^{2}} \approx 1.38889</math> | <math>-31</math> | 568.72 | 569 | TBA | TBA | TBA | TBA |- | 582.5121926 | 7 | 5 | <math>\frac{7}{5}= \frac{7}{5} \approx 1.4</math> | <math>-17</math> | 582.51 | 583 | TBA | TBA | TBA | TBA |- | 590.2237156 | 45 | 32 | <math>\frac{45}{32}= \frac{3^{2}\cdot 5}{2^{5}} \approx 1.40625</math> | <math>-10</math> | 590.22 | 590 | TBA | TBA | TBA | TBA |- | 600 | -1 | 6 | <math> \quad 2^{6/12} \approx1.41421</math> | <math>\;0</math> | 600.00 | 600 | TBA | TBA | TBA | TBA |- | 609.7762844 | 64 | 45 | <math>\frac{64}{45}= \frac{2^{6}}{3^{2}\cdot 5} \approx 1.42222</math> | <math>+10</math> | 609.78 | 610 | TBA | TBA | TBA | TBA |- | 617.4878074 | 10 | 7 | <math>\frac{10}{7}= \frac{2\cdot 5}{7} \approx 1.42857</math> | <math>+17</math> | 617.49 | 617 | TBA | TBA | TBA | TBA |- | 631.282574 | 36 | 25 | <math>\frac{36}{25}= \frac{2^{2}\cdot 3^{2}}{5^{2}} \approx 1.44</math> | <math>+31</math> | 631.28 | 631 | TBA | TBA | TBA | TBA |- | 680.4487113 | 40 | 27 | <math>\frac{40}{27}= \frac{2^{3}\cdot 5}{3^{3}} \approx 1.48148</math> | <math>-20</math> | 680.45 | 680 | TBA | TBA | TBA | TBA |- | 700 | -1 | 7 | <math> \quad 2^{7/12} \approx1.49831</math> | <math>\;0</math> | 700.00 | 700 | TBA | TBA | TBA | TBA |- | 701.9550009 | 3 | 2 | <math>\frac{3}{2}= \frac{3}{2} \approx 1.5</math> | <math>+2</math> | 701.96 | 702 | TBA | TBA | TBA | TBA |- | 764.9159047 | 14 | 9 | <math>\frac{14}{9}= \frac{2\cdot 7}{3^{2}} \approx 1.55556</math> | <math>-35</math> | 764.92 | 765 | TBA | TBA | TBA | TBA |- | 772.6274277 | 25 | 16 | <math>\frac{25}{16}= \frac{5^{2}}{2^{4}} \approx 1.5625</math> | <math>-27</math> | 772.63 | 773 | TBA | TBA | TBA | TBA |- | 800 | -1 | 8 | <math> \quad 2^{8/12} \approx1.5874</math> | <math>\;0</math> | 800.00 | 800 | TBA | TBA | TBA | TBA |- | 813.6862861 | 8 | 5 | <math>\frac{8}{5}= \frac{2^{3}}{5} \approx 1.6</math> | <math>+14</math> | 813.69 | 814 | TBA | TBA | TBA | TBA |- | 835.1925757 | 81 | 50 | <math>\frac{81}{50}= \frac{3^{4}}{2\cdot 5^{2}} \approx 1.62</math> | <math>+35</math> | 835.19 | 835 | TBA | TBA | TBA | TBA |- | 884.358713 | 5 | 3 | <math>\frac{5}{3}= \frac{5}{3} \approx 1.66667</math> | <math>-16</math> | 884.36 | 884 | TBA | TBA | TBA | TBA |- | 900 | -1 | 9 | <math> \quad 2^{9/12} \approx1.68179</math> | <math>\;0</math> | 900.00 | 900 | TBA | TBA | TBA | TBA |- | 905.8650026 | 27 | 16 | <math>\frac{27}{16}= \frac{3^{3}}{2^{4}} \approx 1.6875</math> | <math>+6</math> | 905.87 | 906 | TBA | TBA | TBA | TBA |- | 933.1290944 | 12 | 7 | <math>\frac{12}{7}= \frac{2^{2}\cdot 3}{7} \approx 1.71429</math> | <math>+33</math> | 933.13 | 933 | TBA | TBA | TBA | TBA |- | 968.8259065 | 7 | 4 | <math>\frac{7}{4}= \frac{7}{2^{2}} \approx 1.75</math> | <math>-31</math> | 968.83 | 969 | TBA | TBA | TBA | TBA |- | 996.0899983 | 16 | 9 | <math>\frac{16}{9}= \frac{2^{4}}{3^{2}} \approx 1.77778</math> | <math>-4</math> | 996.09 | 996 | TBA | TBA | TBA | TBA |- | 1000 | -1 | 10 | <math> \quad 2^{10/12} \approx1.7818</math> | <math>\;0</math> | 1000.00 | 1000 | TBA | TBA | TBA | TBA |- | 1017.596288 | 9 | 5 | <math>\frac{9}{5}= \frac{3^{2}}{5} \approx 1.8</math> | <math>+18</math> | 1017.60 | 1018 | TBA | TBA | TBA | TBA |} a9g8ae71mlt0egmus0wrcpyl3h1vm77 2410895 2410894 2022-08-02T04:10:01Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n<br>(cents) ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>-6</math> | 294.13 | 300 | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0</math> | 300.00 | 300 | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>+2</math> | 301.85 | 300 | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>+16</math> | 315.64 | 300 | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>-14</math> | 386.31 | 400 | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0</math> | 400.00 | 400 | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>+8</math> | 407.82 | 400 | TBA | TBA | TBA | TBA |- | 427.3725723 | 32 | 25 | <math>\frac{32}{25}= \frac{2^{5}}{5^{2}} \approx 1.28</math> | <math>+27</math> | 427.37 | 400 | TBA | TBA | TBA | TBA |- | 435.0840953 | 9 | 7 | <math>\frac{9}{7}= \frac{3^{2}}{7} \approx 1.28571</math> | <math>+35</math> | 435.08 | 400 | TBA | TBA | TBA | TBA |- | 498.0449991 | 4 | 3 | <math>\frac{4}{3}= \frac{2^{2}}{3} \approx 1.33333</math> | <math>-2</math> | 498.04 | 500 | TBA | TBA | TBA | TBA |- | 500 | -1 | 5 | <math> \quad 2^{5/12} \approx1.33484</math> | <math>\;0</math> | 500.00 | 500 | TBA | TBA | TBA | TBA |- | 519.5512887 | 27 | 20 | <math>\frac{27}{20}= \frac{3^{3}}{2^{2}\cdot 5} \approx 1.35</math> | <math>+20</math> | 519.55 | 500 | TBA | TBA | TBA | TBA |- | 568.717426 | 25 | 18 | <math>\frac{25}{18}= \frac{5^{2}}{2\cdot 3^{2}} \approx 1.38889</math> | <math>-31</math> | 568.72 | 600 | TBA | TBA | TBA | TBA |- | 582.5121926 | 7 | 5 | <math>\frac{7}{5}= \frac{7}{5} \approx 1.4</math> | <math>-17</math> | 582.51 | 600 | TBA | TBA | TBA | TBA |- | 590.2237156 | 45 | 32 | <math>\frac{45}{32}= \frac{3^{2}\cdot 5}{2^{5}} \approx 1.40625</math> | <math>-10</math> | 590.22 | 600 | TBA | TBA | TBA | TBA |- | 600 | -1 | 6 | <math> \quad 2^{6/12} \approx1.41421</math> | <math>\;0</math> | 600.00 | 600 | TBA | TBA | TBA | TBA |- | 609.7762844 | 64 | 45 | <math>\frac{64}{45}= \frac{2^{6}}{3^{2}\cdot 5} \approx 1.42222</math> | <math>+10</math> | 609.78 | 600 | TBA | TBA | TBA | TBA |- | 617.4878074 | 10 | 7 | <math>\frac{10}{7}= \frac{2\cdot 5}{7} \approx 1.42857</math> | <math>+17</math> | 617.49 | 600 | TBA | TBA | TBA | TBA |- | 631.282574 | 36 | 25 | <math>\frac{36}{25}= \frac{2^{2}\cdot 3^{2}}{5^{2}} \approx 1.44</math> | <math>+31</math> | 631.28 | 600 | TBA | TBA | TBA | TBA |- | 680.4487113 | 40 | 27 | <math>\frac{40}{27}= \frac{2^{3}\cdot 5}{3^{3}} \approx 1.48148</math> | <math>-20</math> | 680.45 | 700 | TBA | TBA | TBA | TBA |- | 700 | -1 | 7 | <math> \quad 2^{7/12} \approx1.49831</math> | <math>\;0</math> | 700.00 | 700 | TBA | TBA | TBA | TBA |- | 701.9550009 | 3 | 2 | <math>\frac{3}{2}= \frac{3}{2} \approx 1.5</math> | <math>+2</math> | 701.96 | 700 | TBA | TBA | TBA | TBA |- | 764.9159047 | 14 | 9 | <math>\frac{14}{9}= \frac{2\cdot 7}{3^{2}} \approx 1.55556</math> | <math>-35</math> | 764.92 | 800 | TBA | TBA | TBA | TBA |- | 772.6274277 | 25 | 16 | <math>\frac{25}{16}= \frac{5^{2}}{2^{4}} \approx 1.5625</math> | <math>-27</math> | 772.63 | 800 | TBA | TBA | TBA | TBA |- | 800 | -1 | 8 | <math> \quad 2^{8/12} \approx1.5874</math> | <math>\;0</math> | 800.00 | 800 | TBA | TBA | TBA | TBA |- | 813.6862861 | 8 | 5 | <math>\frac{8}{5}= \frac{2^{3}}{5} \approx 1.6</math> | <math>+14</math> | 813.69 | 800 | TBA | TBA | TBA | TBA |- | 835.1925757 | 81 | 50 | <math>\frac{81}{50}= \frac{3^{4}}{2\cdot 5^{2}} \approx 1.62</math> | <math>+35</math> | 835.19 | 800 | TBA | TBA | TBA | TBA |- | 884.358713 | 5 | 3 | <math>\frac{5}{3}= \frac{5}{3} \approx 1.66667</math> | <math>-16</math> | 884.36 | 900 | TBA | TBA | TBA | TBA |- | 900 | -1 | 9 | <math> \quad 2^{9/12} \approx1.68179</math> | <math>\;0</math> | 900.00 | 900 | TBA | TBA | TBA | TBA |- | 905.8650026 | 27 | 16 | <math>\frac{27}{16}= \frac{3^{3}}{2^{4}} \approx 1.6875</math> | <math>+6</math> | 905.87 | 900 | TBA | TBA | TBA | TBA |- | 933.1290944 | 12 | 7 | <math>\frac{12}{7}= \frac{2^{2}\cdot 3}{7} \approx 1.71429</math> | <math>+33</math> | 933.13 | 900 | TBA | TBA | TBA | TBA |- | 968.8259065 | 7 | 4 | <math>\frac{7}{4}= \frac{7}{2^{2}} \approx 1.75</math> | <math>-31</math> | 968.83 | 1000 | TBA | TBA | TBA | TBA |- | 996.0899983 | 16 | 9 | <math>\frac{16}{9}= \frac{2^{4}}{3^{2}} \approx 1.77778</math> | <math>-4</math> | 996.09 | 1000 | TBA | TBA | TBA | TBA |- | 1000 | -1 | 10 | <math> \quad 2^{10/12} \approx1.7818</math> | <math>\;0</math> | 1000.00 | 1000 | TBA | TBA | TBA | TBA |- | 1017.596288 | 9 | 5 | <math>\frac{9}{5}= \frac{3^{2}}{5} \approx 1.8</math> | <math>+18</math> | 1017.60 | 1000 | TBA | TBA | TBA | TBA |} ejqmmt44bupx3ivsty0wdsarwnfl2l9 2410896 2410895 2022-08-02T04:15:29Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n<br>(cents) ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>-6</math> | 294.13 | m3 (minor third) | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0</math> | 300.00 | m3 (minor third) | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>+2</math> | 301.85 | m3 (minor third) | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>+16</math> | 315.64 | m3 (minor third) | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>-14</math> | 386.31 | M3 (major third) | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0</math> | 400.00 | M3 (major third) | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>+8</math> | 407.82 | M3 (major third) | TBA | TBA | TBA | TBA |- | 427.3725723 | 32 | 25 | <math>\frac{32}{25}= \frac{2^{5}}{5^{2}} \approx 1.28</math> | <math>+27</math> | 427.37 | M3 (major third) | TBA | TBA | TBA | TBA |- | 435.0840953 | 9 | 7 | <math>\frac{9}{7}= \frac{3^{2}}{7} \approx 1.28571</math> | <math>+35</math> | 435.08 | M3 (major third) | TBA | TBA | TBA | TBA |- | 498.0449991 | 4 | 3 | <math>\frac{4}{3}= \frac{2^{2}}{3} \approx 1.33333</math> | <math>-2</math> | 498.04 | P4 (fourth) | TBA | TBA | TBA | TBA |- | 500 | -1 | 5 | <math> \quad 2^{5/12} \approx1.33484</math> | <math>\;0</math> | 500.00 | P4 (fourth) | TBA | TBA | TBA | TBA |- | 519.5512887 | 27 | 20 | <math>\frac{27}{20}= \frac{3^{3}}{2^{2}\cdot 5} \approx 1.35</math> | <math>+20</math> | 519.55 | P4 (fourth) | TBA | TBA | TBA | TBA |- | 568.717426 | 25 | 18 | <math>\frac{25}{18}= \frac{5^{2}}{2\cdot 3^{2}} \approx 1.38889</math> | <math>-31</math> | 568.72 | TT (tritone) | TBA | TBA | TBA | TBA |- | 582.5121926 | 7 | 5 | <math>\frac{7}{5}= \frac{7}{5} \approx 1.4</math> | <math>-17</math> | 582.51 | TT (tritone) | TBA | TBA | TBA | TBA |- | 590.2237156 | 45 | 32 | <math>\frac{45}{32}= \frac{3^{2}\cdot 5}{2^{5}} \approx 1.40625</math> | <math>-10</math> | 590.22 | TT (tritone) | TBA | TBA | TBA | TBA |- | 600 | -1 | 6 | <math> \quad 2^{6/12} \approx1.41421</math> | <math>\;0</math> | 600.00 | TT (tritone) | TBA | TBA | TBA | TBA |- | 609.7762844 | 64 | 45 | <math>\frac{64}{45}= \frac{2^{6}}{3^{2}\cdot 5} \approx 1.42222</math> | <math>+10</math> | 609.78 | TT (tritone) | TBA | TBA | TBA | TBA |- | 617.4878074 | 10 | 7 | <math>\frac{10}{7}= \frac{2\cdot 5}{7} \approx 1.42857</math> | <math>+17</math> | 617.49 | TT (tritone) | TBA | TBA | TBA | TBA |- | 631.282574 | 36 | 25 | <math>\frac{36}{25}= \frac{2^{2}\cdot 3^{2}}{5^{2}} \approx 1.44</math> | <math>+31</math> | 631.28 | TT (tritone) | TBA | TBA | TBA | TBA |- | 680.4487113 | 40 | 27 | <math>\frac{40}{27}= \frac{2^{3}\cdot 5}{3^{3}} \approx 1.48148</math> | <math>-20</math> | 680.45 | P5 (fifth) | TBA | TBA | TBA | TBA |- | 700 | -1 | 7 | <math> \quad 2^{7/12} \approx1.49831</math> | <math>\;0</math> | 700.00 | P5 (fifth) | TBA | TBA | TBA | TBA |- | 701.9550009 | 3 | 2 | <math>\frac{3}{2}= \frac{3}{2} \approx 1.5</math> | <math>+2</math> | 701.96 | P5 (fifth) | TBA | TBA | TBA | TBA |- | 764.9159047 | 14 | 9 | <math>\frac{14}{9}= \frac{2\cdot 7}{3^{2}} \approx 1.55556</math> | <math>-35</math> | 764.92 | m6 (minor sixth) | TBA | TBA | TBA | TBA |- | 772.6274277 | 25 | 16 | <math>\frac{25}{16}= \frac{5^{2}}{2^{4}} \approx 1.5625</math> | <math>-27</math> | 772.63 | m6 (minor sixth) | TBA | TBA | TBA | TBA |- | 800 | -1 | 8 | <math> \quad 2^{8/12} \approx1.5874</math> | <math>\;0</math> | 800.00 | m6 (minor sixth) | TBA | TBA | TBA | TBA |- | 813.6862861 | 8 | 5 | <math>\frac{8}{5}= \frac{2^{3}}{5} \approx 1.6</math> | <math>+14</math> | 813.69 | m6 (minor sixth) | TBA | TBA | TBA | TBA |- | 835.1925757 | 81 | 50 | <math>\frac{81}{50}= \frac{3^{4}}{2\cdot 5^{2}} \approx 1.62</math> | <math>+35</math> | 835.19 | m6 (minor sixth) | TBA | TBA | TBA | TBA |- | 884.358713 | 5 | 3 | <math>\frac{5}{3}= \frac{5}{3} \approx 1.66667</math> | <math>-16</math> | 884.36 | M6 (major sixth) | TBA | TBA | TBA | TBA |- | 900 | -1 | 9 | <math> \quad 2^{9/12} \approx1.68179</math> | <math>\;0</math> | 900.00 | M6 (major sixth) | TBA | TBA | TBA | TBA |- | 905.8650026 | 27 | 16 | <math>\frac{27}{16}= \frac{3^{3}}{2^{4}} \approx 1.6875</math> | <math>+6</math> | 905.87 | M6 (major sixth) | TBA | TBA | TBA | TBA |- | 933.1290944 | 12 | 7 | <math>\frac{12}{7}= \frac{2^{2}\cdot 3}{7} \approx 1.71429</math> | <math>+33</math> | 933.13 | M6 (major sixth) | TBA | TBA | TBA | TBA |- | 968.8259065 | 7 | 4 | <math>\frac{7}{4}= \frac{7}{2^{2}} \approx 1.75</math> | <math>-31</math> | 968.83 | m7 (minor seventh) | TBA | TBA | TBA | TBA |- | 996.0899983 | 16 | 9 | <math>\frac{16}{9}= \frac{2^{4}}{3^{2}} \approx 1.77778</math> | <math>-4</math> | 996.09 | m7 (minor seventh) | TBA | TBA | TBA | TBA |- | 1000 | -1 | 10 | <math> \quad 2^{10/12} \approx1.7818</math> | <math>\;0</math> | 1000.00 | m7 (minor seventh) | TBA | TBA | TBA | TBA |- | 1017.596288 | 9 | 5 | <math>\frac{9}{5}= \frac{3^{2}}{5} \approx 1.8</math> | <math>+18</math> | 1017.60 | m7 (minor seventh) | TBA | TBA | TBA | TBA |} 6xkxz5mg0dlafafpgptiym3x7p982nc 2410903 2410896 2022-08-02T05:16:55Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n<br>(cents) ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>-6</math> | 294.13 | TBA | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0</math> | 300.00 | TBA | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>+2</math> | 301.85 | TBA | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>+16</math> | 315.64 | TBA | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>-14</math> | 386.31 | TBA | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0</math> | 400.00 | TBA | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>+8</math> | 407.82 | TBA | TBA | TBA | TBA | TBA |- | 427.3725723 | 32 | 25 | <math>\frac{32}{25}= \frac{2^{5}}{5^{2}} \approx 1.28</math> | <math>+27</math> | 427.37 | TBA | TBA | TBA | TBA | TBA |- | 435.0840953 | 9 | 7 | <math>\frac{9}{7}= \frac{3^{2}}{7} \approx 1.28571</math> | <math>+35</math> | 435.08 | TBA | TBA | TBA | TBA | TBA |- | 498.0449991 | 4 | 3 | <math>\frac{4}{3}= \frac{2^{2}}{3} \approx 1.33333</math> | <math>-2</math> | 498.04 | TBA | TBA | TBA | TBA | TBA |- | 500 | -1 | 5 | <math> \quad 2^{5/12} \approx1.33484</math> | <math>\;0</math> | 500.00 | TBA | TBA | TBA | TBA | TBA |- | 519.5512887 | 27 | 20 | <math>\frac{27}{20}= \frac{3^{3}}{2^{2}\cdot 5} \approx 1.35</math> | <math>+20</math> | 519.55 | TBA | TBA | TBA | TBA | TBA |- | 568.717426 | 25 | 18 | <math>\frac{25}{18}= \frac{5^{2}}{2\cdot 3^{2}} \approx 1.38889</math> | <math>-31</math> | 568.72 | TBA | TBA | TBA | TBA | TBA |- | 582.5121926 | 7 | 5 | <math>\frac{7}{5}= \frac{7}{5} \approx 1.4</math> | <math>-17</math> | 582.51 | TBA | TBA | TBA | TBA | TBA |- | 590.2237156 | 45 | 32 | <math>\frac{45}{32}= \frac{3^{2}\cdot 5}{2^{5}} \approx 1.40625</math> | <math>-10</math> | 590.22 | TBA | TBA | TBA | TBA | TBA |- | 600 | -1 | 6 | <math> \quad 2^{6/12} \approx1.41421</math> | <math>\;0</math> | 600.00 | TBA | TBA | TBA | TBA | TBA |- | 609.7762844 | 64 | 45 | <math>\frac{64}{45}= \frac{2^{6}}{3^{2}\cdot 5} \approx 1.42222</math> | <math>+10</math> | 609.78 | TBA | TBA | TBA | TBA | TBA |- | 617.4878074 | 10 | 7 | <math>\frac{10}{7}= \frac{2\cdot 5}{7} \approx 1.42857</math> | <math>+17</math> | 617.49 | TBA | TBA | TBA | TBA | TBA |- | 631.282574 | 36 | 25 | <math>\frac{36}{25}= \frac{2^{2}\cdot 3^{2}}{5^{2}} \approx 1.44</math> | <math>+31</math> | 631.28 | TBA | TBA | TBA | TBA | TBA |- | 680.4487113 | 40 | 27 | <math>\frac{40}{27}= \frac{2^{3}\cdot 5}{3^{3}} \approx 1.48148</math> | <math>-20</math> | 680.45 | TBA | TBA | TBA | TBA | TBA |- | 700 | -1 | 7 | <math> \quad 2^{7/12} \approx1.49831</math> | <math>\;0</math> | 700.00 | TBA | TBA | TBA | TBA | TBA |- | 701.9550009 | 3 | 2 | <math>\frac{3}{2}= \frac{3}{2} \approx 1.5</math> | <math>+2</math> | 701.96 | TBA | TBA | TBA | TBA | TBA |- | 764.9159047 | 14 | 9 | <math>\frac{14}{9}= \frac{2\cdot 7}{3^{2}} \approx 1.55556</math> | <math>-35</math> | 764.92 | TBA | TBA | TBA | TBA | TBA |- | 772.6274277 | 25 | 16 | <math>\frac{25}{16}= \frac{5^{2}}{2^{4}} \approx 1.5625</math> | <math>-27</math> | 772.63 | TBA | TBA | TBA | TBA | TBA |- | 800 | -1 | 8 | <math> \quad 2^{8/12} \approx1.5874</math> | <math>\;0</math> | 800.00 | TBA | TBA | TBA | TBA | TBA |- | 813.6862861 | 8 | 5 | <math>\frac{8}{5}= \frac{2^{3}}{5} \approx 1.6</math> | <math>+14</math> | 813.69 | TBA | TBA | TBA | TBA | TBA |- | 835.1925757 | 81 | 50 | <math>\frac{81}{50}= \frac{3^{4}}{2\cdot 5^{2}} \approx 1.62</math> | <math>+35</math> | 835.19 | TBA | TBA | TBA | TBA | TBA |- | 884.358713 | 5 | 3 | <math>\frac{5}{3}= \frac{5}{3} \approx 1.66667</math> | <math>-16</math> | 884.36 | TBA | TBA | TBA | TBA | TBA |- | 900 | -1 | 9 | <math> \quad 2^{9/12} \approx1.68179</math> | <math>\;0</math> | 900.00 | TBA | TBA | TBA | TBA | TBA |- | 905.8650026 | 27 | 16 | <math>\frac{27}{16}= \frac{3^{3}}{2^{4}} \approx 1.6875</math> | <math>+6</math> | 905.87 | TBA | TBA | TBA | TBA | TBA |- | 933.1290944 | 12 | 7 | <math>\frac{12}{7}= \frac{2^{2}\cdot 3}{7} \approx 1.71429</math> | <math>+33</math> | 933.13 | TBA | TBA | TBA | TBA | TBA |- | 968.8259065 | 7 | 4 | <math>\frac{7}{4}= \frac{7}{2^{2}} \approx 1.75</math> | <math>-31</math> | 968.83 | TBA | TBA | TBA | TBA | TBA |- | 996.0899983 | 16 | 9 | <math>\frac{16}{9}= \frac{2^{4}}{3^{2}} \approx 1.77778</math> | <math>-4</math> | 996.09 | TBA | TBA | TBA | TBA | TBA |- | 1000 | -1 | 10 | <math> \quad 2^{10/12} \approx1.7818</math> | <math>\;0</math> | 1000.00 | TBA | TBA | TBA | TBA | TBA |- | 1017.596288 | 9 | 5 | <math>\frac{9}{5}= \frac{3^{2}}{5} \approx 1.8</math> | <math>+18</math> | 1017.60 | TBA | TBA | TBA | TBA | TBA |} cfvzh14yg3n7q4z57frmlqtns4o59z6 2410904 2410903 2022-08-02T05:22:17Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n<br>(cents) ! n<br>(cents) ! interval ! new ! new ! new ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>-6</math> | 294.13 | m3 (minor third) | TBA | TBA | TBA | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0</math> | 300.00 | m3 (minor third) | TBA | TBA | TBA | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>+2</math> | 301.85 | m3 (minor third) | TBA | TBA | TBA | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>+16</math> | 315.64 | m3 (minor third) | TBA | TBA | TBA | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>-14</math> | 386.31 | M3 (major third) | TBA | TBA | TBA | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0</math> | 400.00 | M3 (major third) | TBA | TBA | TBA | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>+8</math> | 407.82 | M3 (major third) | TBA | TBA | TBA | TBA |- | 427.3725723 | 32 | 25 | <math>\frac{32}{25}= \frac{2^{5}}{5^{2}} \approx 1.28</math> | <math>+27</math> | 427.37 | M3 (major third) | TBA | TBA | TBA | TBA |- | 435.0840953 | 9 | 7 | <math>\frac{9}{7}= \frac{3^{2}}{7} \approx 1.28571</math> | <math>+35</math> | 435.08 | M3 (major third) | TBA | TBA | TBA | TBA |- | 498.0449991 | 4 | 3 | <math>\frac{4}{3}= \frac{2^{2}}{3} \approx 1.33333</math> | <math>-2</math> | 498.04 | P4 (fourth) | TBA | TBA | TBA | TBA |- | 500 | -1 | 5 | <math> \quad 2^{5/12} \approx1.33484</math> | <math>\;0</math> | 500.00 | P4 (fourth) | TBA | TBA | TBA | TBA |- | 519.5512887 | 27 | 20 | <math>\frac{27}{20}= \frac{3^{3}}{2^{2}\cdot 5} \approx 1.35</math> | <math>+20</math> | 519.55 | P4 (fourth) | TBA | TBA | TBA | TBA |- | 568.717426 | 25 | 18 | <math>\frac{25}{18}= \frac{5^{2}}{2\cdot 3^{2}} \approx 1.38889</math> | <math>-31</math> | 568.72 | TT (tritone) | TBA | TBA | TBA | TBA |- | 582.5121926 | 7 | 5 | <math>\frac{7}{5}= \frac{7}{5} \approx 1.4</math> | <math>-17</math> | 582.51 | TT (tritone) | TBA | TBA | TBA | TBA |- | 590.2237156 | 45 | 32 | <math>\frac{45}{32}= \frac{3^{2}\cdot 5}{2^{5}} \approx 1.40625</math> | <math>-10</math> | 590.22 | TT (tritone) | TBA | TBA | TBA | TBA |- | 600 | -1 | 6 | <math> \quad 2^{6/12} \approx1.41421</math> | <math>\;0</math> | 600.00 | TT (tritone) | TBA | TBA | TBA | TBA |- | 609.7762844 | 64 | 45 | <math>\frac{64}{45}= \frac{2^{6}}{3^{2}\cdot 5} \approx 1.42222</math> | <math>+10</math> | 609.78 | TT (tritone) | TBA | TBA | TBA | TBA |- | 617.4878074 | 10 | 7 | <math>\frac{10}{7}= \frac{2\cdot 5}{7} \approx 1.42857</math> | <math>+17</math> | 617.49 | TT (tritone) | TBA | TBA | TBA | TBA |- | 631.282574 | 36 | 25 | <math>\frac{36}{25}= \frac{2^{2}\cdot 3^{2}}{5^{2}} \approx 1.44</math> | <math>+31</math> | 631.28 | TT (tritone) | TBA | TBA | TBA | TBA |- | 680.4487113 | 40 | 27 | <math>\frac{40}{27}= \frac{2^{3}\cdot 5}{3^{3}} \approx 1.48148</math> | <math>-20</math> | 680.45 | P5 (fifth) | TBA | TBA | TBA | TBA |- | 700 | -1 | 7 | <math> \quad 2^{7/12} \approx1.49831</math> | <math>\;0</math> | 700.00 | P5 (fifth) | TBA | TBA | TBA | TBA |- | 701.9550009 | 3 | 2 | <math>\frac{3}{2}= \frac{3}{2} \approx 1.5</math> | <math>+2</math> | 701.96 | P5 (fifth) | TBA | TBA | TBA | TBA |- | 764.9159047 | 14 | 9 | <math>\frac{14}{9}= \frac{2\cdot 7}{3^{2}} \approx 1.55556</math> | <math>-35</math> | 764.92 | m6 (minor sixth) | TBA | TBA | TBA | TBA |- | 772.6274277 | 25 | 16 | <math>\frac{25}{16}= \frac{5^{2}}{2^{4}} \approx 1.5625</math> | <math>-27</math> | 772.63 | m6 (minor sixth) | TBA | TBA | TBA | TBA |- | 800 | -1 | 8 | <math> \quad 2^{8/12} \approx1.5874</math> | <math>\;0</math> | 800.00 | m6 (minor sixth) | TBA | TBA | TBA | TBA |- | 813.6862861 | 8 | 5 | <math>\frac{8}{5}= \frac{2^{3}}{5} \approx 1.6</math> | <math>+14</math> | 813.69 | m6 (minor sixth) | TBA | TBA | TBA | TBA |- | 835.1925757 | 81 | 50 | <math>\frac{81}{50}= \frac{3^{4}}{2\cdot 5^{2}} \approx 1.62</math> | <math>+35</math> | 835.19 | m6 (minor sixth) | TBA | TBA | TBA | TBA |- | 884.358713 | 5 | 3 | <math>\frac{5}{3}= \frac{5}{3} \approx 1.66667</math> | <math>-16</math> | 884.36 | M6 (major sixth) | TBA | TBA | TBA | TBA |- | 900 | -1 | 9 | <math> \quad 2^{9/12} \approx1.68179</math> | <math>\;0</math> | 900.00 | M6 (major sixth) | TBA | TBA | TBA | TBA |- | 905.8650026 | 27 | 16 | <math>\frac{27}{16}= \frac{3^{3}}{2^{4}} \approx 1.6875</math> | <math>+6</math> | 905.87 | M6 (major sixth) | TBA | TBA | TBA | TBA |- | 933.1290944 | 12 | 7 | <math>\frac{12}{7}= \frac{2^{2}\cdot 3}{7} \approx 1.71429</math> | <math>+33</math> | 933.13 | M6 (major sixth) | TBA | TBA | TBA | TBA |- | 968.8259065 | 7 | 4 | <math>\frac{7}{4}= \frac{7}{2^{2}} \approx 1.75</math> | <math>-31</math> | 968.83 | m7 (minor seventh) | TBA | TBA | TBA | TBA |- | 996.0899983 | 16 | 9 | <math>\frac{16}{9}= \frac{2^{4}}{3^{2}} \approx 1.77778</math> | <math>-4</math> | 996.09 | m7 (minor seventh) | TBA | TBA | TBA | TBA |- | 1000 | -1 | 10 | <math> \quad 2^{10/12} \approx1.7818</math> | <math>\;0</math> | 1000.00 | m7 (minor seventh) | TBA | TBA | TBA | TBA |- | 1017.596288 | 9 | 5 | <math>\frac{9}{5}= \frac{3^{2}}{5} \approx 1.8</math> | <math>+18</math> | 1017.60 | m7 (minor seventh) | TBA | TBA | TBA | TBA |} 6xkxz5mg0dlafafpgptiym3x7p982nc 2410931 2410904 2022-08-02T06:23:14Z Guy vandegrift 813252 Undo all revisions. Resource is empty, but not [[Wikiversity:Deletions|deleted]]. wikitext text/x-wiki phoiac9h4m842xq45sp7s6u21eteeq1 2410933 2410931 2022-08-02T06:38:07Z Guy vandegrift 813252 wikitext text/x-wiki {| class="wikitable" style="" |- ! cents ! p ! q ! ratio ! &Delta;n<br>(cents) ! n<br>(cents) ! interval ! new |- | 294.1349974 | 32 | 27 | <math>\frac{32}{27}= \frac{2^{5}}{3^{3}} \approx 1.18519</math> | <math>-6</math> | 294.13 | m3 (minor third) | TBA |- | 300 | -1 | 3 | <math> \quad 2^{3/12} \approx1.18921</math> | <math>\;0</math> | 300.00 | m3 (minor third) | TBA |- | 301.8465204 | 25 | 21 | <math>\frac{25}{21}= \frac{5^{2}}{3\cdot 7} \approx 1.19048</math> | <math>+2</math> | 301.85 | m3 (minor third) | TBA |- | 315.641287 | 6 | 5 | <math>\frac{6}{5}= \frac{2\cdot 3}{5} \approx 1.2</math> | <math>+16</math> | 315.64 | m3 (minor third) | TBA |- | 386.3137139 | 5 | 4 | <math>\frac{5}{4}= \frac{5}{2^{2}} \approx 1.25</math> | <math>-14</math> | 386.31 | M3 (major third) | TBA |- | 400 | -1 | 4 | <math> \quad 2^{4/12} \approx1.25992</math> | <math>\;0</math> | 400.00 | M3 (major third) | TBA |- | 407.8200035 | 81 | 64 | <math>\frac{81}{64}= \frac{3^{4}}{2^{6}} \approx 1.26562</math> | <math>+8</math> | 407.82 | M3 (major third) | TBA |- | 427.3725723 | 32 | 25 | <math>\frac{32}{25}= \frac{2^{5}}{5^{2}} \approx 1.28</math> | <math>+27</math> | 427.37 | M3 (major third) | TBA |- | 435.0840953 | 9 | 7 | <math>\frac{9}{7}= \frac{3^{2}}{7} \approx 1.28571</math> | <math>+35</math> | 435.08 | M3 (major third) | TBA |- | 498.0449991 | 4 | 3 | <math>\frac{4}{3}= \frac{2^{2}}{3} \approx 1.33333</math> | <math>-2</math> | 498.04 | P4 (fourth) | TBA |- | 500 | -1 | 5 | <math> \quad 2^{5/12} \approx1.33484</math> | <math>\;0</math> | 500.00 | P4 (fourth) | TBA |- | 519.5512887 | 27 | 20 | <math>\frac{27}{20}= \frac{3^{3}}{2^{2}\cdot 5} \approx 1.35</math> | <math>+20</math> | 519.55 | P4 (fourth) | TBA |- | 568.717426 | 25 | 18 | <math>\frac{25}{18}= \frac{5^{2}}{2\cdot 3^{2}} \approx 1.38889</math> | <math>-31</math> | 568.72 | TT (tritone) | TBA |- | 582.5121926 | 7 | 5 | <math>\frac{7}{5}= \frac{7}{5} \approx 1.4</math> | <math>-17</math> | 582.51 | TT (tritone) | TBA |- | 590.2237156 | 45 | 32 | <math>\frac{45}{32}= \frac{3^{2}\cdot 5}{2^{5}} \approx 1.40625</math> | <math>-10</math> | 590.22 | TT (tritone) | TBA |- | 600 | -1 | 6 | <math> \quad 2^{6/12} \approx1.41421</math> | <math>\;0</math> | 600.00 | TT (tritone) | TBA |- | 609.7762844 | 64 | 45 | <math>\frac{64}{45}= \frac{2^{6}}{3^{2}\cdot 5} \approx 1.42222</math> | <math>+10</math> | 609.78 | TT (tritone) | TBA |- | 617.4878074 | 10 | 7 | <math>\frac{10}{7}= \frac{2\cdot 5}{7} \approx 1.42857</math> | <math>+17</math> | 617.49 | TT (tritone) | TBA |- | 631.282574 | 36 | 25 | <math>\frac{36}{25}= \frac{2^{2}\cdot 3^{2}}{5^{2}} \approx 1.44</math> | <math>+31</math> | 631.28 | TT (tritone) | TBA |- | 680.4487113 | 40 | 27 | <math>\frac{40}{27}= \frac{2^{3}\cdot 5}{3^{3}} \approx 1.48148</math> | <math>-20</math> | 680.45 | P5 (fifth) | TBA |- | 700 | -1 | 7 | <math> \quad 2^{7/12} \approx1.49831</math> | <math>\;0</math> | 700.00 | P5 (fifth) | TBA |- | 701.9550009 | 3 | 2 | <math>\frac{3}{2}= \frac{3}{2} \approx 1.5</math> | <math>+2</math> | 701.96 | P5 (fifth) | TBA |- | 764.9159047 | 14 | 9 | <math>\frac{14}{9}= \frac{2\cdot 7}{3^{2}} \approx 1.55556</math> | <math>-35</math> | 764.92 | m6 (minor sixth) | TBA |- | 772.6274277 | 25 | 16 | <math>\frac{25}{16}= \frac{5^{2}}{2^{4}} \approx 1.5625</math> | <math>-27</math> | 772.63 | m6 (minor sixth) | TBA |- | 800 | -1 | 8 | <math> \quad 2^{8/12} \approx1.5874</math> | <math>\;0</math> | 800.00 | m6 (minor sixth) | TBA |- | 813.6862861 | 8 | 5 | <math>\frac{8}{5}= \frac{2^{3}}{5} \approx 1.6</math> | <math>+14</math> | 813.69 | m6 (minor sixth) | TBA |- | 835.1925757 | 81 | 50 | <math>\frac{81}{50}= \frac{3^{4}}{2\cdot 5^{2}} \approx 1.62</math> | <math>+35</math> | 835.19 | m6 (minor sixth) | TBA |- | 884.358713 | 5 | 3 | <math>\frac{5}{3}= \frac{5}{3} \approx 1.66667</math> | <math>-16</math> | 884.36 | M6 (major sixth) | TBA |- | 900 | -1 | 9 | <math> \quad 2^{9/12} \approx1.68179</math> | <math>\;0</math> | 900.00 | M6 (major sixth) | TBA |- | 905.8650026 | 27 | 16 | <math>\frac{27}{16}= \frac{3^{3}}{2^{4}} \approx 1.6875</math> | <math>+6</math> | 905.87 | M6 (major sixth) | TBA |- | 933.1290944 | 12 | 7 | <math>\frac{12}{7}= \frac{2^{2}\cdot 3}{7} \approx 1.71429</math> | <math>+33</math> | 933.13 | M6 (major sixth) | TBA |- | 968.8259065 | 7 | 4 | <math>\frac{7}{4}= \frac{7}{2^{2}} \approx 1.75</math> | <math>-31</math> | 968.83 | m7 (minor seventh) | TBA |- | 996.0899983 | 16 | 9 | <math>\frac{16}{9}= \frac{2^{4}}{3^{2}} \approx 1.77778</math> | <math>-4</math> | 996.09 | m7 (minor seventh) | TBA |- | 1000 | -1 | 10 | <math> \quad 2^{10/12} \approx1.7818</math> | <math>\;0</math> | 1000.00 | m7 (minor seventh) | TBA |- | 1017.596288 | 9 | 5 | <math>\frac{9}{5}= \frac{3^{2}}{5} \approx 1.8</math> | <math>+18</math> | 1017.60 | m7 (minor seventh) | TBA |} gao7ibcgsx1g1kwhd0tezuimzq467zb 0 215892 2410866 2394961 2022-08-02T02:22:21Z 172.89.36.50 /* Suicide should be legal */ wikitext text/x-wiki {{Wikidebate}} {{Law}} '''Suicide''' is the act of intentionally causing one's own death. Following the [[Should voluntary euthanasia be legal?|debate about euthanasia]], should suicide be legal as well? Do we own our own lives so completely that we have a right to end them? Or do we owe certain obligations to our communities and societies that we should stay alive even when we may want to die? Suicide is legal. It is effectively illegal. ==Suicide should be legal== ===Arguments for=== * *{{Argument for}} In a free society, we have the right to do what we want with our lives as long as we don't hurt others. Suicide can be what a rational adult chooses for himself. **{{Objection}} Suicide does hurt others—maybe not physically but the emotional and social costs of losing a loved one to suicide are huge. ***{{Objection}} No individual provides informed consent to being born, so no individual is under any obligation to remain alive for the emotional benefit of others. ***{{Objection}} Many other life events cause emotional harm but are not criminalized. For example, a son cutting off all contact with their parents for life (this can be as traumatizing as death), or bad/negligent parenting leading to children developing psychological issues like extreme attention seeking behavior (which can lead to school shootings). ***{{Objection}} Even if a person dies naturally, it will still cause emotional harm to others. *{{Argument for}} It is impossible to enforce a law against suicide. **{{Objection}} But it's possible to punish those who attempt suicide and fail. ***{{Objection}} Forcing someone to live an unwished life is equivalent to torture. ***{{Objection}} Such punishments probably serve no useful purpose. ***{{Objection}} Punishment is a dysfunctional concept altogether. With suicide, enforcement is a dysfunctional concept as well. Preventing physical harm and/or death from being perpetrated by any person to any person (including the self to the self) without the perpetrator having just logical reason to is appropriate. It's not a for the courts to prosecute as a matter requiring punitive measures that are practically completely dysfunctional even when punitive measures are appropriate. Is it logical to punish someone who is trying to give themselves the death sentence? ***{{Argument for}} Gandhi would likely have been force-fed in a prison hospital if his culture had the same attitudes toward mortality as ours during his hunger strikes. I suspect that laws against suicide are written more to prevent its use as a political tool than out of concern for our well-being or eternal souls. They can't punish someone who's dead, but they can haul an emaciated activist out of the public eye and put him in a mental hospital. Further, I see no reason why someone should suffer a protracted death while waiting to die in hospice. The state should not have title over one's life. *{{Argument for}} We do not know what happens after death. It could be better after death for one who engages in suicide. By prohibiting suicide we are imposing implicit theological beliefs on others. Prohibitions on suicide impose the implicit theological belief that death is a worse consequences than life on one who attempts to engage in suicide. This is not morally right. Those willing to engage in suicide also embrace certain implicit theological beliefs. **{{Objection}} Precisely since we don't know what happens next, we can just wait and see until later—there is no rush to die now, since anyone who would commit suicide will die of some other cause anyway. ***{{Objection}} Not feasible for someone suffering from mental anguish or any other sickness making life unbearable. **{{Objection}} The ambiguity of what happens after suicide isn't a sound argument for why suicide should be legal. If someone has made the sound logical decision that their fear of death is less emotionally painful than their fear of remaining alive with their current perception of reality, making it worse with legal complications doesn't make sense. Unless we are all explicitly required to literally bestow the accountability and authority in our lives for our lives to something external, then it is a decision that one should be allowed to make without requiring justification if it is not going to hurt anyone else against their will. Especially if the strongest reason against allowing it is OUR inability to be emotionally balanced when confronted with the reality that someone else did something they wanted to with themselves. So we literally insist at face value that, at all cost, suicide must be prevented only because of our own inability to process our own emotions about it. We are literally requiring them to remain in agony to support our codependently narcissistic ego while knowing and admitting that there is nothing we can do to help besides falsely imprison, forcibly medicate, and haphazardly attempt to counsel. But we'll feel better about ourselves, like all narcissists, when we try to force our (unwittingly) false perception of reality onto another and literally force them to bend to OUR will. But that requires understanding narcissism, and only painfully few do. ***{{Objection}} Forcing adults to wait is dehumanizing and robbing adults of personal responsibility and freedom. We should use persuasion, reason, and kindness to entice adults to remain on Earth, not coercion, force, and confinement. **{{Objection}} Most arguments against suicide don't have theological assumptions. ***{{Objection}} Arguments against suicide always rely on implicit theological assumptions. ****{{Objection}} Several on this page don't. *****{{Objection}} It doesn't change the fact that the majority of arguments are based on theological beliefs. *{{Argument for}} Coercion, force, and confinement that inevitably is involved with suicide prohibitions have too many negative unintended consequences. *{{Argument for}} Suicide can be seen as a form of succession or migration. Prohibiting suicide is like a nation stating that its citizens cannot leave its borders. **{{Objection}} Migration is impermanent, suicide is permanent. ***{{Objection}} Adults should have the right to make permanent decisions for themselves. Taking away such choices is patronizing and dehumanizing. ****{{Objection}} Helping adults seek help for a severe mental illness is not patronizing. Suicide is not simply a "decision," it is murder, and while it shouldn't be criminalized like regular murder, it should not be supported or promoted by the law. *****{{Objection}} Helping adults who want help or encouraging adults to seek help is not patronizing. It is the forcing of "help" which dehumanizing; forced psychiatric "help" is not help when an adult is actively declining such services. Might such "help" be what psychiatrist Dr. Thomas Szasz refereed to as cruel compassion in his book Cruel Compassion? *****{{Objection}} Sick people —especially mentally ill ones— have no right to force their illness on someone else by having children. Those opposed to suicide need to support eugenics (anti-dysgenics) at a minimum. Cf. Fredrick "hotwheels" Brennan. *****{{Objection}} Whether or not to engage in suicide is ultimately a personal decision. Suicide should not be promoted, however, it should not be prohibited through state sanctioned psychiatric coercion, force, and confinement. Voluntary/consensual psychiatry should remain legal. Nonconsensual psychiatry should be illegal. This thread is deviating from the original argument/reasoning that suicide can be seen as a form of succession or migration. *{{Argument for}} Adults should be able to do what they want with their own body. This includes suicide. Just as birth control is legal (influencing when one procreates), death control should be legal too (influencing when one dies). **{{Objection}} Comparing the legality of birth control to the legality of suicide is a false analogy, and thus a logical fallacy. Taking birth control does not inherently harm anyone, killing yourself is inherently harmful. The government should not criminalize suicide, but it should not promote it either. Those suffering from suicide ideation should be encouraged —or in extremely severe cases, required— to seek whichever method of help is the most beneficial and comfortable for the individual. ***{{Objection}} How is killing oneself own self is inherently harmful? Suicide may or not be a harmful act. Who is harmed? How are others harmed by suicide? It will still be good if we can reduce suicides to zero; we should just only use persuasion, kindness, and reason to reduce suicides, not psychiatric force, confinement, and coercion. Nonconsensual psychiatry may be traumatizing the individual who experience; it by definition, unwanted intervention. *{{Argument for}} Suicides can help to solve overcrowding problems. One suicide is one less person who won't have children. Moreover, in case the person who commits suicide is depressed, he or she will not negatively influence other people. **{{Objection}} Overcrowding is not created by an increase in population, thus, would not be solved by suicide. Overcrowding is a result of increasing population density due to perception of efficiency and preference. It can be solved through technological advancement. Earth is huge. Depression in one can increase love in another through empathy and compassion. Suicide can negatively influence other people. Whether that is positive or negative is debatable. ***{{Objection}} Earth is not huge, since people live on land, most of earth's surface is water. Further, people mostly live in cities, which increases density. Tokyo is a good example of a city I never would want to live in. True, increase in population is also due to longer lifespans, but especially people in the third world continue to have lots of children of which more and more immigrate into Western first world nations, artificially increasing the otherwise naturally decreasing populations; it also causes a lot of ethnic/religious tensions that otherwise would not exist in the West. My native Germany, which, especially in Western Germany, is way too densly populates. *{{Argument for}} Not only should suicide be ‘legal’ it should be available to adults. Counseling, therapy, and other forms of help should be encouraged. Only through these safe spaces should they be forced... maybe part of a plan before the procedure. But safe areas where one can be euthanized should be legal and available. First, this stops a lot of trauma of those ‘victims’ that find their loved ones whom have committed suicide. Second, it stops botched attempts which add even more trauma to the true individual. Third, an actual process that isn’t harmful but also creates a timeline with therapy sessions, could actually save a lot of lives. Going through the process before deciding to not go through with it can easily change someone’s entire perception. People will commit suicide. They should have that right and should be able to do it in a safe area where nobody else gets hurt. Where none of their loved ones will find them. Where they can die peacefully. ===Arguments against=== *{{Argument against}} Suicide creates an emotional toll beyond simply the person who is suffering from depression, anxiety, or another root cause of suicidal ideation. It can create feelings of survivor's remorse and intense mourning and trauma for those who are close to this individual. Any possible form of discouragement should be seen as a positive step. **{{Objection}} Emotional toll is not a logical criterion for the illegality of suicide because the origin of one's own emotion is entirely caused by and within the self (through the perceptions and relationships one has of and with self, logic and reality). **{{Objection}} The right of an adult to end their own life is more important than the feelings of other adults who were not able to use enough persuasion, reason, and kindness to successfully influence other adults to not engage in suicide. **{{Objection}} Reconsider the adjective "more important". Consider that persuasion and kindness without the logic to back it up results in well-intentioned people effectively using the narcissistic abuse cycle to effectively manipulate someone into remaining alive against their will only for the benefit of our own emotions. If you think your emotions are just too difficult to bear when you think of a loved one deciding it's time to die, imagine how emotionally overwhelmed one must be to be suicidal. If you have a level of awareness to where you can use logic to guide a suicidal person to the places in their minds where they will see their own ==Further readings== *{{Cite book |title=Suicide Prohibition: The Shame of Medicine |last=Szasz |first=Thomas}} *{{Cite book |title=Fatal Freedom: The Ethics and Politics of Suicide |last=Szasz |first=Thomas}} *{{Cite book |title=The Contribution of Natural Law Theory to Moral and Legal Debate on Suicide, Assisted Suicide and Euthanasia |last=Paterson |first=Craig}} ==See also== *[[Suicide]] ==External links== *[[Wikipedia:Suicide legislation]] ==References== {{Reflist}} [[Category:Death]] [[Category:Politics]] [[Category:Wikiversity Law Review]] [[Category:Legal ethics]] 4ryw324lb70puyua21w86zk8odr65o3 2410867 2410866 2022-08-02T02:22:48Z 172.89.36.50 /* Suicide should be legal */ wikitext text/x-wiki {{Wikidebate}} {{Law}} '''Suicide''' is the act of intentionally causing one's own death. Following the [[Should voluntary euthanasia be legal?|debate about euthanasia]], should suicide be legal as well? Do we own our own lives so completely that we have a right to end them? Or do we owe certain obligations to our communities and societies that we should stay alive even when we may want to die? Suicide is legal. It is effectively illegal. ==Suicide should be legal== ===Arguments for=== * *{{Argument for}} In a free society, we have the right to do what we want with our lives as long as we don't hurt others. Suicide can be what a rational adult chooses for himself. **{{Objection}} Suicide does hurt others—maybe not physically but the emotional and social costs of losing a loved one to suicide are huge. ***{{Objection}} No individual provides informed consent to being born, so no individual is under any obligation to remain alive for the emotional benefit of others. ***{{Objection}} Many other life events cause emotional harm but are not criminalized. For example, a son cutting off all contact with their parents for life (this can be as traumatizing as death), or bad/negligent parenting leading to children developing psychological issues like extreme attention seeking behavior (which can lead to school shootings). ***{{Objection}} Even if a person dies naturally, it will still cause emotional harm to others. *{{Argument for}} It is impossible to enforce a law against suicide. **{{Objection}} But it's possible to punish those who attempt suicide and fail. ***{{Objection}} Forcing someone to live an unwished life is equivalent to torture. ***{{Objection}} Such punishments probably serve no useful purpose. ***{{Objection}} Punishment is a dysfunctional concept altogether. With suicide, enforcement is a dysfunctional concept as well. Preventing physical harm and/or death from being perpetrated by any person to any person (including the self to the self) without the perpetrator having just logical reason to is appropriate. It's not a for the courts to prosecute as a matter requiring punitive measures that are practically completely dysfunctional even when punitive measures are appropriate. Is it logical to punish someone who is trying to give themselves the death sentence? ***{{Argument for}} Gandhi would likely have been force-fed in a prison hospital if his culture had the same attitudes toward mortality as ours during his hunger strikes. I suspect that laws against suicide are written more to prevent its use as a political tool than out of concern for our well-being or eternal souls. They can't punish someone who's dead, but they can haul an emaciated activist out of the public eye and put him in a mental hospital. Further, I see no reason why someone should suffer a protracted death while waiting to die in hospice. The state should not have title over one's life. *{{Argument for}} We do not know what happens after death. It could be better after death for one who engages in suicide. By prohibiting suicide we are imposing implicit theological beliefs on others. Prohibitions on suicide impose the implicit theological belief that death is a worse consequences than life on one who attempts to engage in suicide. This is not morally right. Those willing to engage in suicide also embrace certain implicit theological beliefs. **{{Objection}} Precisely since we don't know what happens next, we can just wait and see until later—there is no rush to die now, since anyone who would commit suicide will die of some other cause anyway. ***{{Objection}} Not feasible for someone suffering from mental anguish or any other sickness making life unbearable. **{{Objection}} The ambiguity of what happens after suicide isn't a sound argument for why suicide should be legal. If someone has made the sound logical decision that their fear of death is less emotionally painful than their fear of remaining alive with their current perception of reality, making it worse with legal complications doesn't make sense. Unless we are all explicitly required to literally bestow the accountability and authority in our lives for our lives to something external, then it is a decision that one should be allowed to make without requiring justification if it is not going to hurt anyone else against their will. Especially if the strongest reason against allowing it is OUR inability to be emotionally balanced when confronted with the reality that someone else did something they wanted to with themselves. So we literally insist at face value that, at all cost, suicide must be prevented only because of our own inability to process our own emotions about it. We are literally requiring them to remain in agony to support our codependently narcissistic ego while knowing and admitting that there is nothing we can do to help besides falsely imprison, forcibly medicate, and haphazardly attempt to counsel. But we'll feel better about ourselves, like all narcissists, when we try to force our (unwittingly) false perception of reality onto another and literally force them to bend to OUR will. But that requires understanding narcissism, and only painfully few do. ***{{Objection}} Forcing adults to wait is dehumanizing and robbing adults of personal responsibility and freedom. We should use persuasion, reason, and kindness to entice adults to remain on Earth, not coercion, force, and confinement. **{{Objection}} Most arguments against suicide don't have theological assumptions. ***{{Objection}} Arguments against suicide always rely on implicit theological assumptions. ****{{Objection}} Several on this page don't. *****{{Objection}} It doesn't change the fact that the majority of arguments are based on theological beliefs. *{{Argument for}} Coercion, force, and confinement that inevitably is involved with suicide prohibitions have too many negative unintended consequences. *{{Argument for}} Suicide can be seen as a form of succession or migration. Prohibiting suicide is like a nation stating that its citizens cannot leave its borders. **{{Objection}} Migration is impermanent, suicide is permanent. ***{{Objection}} Adults should have the right to make permanent decisions for themselves. Taking away such choices is patronizing and dehumanizing. ****{{Objection}} Helping adults seek help for a severe mental illness is not patronizing. Suicide is not simply a "decision," it is murder, and while it shouldn't be criminalized like regular murder, it should not be supported or promoted by the law. *****{{Objection}} Helping adults who want help or encouraging adults to seek help is not patronizing. It is the forcing of "help" which dehumanizing; forced psychiatric "help" is not help when an adult is actively declining such services. Might such "help" be what psychiatrist Dr. Thomas Szasz refereed to as cruel compassion in his book Cruel Compassion? *****{{Objection}} Sick people —especially mentally ill ones— have no right to force their illness on someone else by having children. Those opposed to suicide need to support eugenics (anti-dysgenics) at a minimum. Cf. Fredrick "hotwheels" Brennan. *****{{Objection}} Whether or not to engage in suicide is ultimately a personal decision. Suicide should not be promoted, however, it should not be prohibited through state sanctioned psychiatric coercion, force, and confinement. Voluntary/consensual psychiatry should remain legal. Nonconsensual psychiatry should be illegal. This thread is deviating from the original argument/reasoning that suicide can be seen as a form of succession or migration. *{{Argument for}} Adults should be able to do what they want with their own body. This includes suicide. Just as birth control is legal (influencing when one procreates), death control should be legal too (influencing when one dies). **{{Objection}} Comparing the legality of birth control to the legality of suicide is a false analogy, and thus a logical fallacy. Taking birth control does not inherently harm anyone, killing yourself is inherently harmful. The government should not criminalize suicide, but it should not promote it either. Those suffering from suicide ideation should be encouraged —or in extremely severe cases, required— to seek whichever method of help is the most beneficial and comfortable for the individual. ***{{Objection}} How is killing oneself own self is inherently harmful? Suicide may or not be a harmful act. Who is harmed? How are others harmed by suicide? It will still be good if we can reduce suicides to zero; we should just only use persuasion, kindness, and reason to reduce suicides, not psychiatric force, confinement, and coercion. Nonconsensual psychiatry may be traumatizing the individual who experience; it by definition, unwanted intervention. *{{Argument for}} Suicides can help to solve overcrowding problems. One suicide is one less person who won't have children. Moreover, in case the person who commits suicide is depressed, he or she will not negatively influence other people. **{{Objection}} Overcrowding is not created by an increase in population, thus, would not be solved by suicide. Overcrowding is a result of increasing population density due to perception of efficiency and preference. It can be solved through technological advancement. Earth is huge. Depression in one can increase love in another through empathy and compassion. Suicide can negatively influence other people. Whether that is positive or negative is debatable. ***{{Objection}} Earth is not huge, since people live on land, most of earth's surface is water. Further, people mostly live in cities, which increases density. Tokyo is a good example of a city I never would want to live in. True, increase in population is also due to longer lifespans, but especially people in the third world continue to have lots of children of which more and more immigrate into Western first world nations, artificially increasing the otherwise naturally decreasing populations; it also causes a lot of ethnic/religious tensions that otherwise would not exist in the West. My native Germany, which, especially in Western Germany, is way too densly populates. *{{Argument for}} Not only should suicide be ‘legal’ it should be available to adults. Counseling, therapy, and other forms of help should be encouraged. Only through these safe spaces should they be forced... maybe part of a plan before the procedure. But safe areas where one can be euthanized should be legal and available. First, this stops a lot of trauma of those ‘victims’ that find their loved ones whom have committed suicide. Second, it stops botched attempts which add even more trauma to the true individual. Third, an actual process that isn’t harmful but also creates a timeline with therapy sessions, could actually save a lot of lives. Going through the process before deciding to not go through with it can easily change someone’s entire perception. People will commit suicide. They should have that right and should be able to do it in a safe area where nobody else gets hurt. Where none of their loved ones will find them. Where they can die peacefully. ===Arguments against=== *{{Argument against}} Suicide creates an emotional toll beyond simply the person who is suffering from depression, anxiety, or another root cause of suicidal ideation. It can create feelings of survivor's remorse and intense mourning and trauma for those who are close to this individual. Any possible form of discouragement should be seen as a positive step. **{{Objection}} Emotional toll is not a logical criterion for the illegality of suicide because the origin of one's own emotion is entirely caused by and within the self (through the perceptions and relationships one has of and with self, logic and reality). **{{Objection}} The right of an adult to end their own life is more important than the feelings of other adults who were not able to use enough persuasion, reason, and kindness to successfully influence other adults to not engage in suicide. **{{Objection}} Reconsider the adjective "more important". Consider that persuasion and kindness without the logic to back it up results in well-intentioned people effectively using the narcissistic abuse cycle to effectively manipulate someone into remaining alive against their will only for the benefit of our own emotions. If you think your emotions are just too difficult to bear when you think of a loved one deciding it's time to die, imagine how emotionally overwhelmed one must be to be suicidal. If you have a level of awareness to where you can use logic to guide a suicidal person to the places in their minds where they will see their own Abortion rights are similar. ==Further readings== *{{Cite book |title=Suicide Prohibition: The Shame of Medicine |last=Szasz |first=Thomas}} *{{Cite book |title=Fatal Freedom: The Ethics and Politics of Suicide |last=Szasz |first=Thomas}} *{{Cite book |title=The Contribution of Natural Law Theory to Moral and Legal Debate on Suicide, Assisted Suicide and Euthanasia |last=Paterson |first=Craig}} ==See also== *[[Suicide]] ==External links== *[[Wikipedia:Suicide legislation]] ==References== {{Reflist}} [[Category:Death]] [[Category:Politics]] [[Category:Wikiversity Law Review]] [[Category:Legal ethics]] 8t5te8908ml886wact0nao899hfh5ah 2410868 2410867 2022-08-02T02:23:20Z 172.89.36.50 /* Suicide should be legal */ wikitext text/x-wiki {{Wikidebate}} {{Law}} '''Suicide''' is the act of intentionally causing one's own death. Following the [[Should voluntary euthanasia be legal?|debate about euthanasia]], should suicide be legal as well? Do we own our own lives so completely that we have a right to end them? Or do we owe certain obligations to our communities and societies that we should stay alive even when we may want to die? Suicide is legal. It is effectively illegal. ==Suicide should be legal== ===Arguments for=== * *{{Argument for}} In a free society, we have the right to do what we want with our lives as long as we don't hurt others. Suicide can be what a rational adult chooses for himself. **{{Objection}} Suicide does hurt others—maybe not physically but the emotional and social costs of losing a loved one to suicide are huge. ***{{Objection}} No individual provides informed consent to being born, so no individual is under any obligation to remain alive for the emotional benefit of others. ***{{Objection}} Many other life events cause emotional harm but are not criminalized. For example, a son cutting off all contact with their parents for life (this can be as traumatizing as death), or bad/negligent parenting leading to children developing psychological issues like extreme attention seeking behavior (which can lead to school shootings). ***{{Objection}} Even if a person dies naturally, it will still cause emotional harm to others. *{{Argument for}} It is impossible to enforce a law against suicide. **{{Objection}} But it's possible to punish those who attempt suicide and fail. ***{{Objection}} Forcing someone to live an unwished life is equivalent to torture. ***{{Objection}} Such punishments probably serve no useful purpose. ***{{Objection}} Punishment is a dysfunctional concept altogether. With suicide, enforcement is a dysfunctional concept as well. Preventing physical harm and/or death from being perpetrated by any person to any person (including the self to the self) without the perpetrator having just logical reason to is appropriate. It's not a for the courts to prosecute as a matter requiring punitive measures that are practically completely dysfunctional even when punitive measures are appropriate. Is it logical to punish someone who is trying to give themselves the death sentence? ***{{Argument for}} Gandhi would likely have been force-fed in a prison hospital if his culture had the same attitudes toward mortality as ours during his hunger strikes. I suspect that laws against suicide are written more to prevent its use as a political tool than out of concern for our well-being or eternal souls. They can't punish someone who's dead, but they can haul an emaciated activist out of the public eye and put him in a mental hospital. Further, I see no reason why someone should suffer a protracted death while waiting to die in hospice. The state should not have title over one's life. *{{Argument for}} We do not know what happens after death. It could be better after death for one who engages in suicide. By prohibiting suicide we are imposing implicit theological beliefs on others. Prohibitions on suicide impose the implicit theological belief that death is a worse consequences than life on one who attempts to engage in suicide. This is not morally right. Those willing to engage in suicide also embrace certain implicit theological beliefs. **{{Objection}} Precisely since we don't know what happens next, we can just wait and see until later—there is no rush to die now, since anyone who would commit suicide will die of some other cause anyway. ***{{Objection}} Not feasible for someone suffering from mental anguish or any other sickness making life unbearable. **{{Objection}} The ambiguity of what happens after suicide isn't a sound argument for why suicide should be legal. If someone has made the sound logical decision that their fear of death is less emotionally painful than their fear of remaining alive with their current perception of reality, making it worse with legal complications doesn't make sense. Unless we are all explicitly required to literally bestow the accountability and authority in our lives for our lives to something external, then it is a decision that one should be allowed to make without requiring justification if it is not going to hurt anyone else against their will. Especially if the strongest reason against allowing it is OUR inability to be emotionally balanced when confronted with the reality that someone else did something they wanted to with themselves. So we literally insist at face value that, at all cost, suicide must be prevented only because of our own inability to process our own emotions about it. We are literally requiring them to remain in agony to support our codependently narcissistic ego while knowing and admitting that there is nothing we can do to help besides falsely imprison, forcibly medicate, and haphazardly attempt to counsel. But we'll feel better about ourselves, like all narcissists, when we try to force our (unwittingly) false perception of reality onto another and literally force them to bend to OUR will. But that requires understanding narcissism, and only painfully few do. ***{{Objection}} Forcing adults to wait is dehumanizing and robbing adults of personal responsibility and freedom. We should use persuasion, reason, and kindness to entice adults to remain on Earth, not coercion, force, and confinement. **{{Objection}} Most arguments against suicide don't have theological assumptions. ***{{Objection}} Arguments against suicide always rely on implicit theological assumptions. ****{{Objection}} Several on this page don't. *****{{Objection}} It doesn't change the fact that the majority of arguments are based on theological beliefs. *{{Argument for}} Coercion, force, and confinement that inevitably is involved with suicide prohibitions have too many negative unintended consequences. *{{Argument for}} Suicide can be seen as a form of succession or migration. Prohibiting suicide is like a nation stating that its citizens cannot leave its borders. **{{Objection}} Migration is impermanent, suicide is permanent. ***{{Objection}} Adults should have the right to make permanent decisions for themselves. Taking away such choices is patronizing and dehumanizing. ****{{Objection}} Helping adults seek help for a severe mental illness is not patronizing. Suicide is not simply a "decision," it is murder, and while it shouldn't be criminalized like regular murder, it should not be supported or promoted by the law. *****{{Objection}} Helping adults who want help or encouraging adults to seek help is not patronizing. It is the forcing of "help" which dehumanizing; forced psychiatric "help" is not help when an adult is actively declining such services. Might such "help" be what psychiatrist Dr. Thomas Szasz refereed to as cruel compassion in his book Cruel Compassion? *****{{Objection}} Sick people —especially mentally ill ones— have no right to force their illness on someone else by having children. Those opposed to suicide need to support eugenics (anti-dysgenics) at a minimum. Cf. Fredrick "hotwheels" Brennan. *****{{Objection}} Whether or not to engage in suicide is ultimately a personal decision. Suicide should not be promoted, however, it should not be prohibited through state sanctioned psychiatric coercion, force, and confinement. Voluntary/consensual psychiatry should remain legal. Nonconsensual psychiatry should be illegal. This thread is deviating from the original argument/reasoning that suicide can be seen as a form of succession or migration. *{{Argument for}} Adults should be able to do what they want with their own body. This includes suicide. Just as birth control is legal (influencing when one procreates), death control should be legal too (influencing when one dies). **{{Objection}} Comparing the legality of birth control to the legality of suicide is a false analogy, and thus a logical fallacy. Taking birth control does not inherently harm anyone, killing yourself is inherently harmful. The government should not criminalize suicide, but it should not promote it either. Those suffering from suicide ideation should be encouraged —or in extremely severe cases, required— to seek whichever method of help is the most beneficial and comfortable for the individual. ***{{Objection}} How is killing oneself own self is inherently harmful? Suicide may or not be a harmful act. Who is harmed? How are others harmed by suicide? It will still be good if we can reduce suicides to zero; we should just only use persuasion, kindness, and reason to reduce suicides, not psychiatric force, confinement, and coercion. Nonconsensual psychiatry may be traumatizing the individual who experience; it by definition, unwanted intervention. *{{Argument for}} Suicides can help to solve overcrowding problems. One suicide is one less person who won't have children. Moreover, in case the person who commits suicide is depressed, he or she will not negatively influence other people. **{{Objection}} Overcrowding is not created by an increase in population, thus, would not be solved by suicide. Overcrowding is a result of increasing population density due to perception of efficiency and preference. It can be solved through technological advancement. Earth is huge. Depression in one can increase love in another through empathy and compassion. Suicide can negatively influence other people. Whether that is positive or negative is debatable. ***{{Objection}} Earth is not huge, since people live on land, most of earth's surface is water. Further, people mostly live in cities, which increases density. Tokyo is a good example of a city I never would want to live in. True, increase in population is also due to longer lifespans, but especially people in the third world continue to have lots of children of which more and more immigrate into Western first world nations, artificially increasing the otherwise naturally decreasing populations; it also causes a lot of ethnic/religious tensions that otherwise would not exist in the West. My native Germany, which, especially in Western Germany, is way too densly populates. *{{Argument for}} Not only should suicide be ‘legal’ it should be available to adults. Counseling, therapy, and other forms of help should be encouraged. Only through these safe spaces should they be forced... maybe part of a plan before the procedure. But safe areas where one can be euthanized should be legal and available. First, this stops a lot of trauma of those ‘victims’ that find their loved ones whom have committed suicide. Second, it stops botched attempts which add even more trauma to the true individual. Third, an actual process that isn’t harmful but also creates a timeline with therapy sessions, could actually save a lot of lives. Going through the process before deciding to not go through with it can easily change someone’s entire perception. People will commit suicide. They should have that right and should be able to do it in a safe area where nobody else gets hurt. Where none of their loved ones will find them. Where they can die peacefully. ===Arguments against=== *{{Argument against}} Suicide creates an emotional toll beyond simply the person who is suffering from depression, anxiety, or another root cause of suicidal ideation. It can create feelings of survivor's remorse and intense mourning and trauma for those who are close to this individual. Any possible form of discouragement should be seen as a positive step. **{{Objection}} Emotional toll is not a logical criterion for the illegality of suicide because the origin of one's own emotion is entirely caused by and within the self (through the perceptions and relationships one has of and with self, logic and reality). **{{Objection}} The right of an adult to end their own life is more important than the feelings of other adults who were not able to use enough persuasion, reason, and kindness to successfully influence other adults to not engage in suicide. **{{Objection}} Reconsider the adjective "more important". Consider that persuasion and kindness without the logic to back it up results in well-intentioned people effectively using the narcissistic abuse cycle to effectively manipulate someone into remaining alive against their will only for the benefit of our own emotions. If you think your emotions are just too difficult to bear when you think of a loved one deciding it's time to die, imagine how emotionally overwhelmed one must be to be suicidal. If you have a level of awareness to where you can use logic to guide a suicidal person to the places in their minds where they will see their o ==Further readings== *{{Cite book |title=Suicide Prohibition: The Shame of Medicine |last=Szasz |first=Thomas}} *{{Cite book |title=Fatal Freedom: The Ethics and Politics of Suicide |last=Szasz |first=Thomas}} *{{Cite book |title=The Contribution of Natural Law Theory to Moral and Legal Debate on Suicide, Assisted Suicide and Euthanasia |last=Paterson |first=Craig}} ==See also== *[[Suicide]] ==External links== *[[Wikipedia:Suicide legislation]] ==References== {{Reflist}} [[Category:Death]] [[Category:Politics]] [[Category:Wikiversity Law Review]] [[Category:Legal ethics]] iq0cqnxj15kvoztinlgqggvhqjea72a 2410869 2410868 2022-08-02T02:24:46Z 172.89.36.50 /* Suicide should be legal */ wikitext text/x-wiki {{Wikidebate}} {{Law}} '''Suicide''' is the act of intentionally causing one's own death. Following the [[Should voluntary euthanasia be legal?|debate about euthanasia]], should suicide be legal as well? Do we own our own lives so completely that we have a right to end them? Or do we owe certain obligations to our communities and societies that we should stay alive even when we may want to die? Suicide is legal. It is effectively illegal. ==Suicide should be legal== ===Arguments for=== * *{{Argument for}} In a free society, we have the right to do what we want with our lives as long as we don't hurt others. Suicide can be what a rational adult chooses for himself. **{{Objection}} Suicide does hurt others—maybe not physically but the emotional and social costs of losing a loved one to suicide are huge. ***{{Objection}} No individual provides informed consent to being born, so no individual is under any obligation to remain alive for the emotional benefit of others. ***{{Objection}} Many other life events cause emotional harm but are not criminalized. For example, a son cutting off all contact with their parents for life (this can be as traumatizing as death), or bad/negligent parenting leading to children developing psychological issues like extreme attention seeking behavior (which can lead to school shootings). ***{{Objection}} Even if a person dies naturally, it will still cause emotional harm to others. *{{Argument for}} It is impossible to enforce a law against suicide. **{{Objection}} But it's possible to punish those who attempt suicide and fail. ***{{Objection}} Forcing someone to live an unwished life is equivalent to torture. ***{{Objection}} Such punishments probably serve no useful purpose. ***{{Objection}} Punishment is a dysfunctional concept altogether. With suicide, enforcement is a dysfunctional concept as well. Preventing physical harm and/or death from being perpetrated by any person to any person (including the self to the self) without the perpetrator having just logical reason to is appropriate. It's not a for the courts to prosecute as a matter requiring punitive measures that are practically completely dysfunctional even when punitive measures are appropriate. Is it logical to punish someone who is trying to give themselves the death sentence? ***{{Argument for}} Gandhi would likely have been force-fed in a prison hospital if his culture had the same attitudes toward mortality as ours during his hunger strikes. I suspect that laws against suicide are written more to prevent its use as a political tool than out of concern for our well-being or eternal souls. They can't punish someone who's dead, but they can haul an emaciated activist out of the public eye and put him in a mental hospital. Further, I see no reason why someone should suffer a protracted death while waiting to die in hospice. The state should not have title over one's life. *{{Argument for}} We do not know what happens after death. It could be better after death for one who engages in suicide. By prohibiting suicide we are imposing implicit theological beliefs on others. Prohibitions on suicide impose the implicit theological belief that death is a worse consequences than life on one who attempts to engage in suicide. This is not morally right. Those willing to engage in suicide also embrace certain implicit theological beliefs. **{{Objection}} Precisely since we don't know what happens next, we can just wait and see until later—there is no rush to die now, since anyone who would commit suicide will die of some other cause anyway. ***{{Objection}} Not feasible for someone suffering from mental anguish or any other sickness making life unbearable. **{{Objection}} The ambiguity of what happens after suicide isn't a sound argument for why suicide should be legal. If someone has made the sound logical decision that their fear of death is less emotionally painful than their fear of remaining alive with their current perception of reality, making it worse with legal complications doesn't make sense. Unless we are all explicitly required to literally bestow the accountability and authority in our lives for our lives to something external, then it is a decision that one should be allowed to make without requiring justification if it is not going to hurt anyone else against their will. Especially if the strongest reason against allowing it is OUR inability to be emotionally balanced when confronted with the reality that someone else did something they wanted to with themselves. So we literally insist at face value that, at all cost, suicide must be prevented only because of our own inability to process our own emotions about it. We are literally requiring them to remain in agony to support our codependently narcissistic ego while knowing and admitting that there is nothing we can do to help besides falsely imprison, forcibly medicate, and haphazardly attempt to counsel. But we'll feel better about ourselves, like all narcissists, when we try to force our (unwittingly) false perception of reality onto another and literally force them to bend to OUR will. But that requires understanding narcissism, and only painfully few do. ***{{Objection}} Forcing adults to wait is dehumanizing and robbing adults of personal responsibility and freedom. We should use persuasion, reason, and kindness to entice adults to remain on Earth, not coercion, force, and confinement. **{{Objection}} Most arguments against suicide don't have theological assumptions. ***{{Objection}} Arguments against suicide always rely on implicit theological assumptions. ****{{Objection}} Several on this page don't. *****{{Objection}} It doesn't change the fact that the majority of arguments are based on theological beliefs. *{{Argument for}} Coercion, force, and confinement that inevitably is involved with suicide prohibitions have too many negative unintended consequences. *{{Argument for}} Suicide can be seen as a form of succession or migration. Prohibiting suicide is like a nation stating that its citizens cannot leave its borders. **{{Objection}} Migration is impermanent, suicide is permanent. ***{{Objection}} Adults should have the right to make permanent decisions for themselves. Taking away such choices is patronizing and dehumanizing. ****{{Objection}} Helping adults seek help for a severe mental illness is not patronizing. Suicide is not simply a "decision," it is murder, and while it shouldn't be criminalized like regular murder, it should not be supported or promoted by the law. *****{{Objection}} Helping adults who want help or encouraging adults to seek help is not patronizing. It is the forcing of "help" which dehumanizing; forced psychiatric "help" is not help when an adult is actively declining such services. Might such "help" be what psychiatrist Dr. Thomas Szasz refereed to as cruel compassion in his book Cruel Compassion? *****{{Objection}} Sick people —especially mentally ill ones— have no right to force their illness on someone else by having children. Those opposed to suicide need to support eugenics (anti-dysgenics) at a minimum. Cf. Fredrick "hotwheels" Brennan. *****{{Objection}} Whether or not to engage in suicide is ultimately a personal decision. Suicide should not be promoted, however, it should not be prohibited through state sanctioned psychiatric coercion, force, and confinement. Voluntary/consensual psychiatry should remain legal. Nonconsensual psychiatry should be illegal. This thread is deviating from the original argument/reasoning that suicide can be seen as a form of succession or migration. *{{Argument for}} Adults should be able to do what they want with their own body. This includes suicide. Just as birth control is legal (influencing when one procreates), death control should be legal too (influencing when one dies). **{{Objection}} Comparing the legality of birth control to the legality of suicide is a false analogy, and thus a logical fallacy. Taking birth control does not inherently harm anyone, killing yourself is inherently harmful. The government should not criminalize suicide, but it should not promote it either. Those suffering from suicide ideation should be encouraged —or in extremely severe cases, required— to seek whichever method of help is the most beneficial and comfortable for the individual. ***{{Objection}} How is killing oneself own self is inherently harmful? Suicide may or not be a harmful act. Who is harmed? How are others harmed by suicide? It will still be good if we can reduce suicides to zero; we should just only use persuasion, kindness, and reason to reduce suicides, not psychiatric force, confinement, and coercion. Nonconsensual psychiatry may be traumatizing the individual who experience; it by definition, unwanted intervention. *{{Argument for}} Suicides can help to solve overcrowding problems. One suicide is one less person who won't have children. Moreover, in case the person who commits suicide is depressed, he or she will not negatively influence other people. **{{Objection}} Overcrowding is not created by an increase in population, thus, would not be solved by suicide. Overcrowding is a result of increasing population density due to perception of efficiency and preference. It can be solved through technological advancement. Earth is huge. Depression in one can increase love in another through empathy and compassion. Suicide can negatively influence other people. Whether that is positive or negative is debatable. ***{{Objection}} Earth is not huge, since people live on land, most of earth's surface is water. Further, people mostly live in cities, which increases density. Tokyo is a good example of a city I never would want to live in. True, increase in population is also due to longer lifespans, but especially people in the third world continue to have lots of children of which more and more immigrate into Western first world nations, artificially increasing the otherwise naturally decreasing populations; it also causes a lot of ethnic/religious tensions that otherwise would not exist in the West. My native Germany, which, especially in Western Germany, is way too densly populates. *{{Argument for}} Not only should suicide be ‘legal’ it should be available to adults. Counseling, therapy, and other forms of help should be encouraged. Only through these safe spaces should they be forced... maybe part of a plan before the procedure. But safe areas where one can be euthanized should be legal and available. First, this stops a lot of trauma of those ‘victims’ that find their loved ones whom have committed suicide. Second, it stops botched attempts which add even more trauma to the true individual. Third, an actual process that isn’t harmful but also creates a timeline with therapy sessions, could actually save a lot of lives. Going through the process before deciding to not go through with it can easily change someone’s entire perception. People will commit suicide. They should have that right and should be able to do it in a safe area where nobody else gets hurt. Where none of their loved ones will find them. Where they can die peacefully. ===Arguments against=== *{{Argument against}} Suicide creates an emotional toll beyond simply the person who is suffering from depression, anxiety, or another root cause of suicidal ideation. It can create feelings of survivor's remorse and intense mourning and trauma for those who are close to this individual. Any possible form of discouragement should be seen as a positive step. **{{Objection}} Emotional toll is not a logical criterion for the illegality of suicide because the origin of one's own emotion is entirely caused by and within the self (through the perceptions and relationships one has of and with self, logic and reality). **{{Objection}} The right of an adult to end their own life is more important than the feelings of other adults who were not able to use enough persuasion, reason, and kindness to successfully influence other adults to not engage in suicide. **{{Objection}} Reconsider the adjective "more important". Consider that persuasion and kindness without the logic to back it up results in well-intentioned people effectively using the narcissistic abuse cycle to effectively manipulate someone into remaining alive against their will only for the benefit of our own emotions. If you think your emotions are just too difficult to bear when you think of a loved one deciding it's time to die, imagine how emotionally overwhelmed one must be to be suicidal. If you have a level of awareness to where you can use logic to guide a suicidal person to the places in their minds where they will see their own. ==Further readings== *{{Cite book |title=Suicide Prohibition: The Shame of Medicine |last=Szasz |first=Thomas}} *{{Cite book |title=Fatal Freedom: The Ethics and Politics of Suicide |last=Szasz |first=Thomas}} *{{Cite book |title=The Contribution of Natural Law Theory to Moral and Legal Debate on Suicide, Assisted Suicide and Euthanasia |last=Paterson |first=Craig}} ==See also== *[[Suicide]] ==External links== *[[Wikipedia:Suicide legislation]] ==References== {{Reflist}} [[Category:Death]] [[Category:Politics]] [[Category:Wikiversity Law Review]] [[Category:Legal ethics]] p1s1p0yk0ysunam07yqdg7iuxxv1b45 2410870 2410869 2022-08-02T02:27:18Z Koavf 147 Reverted edits by [[Special:Contributions/172.89.36.50|172.89.36.50]] ([[User_talk:172.89.36.50|talk]]) to last version by [[User:Sophivorus|Sophivorus]] using [[Wikiversity:Rollback|rollback]] wikitext text/x-wiki {{Wikidebate}} {{Law}} '''Suicide''' is the act of intentionally causing one's own death. Following the [[Should voluntary euthanasia be legal?|debate about euthanasia]], should suicide be legal as well? Do we own our own lives so completely that we have a right to end them? Or do we owe certain obligations to our communities and societies that we should stay alive even when we may want to die? Suicide is legal. It is effectively illegal. ==Suicide should be legal== ===Arguments for=== * *{{Argument for}} In a free society, we have the right to do what we want with our lives as long as we don't hurt others. Suicide can be what a rational adult chooses for himself. **{{Objection}} Suicide does hurt others—maybe not physically but the emotional and social costs of losing a loved one to suicide are huge. ***{{Objection}} No individual provides informed consent to being born, so no individual is under any obligation to remain alive for the emotional benefit of others. ***{{Objection}} Many other life events cause emotional harm but are not criminalized. For example, a son cutting off all contact with their parents for life (this can be as traumatizing as death), or bad/negligent parenting leading to children developing psychological issues like extreme attention seeking behavior (which can lead to school shootings). ***{{Objection}} Even if a person dies naturally, it will still cause emotional harm to others. *{{Argument for}} It is impossible to enforce a law against suicide. **{{Objection}} But it's possible to punish those who attempt suicide and fail. ***{{Objection}} Forcing someone to live an unwished life is equivalent to torture. ***{{Objection}} Such punishments probably serve no useful purpose. ***{{Objection}} Punishment is a dysfunctional concept altogether. With suicide, enforcement is a dysfunctional concept as well. Preventing physical harm and/or death from being perpetrated by any person to any person (including the self to the self) without the perpetrator having just logical reason to is appropriate. It's not a for the courts to prosecute as a matter requiring punitive measures that are practically completely dysfunctional even when punitive measures are appropriate. Is it logical to punish someone who is trying to give themselves the death sentence? ***{{Argument for}} Gandhi would likely have been force-fed in a prison hospital if his culture had the same attitudes toward mortality as ours during his hunger strikes. I suspect that laws against suicide are written more to prevent its use as a political tool than out of concern for our well-being or eternal souls. They can't punish someone who's dead, but they can haul an emaciated activist out of the public eye and put him in a mental hospital. Further, I see no reason why someone should suffer a protracted death while waiting to die in hospice. The state should not have title over one's life. *{{Argument for}} We do not know what happens after death. It could be better after death for one who engages in suicide. By prohibiting suicide we are imposing implicit theological beliefs on others. Prohibitions on suicide impose the implicit theological belief that death is a worse consequences than life on one who attempts to engage in suicide. This is not morally right. Those willing to engage in suicide also embrace certain implicit theological beliefs. **{{Objection}} Precisely since we don't know what happens next, we can just wait and see until later—there is no rush to die now, since anyone who would commit suicide will die of some other cause anyway. ***{{Objection}} Not feasible for someone suffering from mental anguish or any other sickness making life unbearable. **{{Objection}} The ambiguity of what happens after suicide isn't a sound argument for why suicide should be legal. If someone has made the sound logical decision that their fear of death is less emotionally painful than their fear of remaining alive with their current perception of reality, making it worse with legal complications doesn't make sense. Unless we are all explicitly required to literally bestow the accountability and authority in our lives for our lives to something external, then it is a decision that one should be allowed to make without requiring justification if it is not going to hurt anyone else against their will. Especially if the strongest reason against allowing it is OUR inability to be emotionally balanced when confronted with the reality that someone else did something they wanted to with themselves. So we literally insist at face value that, at all cost, suicide must be prevented only because of our own inability to process our own emotions about it. We are literally requiring them to remain in agony to support our codependently narcissistic ego while knowing and admitting that there is nothing we can do to help besides falsely imprison, forcibly medicate, and haphazardly attempt to counsel. But we'll feel better about ourselves, like all narcissists, when we try to force our (unwittingly) false perception of reality onto another and literally force them to bend to OUR will. But that requires understanding narcissism, and only painfully few do. ***{{Objection}} Forcing adults to wait is dehumanizing and robbing adults of personal responsibility and freedom. We should use persuasion, reason, and kindness to entice adults to remain on Earth, not coercion, force, and confinement. **{{Objection}} Most arguments against suicide don't have theological assumptions. ***{{Objection}} Arguments against suicide always rely on implicit theological assumptions. ****{{Objection}} Several on this page don't. *****{{Objection}} It doesn't change the fact that the majority of arguments are based on theological beliefs. *{{Argument for}} Coercion, force, and confinement that inevitably is involved with suicide prohibitions have too many negative unintended consequences. *{{Argument for}} Suicide can be seen as a form of succession or migration. Prohibiting suicide is like a nation stating that its citizens cannot leave its borders. **{{Objection}} Migration is impermanent, suicide is permanent. ***{{Objection}} Adults should have the right to make permanent decisions for themselves. Taking away such choices is patronizing and dehumanizing. ****{{Objection}} Helping adults seek help for a severe mental illness is not patronizing. Suicide is not simply a "decision," it is murder, and while it shouldn't be criminalized like regular murder, it should not be supported or promoted by the law. *****{{Objection}} Helping adults who want help or encouraging adults to seek help is not patronizing. It is the forcing of "help" which dehumanizing; forced psychiatric "help" is not help when an adult is actively declining such services. Might such "help" be what psychiatrist Dr. Thomas Szasz refereed to as cruel compassion in his book Cruel Compassion? *****{{Objection}} Sick people —especially mentally ill ones— have no right to force their illness on someone else by having children. Those opposed to suicide need to support eugenics (anti-dysgenics) at a minimum. Cf. Fredrick "hotwheels" Brennan. *****{{Objection}} Whether or not to engage in suicide is ultimately a personal decision. Suicide should not be promoted, however, it should not be prohibited through state sanctioned psychiatric coercion, force, and confinement. Voluntary/consensual psychiatry should remain legal. Nonconsensual psychiatry should be illegal. This thread is deviating from the original argument/reasoning that suicide can be seen as a form of succession or migration. *{{Argument for}} Adults should be able to do what they want with their own body. This includes suicide. Just as birth control is legal (influencing when one procreates), death control should be legal too (influencing when one dies). **{{Objection}} Comparing the legality of birth control to the legality of suicide is a false analogy, and thus a logical fallacy. Taking birth control does not inherently harm anyone, killing yourself is inherently harmful. The government should not criminalize suicide, but it should not promote it either. Those suffering from suicide ideation should be encouraged —or in extremely severe cases, required— to seek whichever method of help is the most beneficial and comfortable for the individual. ***{{Objection}} How is killing oneself own self is inherently harmful? Suicide may or not be a harmful act. Who is harmed? How are others harmed by suicide? It will still be good if we can reduce suicides to zero; we should just only use persuasion, kindness, and reason to reduce suicides, not psychiatric force, confinement, and coercion. Nonconsensual psychiatry may be traumatizing the individual who experience; it by definition, unwanted intervention. *{{Argument for}} Suicides can help to solve overcrowding problems. One suicide is one less person who won't have children. Moreover, in case the person who commits suicide is depressed, he or she will not negatively influence other people. **{{Objection}} Overcrowding is not created by an increase in population, thus, would not be solved by suicide. Overcrowding is a result of increasing population density due to perception of efficiency and preference. It can be solved through technological advancement. Earth is huge. Depression in one can increase love in another through empathy and compassion. Suicide can negatively influence other people. Whether that is positive or negative is debatable. ***{{Objection}} Earth is not huge, since people live on land, most of earth's surface is water. Further, people mostly live in cities, which increases density. Tokyo is a good example of a city I never would want to live in. True, increase in population is also due to longer lifespans, but especially people in the third world continue to have lots of children of which more and more immigrate into Western first world nations, artificially increasing the otherwise naturally decreasing populations; it also causes a lot of ethnic/religious tensions that otherwise would not exist in the West. My native Germany, which, especially in Western Germany, is way too densly populates. *{{Argument for}} Not only should suicide be ‘legal’ it should be available to adults. Counseling, therapy, and other forms of help should be encouraged. Only through these safe spaces should they be forced... maybe part of a plan before the procedure. But safe areas where one can be euthanized should be legal and available. First, this stops a lot of trauma of those ‘victims’ that find their loved ones whom have committed suicide. Second, it stops botched attempts which add even more trauma to the true individual. Third, an actual process that isn’t harmful but also creates a timeline with therapy sessions, could actually save a lot of lives. Going through the process before deciding to not go through with it can easily change someone’s entire perception. People will commit suicide. They should have that right and should be able to do it in a safe area where nobody else gets hurt. Where none of their loved ones will find them. Where they can die peacefully. ===Arguments against=== *{{Argument against}} Suicide creates an emotional toll beyond simply the person who is suffering from depression, anxiety, or another root cause of suicidal ideation. It can create feelings of survivor's remorse and intense mourning and trauma for those who are close to this individual. Any possible form of discouragement should be seen as a positive step. **{{Objection}} Emotional toll is not a logical criterion for the illegality of suicide because the origin of one's own emotion is entirely caused by and within the self (through the perceptions and relationships one has of and with self, logic and reality). **{{Objection}} The right of an adult to end their own life is more important than the feelings of other adults who were not able to use enough persuasion, reason, and kindness to successfully influence other adults to not engage in suicide. **{{Objection}} Reconsider the adjective "more important". Consider that persuasion and kindness without the logic to back it up results in well-intentioned people effectively using the narcissistic abuse cycle to effectively manipulate someone into remaining alive against their will only for the benefit of our own emotions. If you think your emotions are just too difficult to bear when you think of a loved one deciding it's time to die, imagine how emotionally overwhelmed one must be to be suicidal. If you have a level of awareness to where you can use logic to guide a suicidal person to the places in their minds where they will see their own logical fallacies, that is the best therapy money shouldn't have to buy. If the suicidal person is logically ready to commit suicide you won't likely be counseling them. If you are counseling them, it is because they are unsure that they are ready. We, caring compassionate loving others, would be more helpful if we were blatantly honest. If you don't know why they shouldn't, say so. If you can't bear the thought of losing them, say so. Be the open honest example you'd expect them to be if you decided it was necessary to force them to do something against their will. **{{Objection}} Suicide creates a self-inflicted death. Just as one's emotional tolls are their own to interpret and act upon, one's emotions are not directly caused by anything outside the self, but by observing something which appears to be outside the self and perceiving it to conflict with one's beliefs or expectations. The emotions in the conflicted one stem from the conflicts per se and not the external catalyst which triggered cognitive dissonance. While we undoubtedly rely on others to at least be alive for support, requiring them to be for our own selfish emotional needs is immoral. *{{Argument against}} Suicide is usually a permanent solution to a temporary problem. Since most suicides are a product of mental illness and mental illnesses can be transitory, treated, or even cured, then we should discourage someone from killing themselves as a way of addressing mental illness. **{{Objection}} Mental illnesses are a social construct and are not real illnesses. They are metaphorical illnesses. The idea of mental illness is often used as a method of labeling and dehumanizing an individual. It is a method of infantilizing and invalidating an adult's decision. There is no objective means of medical diagnosis for any mental illness in the ''{{W|Diagnostic and Statistical Manual of Mental Disorders}}''. They are diagnosed using verbal interviews and subjective judgement. ***{{Objection}} To say that mental illness is a social construct and not a medical reality is to ignore and invalidate the entire psychiatric, psychological, and neurological communities. ****{{Objection}} This is an {{W|argument from authority}}. Authorities have been wrong many times in the past. Other arguments are needed. *****{{Objection}} This is an {{W|argument from fallacy}}. Just because you are able to categorize something as a logical fallacy does not mean it is untrue. You have not shown any evidence at all that the psychiatric, psychological, and neurological communities and other scientists who believe, based on the evidence they have seen, that mental illness is a real thing are incorrect. If you want to argue that mental illness does not exist, when that is the scientific consensus, the burden of proof is on you to provide that evidence. ******{{Objection}} There is no consensus on the diagnostic labelling of psychology. Many in the scientific community view the term "mental illness" as an inappropriate metaphor for what they believe to be emotional or societal problems, and not legitimate illnesses. ***{{Objection}} Mental illnesses are highly prevalent throughout the world, the most common being depression, but also posttraumatic stress disorder, schizophrenia, conduct disorder, bipolar disorder, and drug addiction. ****{{Objection}} The prevalence of diagnoses is irrelevant, the argument was on the validity of the label. ***{{Objection}} Mental illness is not some arbitrary classification decided on by asking someone how they feel on a whim, but rather by observing patterns in mood, behavior, and thought over an extended period of time. ***{{Objection}} All illnesses fit the criterium of "social construct" and are based on recognition of patterns of chronology, symptoms and behavior. This does not mean they are not real illnesses. Finding the etiology of the symptoms sometimes leads to a pathogen that can then be categorized as a pathogenic disease. Sometimes the etiology isn't completely understood, so it remains a group of symptoms recognized as a syndrome. Sometimes the etiology of the etiology of symptoms (especially with emotional and mental illness) is lack of experience with a genuinely understood healing. When we feel content and happy we don't seek to understand how and why we went from anxious to content or went from content to anxious in the first place. We just "accept" that we feel better for now and "move forward". Then hope we do not fall victim to the allegedly mysterious ways of the emotional midbrain. Believing one understands what they see when one is actually in logical and factual error is diagnostic of a narcissistic belief. When a "professional" is not given the narcissistic supply of "compliance" from a patient who rightfully doubts the false perceptions of the professional, and the professional does not see the error in his own perception, the professional is not immune to the tailspin that is implementing a narcissistic abuse cycle. The patient can end up in seriously deeper mental illness in this way. **{{Objection}} Just because something ought to be discouraged does not mean it should be illegal. Many things are illegal and this ends up harming people, for instance laws against drugs end up harming drug addicts by criminalizing their behavior and turning them into convicted felons who get severely punished, rather than treating their disease. If mental illness is truly an illness than it ought to be treated, just like any other illness, and not criminalized. Also, a law against suicide is rather pointless and impossible to enforce, since the only way someone would ever be guilty of committing the crime of suicide is if they end up dead because of it, and it is stupid to waste time prosecuting dead people. **{{Objection}} Suicide is not only a permanent solution to a temporary problem, it's a permanent solution to all problems, forever. ***{{Objection}} This statement minimizes the pain of one who is suicidal, it shows the lack of understanding of the pain a suicidal person is enduring and increases their isolation. It legitimizes suicide as an option to permanently end one's pain. It can cause more harm than good to say this to a suicidal person. ****{{Objection}} To assume how a person would feel after being told suicide is a viable option is irrational. All people are different, and perceive the world and it's contents in different ways. Where one person would see suicide as a poor option for taking care of a "temporary" problem, many people struggle with depression their entire life, and due to financial or personal issues, either simply cannot afford or do not wish to seek medical help. By not providing an open, accessable option for those who suffer from depression forces them to either do it illegally and run the risk of being put in a mental institution, or seek unwanted medical attention with the chance that their illness cannot be permanently treated. Some would rather have a quick, permanent option instead of constant attention, medication, and mental upkeep that a normal person wouldn't undergo, further isolating them from others, as people with depression or similar mental illnesses are already perceiving the world differently and are being treated different from those who were lucky enough to not be born with/develop said illness. *{{Argument against}} If a person is in debt, the act of committing suicide would lead to unpaid debts, this would ultimately hurt someone else financially.<ref>{{Cite web |url=https://suicidepreventioncommunity.wordpress.com/2010/03/05/please-stop-saying-suicide-is-a-permanent-solution/ |title=Please Stop Saying, 'Suicide is a permanent solution...' |date=2010-03-05 |website=Suicide Prevention News and Comment |language=en |access-date=2020-09-09}}</ref> **{{Objection}} If a person committing suicide is incapable of generating income to pay debt, and living increases cost, then suicide would ultimately help someone else financially. **{{Objection}} The creditor could also be hurt financially by the debtor dying prematurely or otherwise being rendered incapable of paying their debt. That is a risk that the creditor takes on when lending them funds in the first place. *{{argument against}} No it shouldn't. Being honest suicide doesn't hurt anyone physically it's thing that should be followed by a law. It's the person's life and law should not interpret or interfere with someone wanting to relieve themselves of eternal pain and suffering. ==Further readings== *{{Cite book |title=Suicide Prohibition: The Shame of Medicine |last=Szasz |first=Thomas}} *{{Cite book |title=Fatal Freedom: The Ethics and Politics of Suicide |last=Szasz |first=Thomas}} *{{Cite book |title=The Contribution of Natural Law Theory to Moral and Legal Debate on Suicide, Assisted Suicide and Euthanasia |last=Paterson |first=Craig}} ==See also== *[[Suicide]] ==External links== *[[Wikipedia:Suicide legislation]] ==References== {{Reflist}} [[Category:Death]] [[Category:Politics]] [[Category:Wikiversity Law Review]] [[Category:Legal ethics]] oxb1bbq2uogvukexh8bi73uw41do9no 0 222723 2410950 2249433 2022-08-02T11:25:26Z CommonsDelinker 9184 Replacing Flag_of_Nizhnevartovsk_(Khanty-Mansia).svg with [[File:Flag_of_Nizhnevartovsk.svg]] (by [[:c:User:CommonsDelinker|CommonsDelinker]] because: [[:c:COM:FR|File renamed]]: [[:c:COM:FR#FR4|Criterion 4]] (harmonizing names of file set)). wikitext text/x-wiki The article is devoted to the study of different types of cities corresponding to the four objects of Wikidata - [[d:Q3957|"Town"]], [[d:Q515|"City"]], [[d:Q1549591|"Big city"]] and [[d:Q1637706 |"City with millions of inhabitants"]]. Using SPARQL queries to Wikidata, data on the number of instances of the objects under study was obtained and the following information was gathered: * Population of different types of cities * Number of cities without sister cities * List of cities ordered by number of sister cities * Number of cities with certain amount of sister cities * Country with most sister cities * Closest neighbours of Russia by number of sister cities == Item lists == === "Town" === * Wikidata element: [[d:Q3957|Q3957]] <syntaxhighlight lang="SPARQL"> SELECT ?city ?cityLabel WHERE { ?city wdt:P31 wd:Q3957. SERVICE wikibase:label { bd:serviceParam wikibase:language "en" } } </syntaxhighlight> [https://query.wikidata.org/#SELECT%20%3Fcity%20%3FcityLabel%0AWHERE%0A%7B%0A%20%20%20%20%3Fcity%20wdt%3AP31%20wd%3AQ3957.%0A%20%20%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%20%7D%0A%7D SPARQL query], 13800 records (2020). === "City" === * Wikidata element: [[d:Q515|Q515]] <syntaxhighlight lang="SPARQL"> SELECT ?city ?cityLabel WHERE { ?city wdt:P31 wd:Q515. SERVICE wikibase:label { bd:serviceParam wikibase:language "en" } } </syntaxhighlight> [https://query.wikidata.org/#SELECT%20%3Fcity%20%3FcityLabel%0AWHERE%0A%7B%0A%20%20%20%20%3Fcity%20wdt%3AP31%20wd%3AQ515.%0A%20%20%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%20%7D%0A%7D SPARQL query], 20800 records (2017), 9260 records (2020). <span style="color:green">Most complete elements include </span>> [[d:Q62|San-Francisco]], [[d:Q64|Berlin]], [[d:Q1895|Petrozavodsk]], … <span style="color:#8e2727">Almost empty elements are </span>> [[d:Q12241857|Madinat Zayed]], [[d:Q1453196|Muzaffarpur]], [[d:Q1815091|Willow-River]], … According to ProWD [[d:Q334|Singapore]] is the leader in terms of the number of properties (104 properties) among cities around the world. [[d:Q15760|Novorossiysk]] contains 31 properties. This is the maximum number of properties for Russian cities. === "Big city" === * Wikidata element: [[d:Q1549591|Q1549591]] <syntaxhighlight lang="SPARQL"> SELECT ?city ?cityLabel WHERE { ?city wdt:P31 wd:Q1549591. SERVICE wikibase:label { bd:serviceParam wikibase:language "en" } } </syntaxhighlight> [https://query.wikidata.org/#SELECT%20%3Fcity%20%3FcityLabel%0AWHERE%0A%7B%0A%20%20%20%20%3Fcity%20wdt%3AP31%20wd%3AQ1549591.%0A%20%20%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%20%7D%0A%7D SPARQL query], 198 records (2017), 3075 records (2020). <span style="color:green">Most complete elements include </span>> [[d:Q70|Bern]], [[d:Q64|Berlin]], [[d:Q71|Geneva]], … <span style="color:#8e2727">Almost empty elements are </span>> [[d:Q4374738|Balanga (Nigeria)]], [[d:Q3183016|Ungaran]], [[d:Q325164|Kayes]], … According to ProWD [[d:Q334|Singapore]] is the leader in terms of the number of properties (104 properties) among big cities around the world. [[d:Q649|Moscow]] contains 76 properties. This is the maximum number of properties for Russian big cities. === "City with millions of inhabitants" === * Wikidata element: [[d:Q1637706 | Q1637706]] <syntaxhighlight lang="SPARQL"> SELECT ?city ?cityLabel WHERE { ?city wdt:P31 wd:Q1637706. SERVICE wikibase:label { bd:serviceParam wikibase:language "en" } } </syntaxhighlight> [https://query.wikidata.org/#SELECT%20%3Fcity%20%3FcityLabel%0AWHERE%20%0A%7B%0A%20%20%20%20%3Fcity%20wdt%3AP31%20wd%3AQ1637706.%0A%20%20%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%20%7D%0A%7D SPARQL query], 616 records (2020). === Different types of cities === <syntaxhighlight lang="SPARQL"> SELECT ?city ?cityLabel WHERE { # Selecting items which are ... { ?city wdt:P31 wd:Q3957 } UNION # ... instances of "town" ... { ?city wdt:P31 wd:Q515 } UNION # ... instances of "city" ... { ?city wdt:P31 wd:Q1549591 } UNION # ... instances of "big city" ... { ?city wdt:P31 wd:Q1637706 } # ... instances of "city with millions of inhabitants" SERVICE wikibase:label { bd:serviceParam wikibase:language "en". } } </syntaxhighlight> [https://query.wikidata.org/#SELECT%20%3Fcity%20%3FcityLabel%20WHERE%20%7B%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Selecting%20items%20which%20are%20...%0A%20%20%7B%20%3Fcity%20wdt%3AP31%20wd%3AQ3957%20%7D%20UNION%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20instances%20of%20%22town%22%20...%0A%20%20%7B%20%3Fcity%20wdt%3AP31%20wd%3AQ515%20%7D%20UNION%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20instances%20of%20%22city%22%20...%0A%20%20%7B%20%3Fcity%20wdt%3AP31%20wd%3AQ1549591%20%7D%20UNION%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20instances%20of%20%22big%20city%22%20...%0A%20%20%7B%20%3Fcity%20wdt%3AP31%20wd%3AQ1637706%20%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20instances%20of%20%22city%20with%20millions%20of%20inhabitans%22%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%0A%7D SPARQL query], 26751 records (2020). == Population == === "Town" === Used: * Object: [[d:Q3957 |town (Q3957)]] * Property: [[d:Property:P31|instance of (P31)]] * Property: [[d:Property: P1082 |population (P1082)]] <syntaxhighlight lang="SPARQL"> SELECT (SUM(?population_city) as ?sum) WHERE { # Selecting total population of items which are SELECT (MAX(xsd:integer(REPLACE(STR(?population),"\\.",""))) as ?population_city) ?city WHERE { ?city wdt:P31 wd:Q3957. # ... instances of "town" ... ?city wdt:P1082 ?population # ... with filled property "population" } GROUP BY ?city } </syntaxhighlight> [https://query.wikidata.org/#SELECT%20%28SUM%28%3Fpopulation_city%29%20as%20%3Fsum%29%20WHERE%20%7B%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Selecting%20total%20population%20of%20items%20which%20are%0A%20%20SELECT%20%28MAX%28xsd%3Ainteger%28REPLACE%28STR%28%3Fpopulation%29%2C%22%5C%5C.%22%2C%22%22%29%29%29%20as%20%3Fpopulation_city%29%20%3Fcity%20WHERE%20%7B%0A%20%20%20%20%3Fcity%20wdt%3AP31%20wd%3AQ3957.%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20instances%20of%20%22town%22%20...%0A%20%20%20%20%3Fcity%20wdt%3AP1082%20%3Fpopulation%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20with%20filled%20property%20%22population%22%0A%20%20%7D%0A%20%20GROUP%20BY%20%3Fcity%0A%7D SPARQL query], 53,30 million people (2020). === "City" === Used: * Object: [[d:Q515 | city (Q515)]] * Property: [[d:Property:P31|instance of (P31)]] * Property: [[d:Property: P1082 |population (P1082)]] <syntaxhighlight lang="SPARQL"> SELECT (SUM(?population_city) as ?sum) WHERE { # Selecting total population of items which are SELECT (MAX(xsd:integer(REPLACE(STR(?population),"\\.",""))) as ?population_city) ?city WHERE { ?city wdt:P31 wd:Q515. # ... instances of "city" ... ?city wdt:P1082 ?population # ... with filled property "population" } GROUP BY ?city } </syntaxhighlight> [https://query.wikidata.org/#SELECT%20%28SUM%28%3Fpopulation_city%29%20as%20%3Fsum%29%20WHERE%20%7B%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Selecting%20total%20population%20of%20items%20which%20are%0A%20%20SELECT%20%28MAX%28xsd%3Ainteger%28REPLACE%28STR%28%3Fpopulation%29%2C%22%5C%5C.%22%2C%22%22%29%29%29%20as%20%3Fpopulation_city%29%20%3Fcity%20WHERE%20%7B%0A%20%20%20%20%3Fcity%20wdt%3AP31%20wd%3AQ515.%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20instances%20of%20%22city%22%20...%0A%20%20%20%20%3Fcity%20wdt%3AP1082%20%3Fpopulation%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20with%20filled%20property%20%22population%22%0A%20%20%7D%0A%20%20GROUP%20BY%20%3Fcity%0A%7D SPARQL query], 1 133,56 million people (2020). === "Big city" === Used: * Object: [[d:Q1549591 |big city (Q1549591)]] * Property: [[d:Property:P31|instance of (P31)]] * Property: [[d:Property: P1082 |population (P1082)]] <syntaxhighlight lang="SPARQL"> SELECT (SUM(?population_city) as ?sum) WHERE { # Selecting total population of items which are SELECT (MAX(xsd:integer(REPLACE(STR(?population),"\\.",""))) as ?population_city) ?city WHERE { ?city wdt:P31 wd:Q1549591. # ... instances of "big city" ... ?city wdt:P1082 ?population # ... with filled property "population" } GROUP BY ?city } </syntaxhighlight> [https://query.wikidata.org/#SELECT%20%28SUM%28%3Fpopulation_city%29%20as%20%3Fsum%29%20WHERE%20%7B%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Selecting%20total%20population%20of%20items%20which%20are%0A%20%20SELECT%20%28MAX%28xsd%3Ainteger%28REPLACE%28STR%28%3Fpopulation%29%2C%22%5C%5C.%22%2C%22%22%29%29%29%20as%20%3Fpopulation_city%29%20%3Fcity%20WHERE%20%7B%0A%20%20%20%20%3Fcity%20wdt%3AP31%20wd%3AQ1549591.%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20instances%20of%20%22big%20city%22%20...%0A%20%20%20%20%3Fcity%20wdt%3AP1082%20%3Fpopulation%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20with%20filled%20property%20%22population%22%0A%20%20%7D%0A%20%20GROUP%20BY%20%3Fcity%0A%7D SPARQL query], 2 538,49 million people (2020). === "City with millions of inhabitants" === Used: * Object: [[d:Q1637706 |city with millions of inhabitants (Q1637706)]] * Property: [[d:Property:P31|instance of (P31)]] * Property: [[d:Property: P1082 |population (P1082)]] <syntaxhighlight lang="SPARQL"> SELECT (SUM(?population_city) as ?sum) WHERE { # Selecting total population of items which are SELECT (MAX(xsd:integer(REPLACE(STR(?population),"\\.",""))) as ?population_city) ?city WHERE { ?city wdt:P31 wd:Q1637706. # ... instances of "city with millions of inhabitants" ... ?city wdt:P1082 ?population # ... with filled property "population" } GROUP BY ?city } </syntaxhighlight> [https://query.wikidata.org/#SELECT%20%28SUM%28%3Fpopulation_city%29%20as%20%3Fsum%29%20WHERE%20%7B%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Selecting%20total%20population%20of%20items%20which%20are%0A%20%20SELECT%20%28MAX%28xsd%3Ainteger%28REPLACE%28STR%28%3Fpopulation%29%2C%22%5C%5C.%22%2C%22%22%29%29%29%20as%20%3Fpopulation_city%29%20%3Fcity%20WHERE%20%7B%0A%20%20%20%20%3Fcity%20wdt%3AP31%20wd%3AQ1637706.%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20instances%20of%20%22city%20with%20millions%20of%20inhabitants%22%20...%0A%20%20%20%20%3Fcity%20wdt%3AP1082%20%3Fpopulation%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20with%20filled%20property%20%22population%0A%20%20%7D%0A%20%20GROUP%20BY%20%3Fcity%0A%7D SPARQL query], 2 118,39 million people (2020). === Analysis === Different characters, such as point, comma, or space, are used as separators in different countries. As a result, the variants of representing the value of the [[d:Property:P1082|population]] property can also be different. Problems arise when using a point, because in Wikidata this character is the separator between the integer and decimal parts of a number. To disambiguate, [https://en.wikibooks.org/wiki/SPARQL/Expressions_and_Functions#REPLACE REPLACE] function to remove the specified character should be used. This conversion does not affect the value itself, since the population is an integer, and the separators are used solely for ease of reading. The table below shows a summary of the population of different types of cities, as well as the proportion of the population per type of city of the [[w:en:World_population|world population]], which reached approximately 7,8 billion people in 2020{{sfn|Growth is not endless: what awaits the world's population by the end of the century|2020}}. According to Wikidata, almost three quarters of the world's population live in cities. {| class="wikitable" |- ! City type !! Population<br>(million people) !! % of <br>world |- | [[d:Q3957 |"Town"]] ||align="right"| 53,30 ||align="right"| 0,7 % |- | [[d:Q515 |"City"]] ||align="right"| 1 133,56 ||align="right"| 14,5 % |- | [[d:Q1549591 |"Big city"]] ||align="right"| 2 538,49 ||align="right"| 32,5 % |- | [[d:Q1637706 |"City with millions of inhabitants"]] ||align="right"| 2 118,39 ||align="right"| 27,1 % |- ! Total !! 5 843,74 !! 74,8 % |} == Sister cities == [[w:en:Sister city|Sister cities]] are cities of different states that have established permanent friendly relations with each other in order to strengthen international relationship in the fields of culture, economics, creation and management of urban infrastructure, the functioning of civil society, and so on{{sfn|Sister cities|2018}}. === How many cities don't have a single sister city? === Used: * Object: [[d:Q3957|town (Q3957)]] * Object: [[d:Q515|city (Q515)]] * Object: [[d:Q1549591|big city (Q1549591)]] * Object: [[d:Q1637706 |city with millions of inhabitants (Q1637706)]] * Property: [[d:Property:P31|instance of (P31)]] * Property: [[d:Property:P190|sister city (P190)]] <syntaxhighlight lang="SPARQL"> SELECT (COUNT(?city) as ?count) WHERE { # Counting items which are ... { ?city wdt:P31 wd:Q3957 } UNION # ... instances of "town" ... { ?city wdt:P31 wd:Q515 } UNION # ... OR instances of "city" ... { ?city wdt:P31 wd:Q1549591 } UNION # ... OR instances of "big city" ... { ?city wdt:P31 wd:Q1637706 } # ... OR instances of "city with millions of inhabitants" FILTER NOT EXISTS { ?city wdt:P190 [] } # ... with unfilled property "sister city" } </syntaxhighlight> [https://query.wikidata.org/#SELECT%20%28COUNT%28%3Fcity%29%20as%20%3Fcount%29%20WHERE%20%7B%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Counting%20items%20which%20are%20...%20%0A%20%20%7B%20%3Fcity%20wdt%3AP31%20wd%3AQ3957%20%7D%20UNION%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20instances%20of%20%22town%22%20...%0A%20%20%7B%20%3Fcity%20wdt%3AP31%20wd%3AQ515%20%7D%20UNION%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20OR%20instances%20of%20%22city%22%20...%0A%20%20%7B%20%3Fcity%20wdt%3AP31%20wd%3AQ1549591%20%7D%20UNION%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20OR%20instances%20of%20%22big%20city%22%20...%0A%20%20%7B%20%3Fcity%20wdt%3AP31%20wd%3AQ1637706%20%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20OR%20instances%20of%20%22city%20with%20millions%20of%20inhabitants%22%0A%20%20FILTER%20NOT%20EXISTS%20%7B%20%3Fcity%20wdt%3AP190%20%5B%5D%20%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20with%20unfilled%20property%20%22sister%20city%22%0A%7D SPARQL query], 21479 cities (2020). There are 26751 cities of four types known by Wikidata for 2020. Thus, sister cities are known only for 20% of cities. === List of cities ordered by number of sister cities === ==== All ==== Used: * Object: [[d:Q3957|town (Q3957)]] * Object: [[d:Q515|city (Q515)]] * Object: [[d:Q1549591|big city (Q1549591)]] * Object: [[d:Q1637706 |city with millions of inhabitants (Q1637706)]] * Property: [[d:Property:P31|instance of (P31)]] * Property: [[d:Property:P190|sister city (P190)]] <syntaxhighlight lang="SPARQL"> SELECT ?city ?cityLabel (COUNT(?sister) AS ?sisterCount) WHERE { # Counting sister cities of cities which are ... { ?city wdt:P31 wd:Q3957 } UNION # ... instances of "town" ... { ?city wdt:P31 wd:Q515 } UNION # ... OR instances of "city" ... { ?city wdt:P31 wd:Q1549591 } UNION # ... OR instances of "big city" ... { ?city wdt:P31 wd:Q1637706 } # ... OR instances of "city with millions of inhabitants" ?city wdt:P190 ?sister. # ... with filled property "sister city" SERVICE wikibase:label { bd:serviceParam wikibase:language "en". } } GROUP BY ?city ?cityLabel # Grouping by city ORDER BY DESC(?sisterCount) # Sorting by number of sister cities (descending) </syntaxhighlight> [https://w.wiki/t44 SPARQL query], 4046 cities with sister cities (2020). ==== Russia ==== Used: * Object: [[d:Q3957|town (Q3957)]] * Object: [[d:Q515|city (Q515)]] * Object: [[d:Q1549591|big city (Q1549591)]] * Object: [[d:Q1637706 |city with millions of inhabitants (Q1637706)]] * Object: [[d:Q159|Russia (Q159)]] * Property: [[d:Property:P31|instance of (P31)]] * Property: [[d:Property:P17|country (P17)]] * Property: [[d:Property:P190|sister city (P190)]] <syntaxhighlight lang="SPARQL"> SELECT ?city ?cityLabel (COUNT(?sister) AS ?sisterCount) WHERE { # Counting sister cities of cities which are ... VALUES ?cityTypes {wd:Q3957 wd:Q515 wd:Q1549591 wd:Q1637706} ?city wdt:P31 ?cityTypes. # ... instances of different types of cities ... ?city wdt:P17 wd:Q159. # ... belonging to Russia ... ?city wdt:P190 ?sister. # ... with filled property "sister city" SERVICE wikibase:label { bd:serviceParam wikibase:language "en". } } GROUP BY ?city ?cityLabel # Grouping by city ORDER BY DESC(?sisterCount) # Sorting by number of sister cities (descending) </syntaxhighlight> [https://w.wiki/t47 SPARQL query], 82 cities with sister cities (2020). There were more cities wishing to be friends with the cultural capital of Russia (Saint Petersburg, 230 sister cities) than with the official capital (Moscow, 134 sister cities) for 2020. Omsk (58), Volgograd (56) and Kaliningrad (54) had almost the same number of sister cities. Petrozavodsk, Perm, Vladimir and Belgorod each had 14 sister cities. === Number of cities with certain amount of sister cities === ==== All ==== Used: * Object: [[d:Q3957|town (Q3957)]] * Object: [[d:Q515|city (Q515)]] * Object: [[d:Q1549591|big city (Q1549591)]] * Object: [[d:Q1637706 |city with millions of inhabitants (Q1637706)]] * Property: [[d:Property:P31|instance of (P31)]] * Property: [[d:Property:P190|sister city (P190)]] <syntaxhighlight lang="SPARQL"> #defaultView:LineChart # Do line chart as result representation SELECT ?sisterCount (COUNT(?sisterCount) AS ?FreqNSister) WHERE { # Count No. of cities having ?sisterCount sister cities # and number of sister cities themselves { SELECT (COUNT(?sister) AS ?sisterCount) WHERE { # Count sister cities of cities which are ... VALUES ?cityTypes {wd:Q3957 wd:Q515 wd:Q1549591 wd:Q1637706} ?city wdt:P31 ?cityTypes. # ... instances of different types of cities ... ?city wdt:P190 ?sister. # ... with filled property "sister city" } GROUP BY ?city # Group list by city } } GROUP BY ?sisterCount # Group by number of sister cities ORDER BY DESC(?sisterCount) # Order by number of sister cities (descending) </syntaxhighlight> [https://w.wiki/pM9 SPARQL query], 90 variants of sister cities amount (2020). [[File:Number of cities having given number of sister cities according to Wikidata, 2020.png|700px|thumb|left|Relation between number of sister cities the city have (S) and number of world cities which have this amount of sister cities (N), 2020]] [[File:Logarithm of the number of cities having given number of sister cities according to Wikidata, 2020.png|700px|thumb|left|Relation between number of sister cities the city have (S) and logarithm of the number of world cities which have this amount of sister cities (N), 2020]] <br clear=all> A little more than four thousand cities (4046 cities) have at least one sister city, of which: * 32% (1314 cities) have relations with more than five cities; * 18% (728 cities) have at least 11 sister cities; * 9% (345 cities) friends with more than 20 cities; * 2% (94 cities) have 50 or more sister cities. It can be concluded that the relation between number of sister cities the city have and number of cities which have this amount of sister cities has a distribution close to a power law. ==== Russia ==== Used: * Object: [[d:Q3957|town (Q3957)]] * Object: [[d:Q515|city (Q515)]] * Object: [[d:Q1549591|big city (Q1549591)]] * Object: [[d:Q1637706 |city with millions of inhabitants (Q1637706)]] * Object: [[d:Q159|Russia (Q159)]] * Property: [[d:Property:P31|instance of (P31)]] * Property: [[d:Property:P17|country (P17)]] * Property: [[d:Property:P190|sister city (P190)]] <syntaxhighlight lang="SPARQL"> #defaultView:LineChart # Do line chart as result representation SELECT ?sisterCount (COUNT(?sisterCount) AS ?FreqNSister) WHERE { # Count No. of cities having ?sisterCount sister cities # and number of sister cities themselves { SELECT (COUNT(?sister) AS ?sisterCount) WHERE { # Count sister cities of cities which are ... VALUES ?cityTypes {wd:Q3957 wd:Q515 wd:Q1549591 wd:Q1637706} ?city wdt:P31 ?cityTypes. # ... instances of different types of cities ... ?city wdt:P17 wd:Q159. # ... belonging to Russia ... ?city wdt:P190 ?sister. # ... with filled property "sister city" } GROUP BY ?city # Group list by city } } GROUP BY ?sisterCount # Group by number of sister cities ORDER BY DESC(?sisterCount) # Order by number of sister cities (descending) </syntaxhighlight> [https://w.wiki/pMA SPARQL query], 24 variants of sister cities amount (2020). [[File:Number of Russian cities having given number of sister cities according to Wikidata, 2020.png|700px|thumb|left|Relation between number of sister cities the Russian city have (S) and number of Russian cities which have this amount of sister cities (N), 2020]] <br clear=all> A little less than a hundred Russian cities (82 cities) have at least one sister city, of which only 48% (39 cities) are connected with over than five cities. === Which country has the most sister cities? === Used: * Object: [[d:Q3957|town (Q3957)]] * Object: [[d:Q515|city (Q515)]] * Object: [[d:Q1549591|big city (Q1549591)]] * Object: [[d:Q1637706 |city with millions of inhabitants (Q1637706)]] * Property: [[d:Property:P31|instance of (P31)]] * Property: [[d:Property:P17|country (P17)]] * Property: [[d:Property:P190|sister city (P190)]] <syntaxhighlight lang="SPARQL"> #defaultView:BubbleChart SELECT ?countryLabel (COUNT(?sister) as ?sisterCount) WHERE { # Selecting number of distinct sister cities of particular country cities which are ... SELECT DISTINCT ?countryLabel ?sister WHERE { VALUES ?cityTypes {wd:Q3957 wd:Q515 wd:Q1549591 wd:Q1637706} ?city wdt:P31 ?cityTypes. # ... instances of different types of cities ... ?city wdt:P17 ?country. # ... with filled property "country" ... ?city wdt:P190 ?sister. # ... with filled property "sister city" SERVICE wikibase:label { bd:serviceParam wikibase:language "en". } } } GROUP BY ?countryLabel ORDER BY DESC(?sisterCount) </syntaxhighlight> [https://w.wiki/pMT SPARQL query], 208 countries (2020). [[File:Bubble chart number of sister cities of countries according to Wikidata, 2020 (EN).svg|700px|thumb|left|Bubble chart of the countries, the size of the ball - the sister cities number of certain country cities, 2020]] <br clear="all"> Germany had the largest number of sister cities (1375 cities) for 2020. ==== List of countries having sister cities with Germany ==== Used: * Object: [[d:Q3957|town (Q3957)]] * Object: [[d:Q515|city (Q515)]] * Object: [[d:Q1549591|big city (Q1549591)]] * Object: [[d:Q1637706 |city with millions of inhabitants (Q1637706)]] * Object: [[d:Q183|Germany (Q183)]] * Property: [[d:Property:P31|instance of (P31)]] * Property: [[d:Property:P17|country (P17)]] * Property: [[d:Property:P190|sister city (P190)]] <syntaxhighlight lang="SPARQL"> SELECT ?country ?countryLabel (COUNT(DISTINCT ?sister) as ?sisterCount) WHERE { # Selecting number of distinct particular country sister cities of cities which are ... VALUES ?cityTypes {wd:Q3957 wd:Q515 wd:Q1549591 wd:Q1637706} ?city wdt:P31 ?cityTypes. # ... instances of different types of cities ... ?city wdt:P17 wd:Q183. # ... belonging to Germany ... ?city wdt:P190 ?sister. # ... with filled property "sister city" which are ... ?sister wdt:P17 ?country. # ... with filled property "country" ... SERVICE wikibase:label { bd:serviceParam wikibase:language "en". } } GROUP BY ?country ?countryLabel ORDER BY DESC(?sisterCount) </syntaxhighlight> [https://w.wiki/pMV SPARQL query], 93 countries (2020). The table shows a list of ten countries that have the largest number of sister cities with Germany (2020). {| class="wikitable" |- ! # !! Country !! Number of<br>sister cities!! % of total |- | 1 || [[d:Q142| France]] ||align="right"| 247 ||align="right"| 18,0 % |- | 2 || [[d:Q183| Germany]] ||align="right"| 195 ||align="right"| 14,2 % |- | 3 || [[d:Q145| United Kingdom]] ||align="right"| 120 ||align="right"| 8,7 % |- | 4 || [[d:Q38| Italy]] ||align="right"| 86 ||align="right"| 6,3 % |- | 5 || [[d:Q36| Poland]] ||align="right"| 81 ||align="right"| 5,9 % |- | 6 || [[d:Q30| United States of America]] ||align="right"| 60 ||align="right"| 4,4 % |- | 7 || [[d:Q40| Austria]] ||align="right"| 41 ||align="right"| 3,0 % |- | 8 || [[d:Q159| Russia]] ||align="right"| 39 ||align="right"| 2,8 % |- | 9 || [[d:Q28|Hungary]] ||align="right"| 39 ||align="right"| 2,8 % |- | 10 || [[d:Q31| Belgium]] ||align="right"| 33 ||align="right"| 2,4 % |} === Closest neighbours of Russia by number of sister cities === Used: * Object: [[d:Q3957|town (Q3957)]] * Object: [[d:Q515|city (Q515)]] * Object: [[d:Q1549591|big city (Q1549591)]] * Object: [[d:Q1637706 |city with millions of inhabitants (Q1637706)]] * Object: [[d:Q159|Russia (Q159)]] * Property: [[d:Property:P31|instance of (P31)]] * Property: [[d:Property:P17|country (P17)]] * Property: [[d:Property:P190|sister city (P190)]] * Property: [[d:Property:P3896|geoshape (P3896)]] <syntaxhighlight lang="SPARQL"> #defaultView:Map SELECT ?country ?countryLabel ?sisterCount ?shape ?layer WHERE { { # Selecting number of distinct particular country sister cities of cities which are ... SELECT ?country ?countryLabel (COUNT(DISTINCT ?sister) as ?sisterCount) WHERE { VALUES ?cityTypes {wd:Q3957 wd:Q515 wd:Q1549591 wd:Q1637706} ?city wdt:P31 ?cityTypes. # instances of different types of cities ?city wdt:P17 wd:Q159. # city belongs to Russia ?city wdt:P190 ?sister. # city has "sister city" ?sister wdt:P17 ?country. # which belongs to "country" FILTER(?country NOT IN(wd:Q159)) # except the Russia SERVICE wikibase:label {bd:serviceParam wikibase:language "en"} } GROUP BY ?country ?countryLabel ORDER BY DESC(?sisterCount) } OPTIONAL {?country wdt:P3896 ?shape.} # country has "geoshape" BIND( IF(?sisterCount < 5, "<5", IF(?sisterCount <= 10, "5-10", IF(?sisterCount <= 20, "11-20", IF(?sisterCount <= 30, "21-30", IF(?sisterCount <= 40, "31-40", ">40"))))) AS ?layer). } </syntaxhighlight> [https://w.wiki/q5d SPARQL query], 102 countries (2020). [[File:Map of closest neighbours of Russia by number of sister cities, 2020.png|700px|thumb|left|Map of closest neighbours of Russia by number of sister cities, 2020]] <br clear="all"> Russia has more than twenty sister cities with countries such as United States of America (46), China (46), Germany (44), Ukraine (28), Bulgaria (25), Poland (24), France (23) and Italy (22). == Wikidata completeness and disadvantages == City is a type of human settlement with people not occupied with agriculture. At the same time, different countries use different criteria when assigning city status to settlements, the main of which is population. Some countries don't define a term "city" at all. So, in France, only one geographic unit of this kind is used &mdash; a commune, regardless of the number of people living in it and the type of their activity. Therefore, it can be difficult to clearly determine which settlement is classified as a city and which is not. In practice, some Wikidata objects can simultaneously be instances of different types of cities. For example, [[d:Q8686|Shanghai]] is assigned to three objects under study: [[d:Q515|city]], [[d:Q1549591|big city]], [[d:Q1637706|city with millions of inhabitants]]. It is easy to guess that such multiple assignment affects the results of SPARQL queries, in particular, using the UNION construction. This can be verified by running, for example, [https://w.wiki/kCu SPARQL query for finding different types of cities]. [[d:Q8686|Shanghai]] is found in the results for three times. Wikidata has an inheritance mechanism expressed in the [[d:Property:P279|subclass of]] property. This mechanism consists in the fact that if an object is an instance of [[d:Q1549591|big city]], then it is also an instance of [[d:Q515|city]], since [[d:Q1549591|big city]] is a subclass of [[d:Q515|city]]. Thus, the situation described above with [[d:Q8686|Shanghai]] can be resolved by leaving only one class — [[d:Q1637706|city with millions of inhabitants]]. It should be noted that replacing a UNION construction with a subclassing construction is not equivalent. <syntaxhighlight lang=sparql> # Selecting items which are ... SELECT ?city ?cityLabel WHERE { # ... instances of "city" ... { ?city wdt:P31 wd:Q515 } UNION # ... instances of "big city" ... { ?city wdt:P31 wd:Q1549591 } UNION # ... instances of "city with millions of inhabitans" { ?city wdt:P31 wd:Q1637706 } SERVICE wikibase:label { bd:serviceParam wikibase:language "en". } } </syntaxhighlight> [https://w.wiki/kCn SPARQL query] <syntaxhighlight lang=sparql> # Selecting items which are ... SELECT ?city ?cityLabel WHERE { # ... instances of "city" subclasses ?city wdt:P31/wdt:P279* wd:Q515 SERVICE wikibase:label { bd:serviceParam wikibase:language "en". } } </syntaxhighlight> [https://w.wiki/kCo SPARQL query] [[d:Q8686|Shanghai]], considered earlier, can be found four times in the new query results. The fact is that in addition to some of the objects under study, there are other classes inherited from [[d:Q515|city]]. For example, [[d:Q2974842|lost city]], [[d:Q57318|free imperial city]], [[d:Q1094397|autonomous city]] and even [[d:Q1656724|ideal city]]. Also, probably due to the ambiguity in the criteria for assigning city status, subclasses were created for specific countries — [[d:Q25412763|city in Chile]], [[d:Q29556224|city in Cyprus]], [[d:Q494721|city of Japan]] and so on. This tendency was not spared by the cities of Russia, which could be noticed when comparing the results of a SPARQL query to find instances of the "City" object. For 2020, most of them belong to the [[d:Q7930989|city/town]] class. According to the Russian Census (2010){{sfn|Russian Census|2010}} and the Crimean Federal District Census (2014){{sfn|Crimean Federal District Census|2014}} , the total number of Russian cities was '''1117''' in 2014. All cities in Russia have an article in both Russian and English Wikipedia. Number of Wikidata elements which are Russian cities equals to '''1126'''{{sfn|Russian cities query at WDQS}}. It can be assumed that Wikidata completely covers, at least, Russian cities. == Future work == # Construct a graph of Russian sister cities. # Get list of Russian cities situated beyond the Arctic circle. # On which river in Russia is the largest number of cities located? # Which country has the largest proportion of sister cities within a country relative to the number of sister cities that relate that country to other countries? == Tests == <quiz display=simple> {Which of the following cities were named after toponyms? |type="[]"} - [[:en:w:Tolyatti|Tolyatti]] + [[:en:w:Tula, Russia|Tula]] - [[:en:w:Chernyakhovsk|Chernyakhovsk]] + [[:en:w:Kurilsk|Kurilsk]] + [[:en:w:Vologda|Vologda]] - [[:en:w:Obninsk|Obninsk]] {Which of the following flags are belonging to these cities: [[:en:w:Nizhnevartovsk|Nizhnevartovsk]], [[:en:w:Petropavlovsk-Kamchatsky|Petropavlovsk-Kamchatsky]], [[:en:w:Neftekamsk|Neftekamsk]], [[:en:w:Karabulak, Republic of Ingushetia|Karabulak]]?} + [[Image:Flag%20of%20Neftekamsk.svg|300px|link=]] + [[File:Flag of Karabulak (Ingushetia).png|300px|link=]] - [[File:Flag of Anenii Noi.svg|300px|link=]] + [[File:Flag of Nizhnevartovsk.svg|300px|link=]] - [[File:Flag of Pastavy.svg|300px|link=]] + [[File:Flag of Petropavlovsk-Kamchatsky (Kamchatka krai).png|300px|link=]] {Which of the following cities were founded more than 400 years ago? |type="[]"} + [[:en:w:Moscow|Moscow]] - [[:en:w:Sarov|Sarov]] + [[:en:w:Kazan|Kazan]] + [[:en:w:Astrakhan|Astrakhan]] + [[:en:w:Samara, Russia|Samara]] + [[:en:w:Voronezh|Voronezh]] </quiz> Check yourself: #[https://w.wiki/otn cities named after toponyms] #[https://w.wiki/ots flags of cities] #[https://w.wiki/t5v founded more than 400 years ago] == Addon == #[https://query.wikidata.org/#%23%20total%20number%20of%20sister%20cities%20per%20country%0ASELECT%20%3Fcountry%20%3FcountryLabel%20%3FcityCount%20WITH%20%7B%0A%20%20SELECT%20%3Fcountry%20%28COUNT%28%3FsisterCity%29%20AS%20%3FcityCount%29%20WHERE%20%7B%0A%20%20%20%20%3Fcity%20wdt%3AP17%20%3Fcountry%3B%0A%20%20%20%20%20%20%20%20%20%20wdt%3AP190%20%3FsisterCity.%0A%20%20%7D%0A%20%20GROUP%20BY%20%3Fcountry%0A%7D%20AS%20%25results%20WHERE%20%7B%0A%20%20INCLUDE%20%25results.%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22%5BAUTO_LANGUAGE%5D%2Cen%22.%20%7D%0A%7D%0AORDER%20BY%20DESC%28%3FcityCount%29 Total number of sister city statements per country] == References == {{References}} * {{cite web |last1 = Menshikova E. | url = https://prowd.id/dashboards/fa0470eaf5f5/profile | title = Cities in Russia | publisher = ProWD | year = 2020 }} * {{cite web |last1 = Menshikova E. | url = https://prowd.id/dashboards/bd326207c820/profile | title = Big cities in Russia | publisher = ProWD | year = 2020 }} == Links == * {{cite web | author = Alla Salkova | date = 16 April 2020 | url = https://www.gazeta.ru/science/2020/07/16_a_13154515.shtml | title = Growth is not endless: what awaits the world's population by the end of the century | publisher = [[w:en:Gazeta.Ru|Gazeta.Ru]] | accessdate = 2020-12-06 | lang = ru | ref = {{SfnRef|Growth is not endless: what awaits the world's population by the end of the century|2020}} }} * {{cite web | author = The International Association "Sister cities" | date = 2018 | url = http://goroda-pobratimy.ru/porodnennye-goroda | title = Sister cities | accessdate = 2020-12-06 | lang = ru | ref = {{SfnRef|Sister cities|2018}} }} * {{cite web | author = Federal State Statistics Service | authorlink = | date = 2010 | url = http://www.gks.ru/free_doc/new_site/perepis2010/croc/Documents/Vol1/pub-01-03.pdf | title = Russian Census &mdash; 2010 | format = pdf | work = | publisher = | accessdate = 2017-03-08 | language = Russian | ref = {{SfnRef|Russian Census|2010}} }} * {{cite web | author = Federal State Statistics Service | authorlink = | date = 2014 | url = https://rosstat.gov.ru/free_doc/new_site/population/demo/perepis_krim/KRUM_2015.pdf | title = Crimean Federal District Census &mdash; 2014 | format = pdf | work = | publisher = | accessdate = 2020-11-05 | language = Russian | ref = {{SfnRef|Crimean Federal District Census|2014}} }} * {{cite web | author = | authorlink = | date = | url = https://query.wikidata.org/#SELECT%20DISTINCT%20%3Fcity%20%3FcityLabel%20WHERE%20%7B%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Selecting%20items%20which%20are%20...%0A%20%20%7B%20%3Fcity%20wdt%3AP31%20wd%3AQ3957%20%7D%20UNION%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20instances%20of%20%22town%22%20...%0A%20%20%7B%20%3Fcity%20wdt%3AP31%20wd%3AQ515%20%7D%20UNION%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20instances%20of%20%22city%22%20...%0A%20%20%7B%20%3Fcity%20wdt%3AP31%20wd%3AQ1549591%20%7D%20UNION%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20instances%20of%20%22big%20city%22%20...%0A%20%20%7B%20%3Fcity%20wdt%3AP31%20wd%3AQ1637706%20%7D%20UNION%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20instances%20of%20%22city%20with%20millions%20of%20inhabitans%22%20...%0A%20%20%7B%20%3Fcity%20wdt%3AP31%20wd%3AQ7930989%20%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20instances%20of%20%22city%2Ftown%22%20...%0A%20%20%3Fcity%20wdt%3AP17%20wd%3AQ159%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20...%20belonging%20to%20Russia%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%0A%7D | title = Wikidata Query Service | format = | work = | publisher = | accessdate = 2020-11-05 | language = | ref = {{SfnRef|Russian cities query at WDQS}} }} * {{cite web | author = Andrew Krizhanovsky, Alexander Kurbeev | authorlink = | date = 2017-03-13 | url = https://www.authorea.com/users/148409/articles/172518 | title = Исследование городов-побратимов с использованием Викиданных - Authorea | format = | work = | publisher = | accessdate = 2017-05-03 | language = Russian }} [[Category:Research in programming Wikidata|{{SUBPAGENAME}}]] [[ru:Программирование Викиданных/Города]] lc5pqymlrk2grhtt0m4qu2a4z2fsmzv 0 223757 2410811 2410224 2022-08-01T19:26:24Z Sophivorus 85690 Move objection back in place wikitext text/x-wiki {{Wikidebate}} {{Philosophy}} Most people assume that there's an objective reality in which all of us exist, with facts that are universally true and more than just a matter of opinion. However, others argue that there is no such thing as objective reality, that everything is subjective and that anything can be questioned or legitimately disagreed with, even facts that used to have near-universal agreement. So who is right about the nature of reality? Is there an objective reality that we are all a part of? Does objective reality truly exist? == Definitions == * '''Subjective reality''' means that something is actual depending on the mind. For example: someone walks by a flower and experiences the beauty of the flower. Would you say that the experience of beauty is dependent or independent of the mind? Would the experience of the form in general of the flower be dependent on the mind? * '''Objective reality''' means that something is actual (so it exists) independent of the mind. For example: while no one is nearby, a meteor crashes into a car, putting it on flames, leaving only a pile of ashes. Are any of these events depending on some mind? It is actually hard to state a purely objective reality example, as one needs to describe it in concepts that are interpreted by the mind. You see the difficulty? Anything related to experience, like form, weight, heat, color, beauty, etc, etc is dependent on a mind. So we could say, that objective reality is formless. Only when observed by a mind, there is form. This has lots of similarities with a computer rendered game. The scenario is there, but unless it is rendered on the screen, it is formless. So objective reality is here, but unless it is 'rendered' on consciousness, it is formless. == Objective reality exists == === Arguments for === * {{Argument for}} Humans have physically similar brains and nervous systems. These similarities result in similar interpretations of similar stimuli. The fact that two people, located side by side, and looking in the same direction, describe similarly what they see, reasonably indicates that they are reacting to a single external reality. In the case of hallucinations, radical differences in two people's descriptions of a single aspect of a presumed single external reality, can occur. Some such differences can repeatably be demonstrated to coincide with chemical or structural differences in one nervous system as compared to the other. When such demonstrably dissimilar chemistry or structure are returned to similar states, the re-emergence of the similar descriptions are likely to return as well. Again, this fact is reasonably attributable to there being a single external reality, and multiple various internal realities whose differences are a result of various nervous systems functioning with various abilities, structures, chemistries, locations, experience and knowledge. ** {{Objection}} Does agreement of description necessarily imply objectivity? Imagine if half of humankind saw the color "green" as the color "red". Regardless of how true this scenario is, note that there would be a clear subjectivity of how we view the environment. Half the population would see a different color than the other half. We would no longer agree, and couldn't agree. **{{Objection}} Is it necessarily true that we share a physical brain structure? Certainly, according to science, this is very true, but that says nothing about the existence of an objective reality, because where science comes from is perceived, whether directly by us or through the measurements and assumptions we make, which is an inherently subjective process. This argument takes subjective ideas to be objective, and draws assumptions from them. * {{Argument for}} There is no evidence that objective reality doesn't exist, so [[Wikipedia:Occam's razor|Occam's razor]] suggests we ought to accept it as the simplest possible explanation for reality. ** {{Objection}} If a particle or set of particles in quantum physics can be in 2 different contradictory states at the same time, as in the [[Wikipedia:Schrödinger's cat|Schrödinger's cat]] paradox, that could be considered evidence to the contrary. *** {{Objection}} Quantum physics might seem contradictory, and in fact it is with traditional logic, but if one uses [[Wikipedia:Quantum logic|quantum logic]], which theoretical physicists have developed and which is the same as traditional logic other than abandoning the law of commutativity, there are not any contradictions in it at all. ** {{Objection}} Occam's Razor is not necessarily true, it is just a general guideline to help people guess the most likely answer to a question. It's still wrong sometimes. In the case of objective reality, at best it proves that it's "more likely" to exist than not, whatever that means. **{{Objection}} The concept of "simple" is subjective due to the subjectivity of the construction of definitions, so it cannot be used to prove the existence of objectivity, which requires the definitions to be true regardless of perspective (or lack of perspective). * {{Argument for}} The theory that we all exist in an objective reality is the best theory anyone has ever come up with to describe the conditions in which we appear to exist, so unless anyone comes up with a better theory we should stick with it. ** {{Objection}} We don't necessarily need to assume anything at all regarding reality and whether or not an objective reality exists. We can remain agnostic on the subject until there is sufficient evidence to prove things one way or the other. *** {{Objection}} Being agnostic on the subject is not a position against the existence of objective reality, but instead a neutral position that doesn't take either side in the argument. I have not entirely made up my mind 100% sure either way on objective reality myself, but I am about 99% sure that objective reality exists. I could call myself agnostic on the subject too, but those of us who are agnostic on the subject ought to lean in favor of whatever seems to be the most probable explanation. ** {{Objection}} This just assumes that it exists for the sake of ease (similar to the Occam's Razor proof above). As with that one, I point out that "best" is subjective in nature, and what you consider "best" may be different from what I consider "best". I too, think that there exists an objective reality, but to assume it just exists is just circumventing the purpose of debating this. That is not to say that you cannot have your own opinion on this matter, only that your opinion is not sufficient for a proof. * {{Argument for}} All realities exist, including both objective and subjective realities, which exist as parallel realities or as a kind of multiverse. In other words, the philosophy of {{W|modal realism}} is correct. ** {{Objection}} Modal realism is too radical of a multiverse theory to take seriously. It proposes that all proposed fictional worlds (not just proposed real worlds) actually exist. *** {{Objection}} This is an attempt to shame people into not believing the argument. Sometimes radical viewpoints turn out to be true. ** {{Objection}} This doesn't prove objective reality, because it doesn't prove that all realities exist. If all realities existed, then objective reality would exist too, but it isn't necessarily true that all realities exist. * {{Argument for}} If objective truth doesn't exist, that fact would be in and of itself an objective truth, disproving itself. ** {{Objection}} This only proves the existence of an objective truth, not of an objective reality. No argument is given to prove that an objective truth implies an objective reality. * {{Argument for}} Some facts have no non-objective interpretation, and are universally accepted as objectively true. Example: ''The Statue of Liberty exists in Upper New York Bay.'' There is no alternative, subjective interpretation possible. ** {{Objection}} The statue and location are known by other names in other languages. There are no universally agreed names for anything. *** {{Objection}} Changing the name of the statue or location does not change the fact that everyone can perceive that the statue does indeed exist in objective reality. ****{{Objection}} Blind people can not see the statue in the same way. Nor can people who have lost their sense of touch feel the statue in the same way. Our perception depends on our available biological configuration as it were. *{{Argument for}} Since our realities all interact and we are able to independently observe the same things and do experiments that have reproducible results, clearly we are all part of the same reality. ** {{Objection}} Not necessarily, you could be a figment of my imagination, or we could both be figments of somebody else's imagination. ***{{Objection}} The human mind is not creative enough to imagine the reality we find ourselves in. **{{Objection}} Maybe nobody exists at all and everything is imaginary. *** {{Objection}} As Descartes said, "I think, therefore I am." **** {{Objection}} "I think, therefore I am" is a logical fallacy because fictional characters in a story can say that, but it doesn't turn them into real people. ***** {{Objection}} Simply because a fictional representation of an actual phenomena is presented to the reader does not invalidate it as an actual phenomena outside the fiction. ****{{Objection}} The idea that "I think therefore I am" is based off of a certain assumption: that the "I" in question really does think. I'm sure that just about anyone will say that they have subjective (though objective can be argued) proof to themselves that they indeed do think, simply by being able to wonder if they do. But in this case, sentience is being questioned, and thus the ability to fulfil the criteria of thinking in the first place is as well. === Arguments against === * {{Argument against}} Starting from an objective perspective, the objective world exists and the subjective one does not. "Feelings", for instance, exist only as neuron firing patterns and not as they are felt subjectively, because that would not fit into the objective world view. Similarly, starting from a subjective perspective, the objective reality does not exist. Objects exist only as their perceptions and affects on subjective reality, not as a thing-in-itself. Because I surely perceive from a subjective perspective, I assert the denial of an objective reality. So both realities are mutually exclusive. Any starting perception would deny the other reality's existence. * {{Argument against}} The existence of an objective reality has never been proven, and all of what we think of reality could just be something similar to the Matrix or a computer simulation or we could all just be characters in someone's dream. ** {{Objection}} Those are all unlikely scenarios which are similarly unproven. **{{Objection}} Even if the Matrix scenario were real, the Matrix itself exists within a larger reality that actually is objective, and in that reality, the people who exist in the Matrix all have physical bodies hooked up to the Matrix, so even if the Matrix turned out to be real, we would still be living in an objective reality. Similarly with the other scenarios. * {{Argument against}} There are multiple possible scenarios in which objective reality is wrong, and only one in which it is correct. If we assign an equal level of probability to each one, objective reality is actually highly unlikely. ** {{Objection}} There is no rational basis for assigning an equal level of probability to the idea that we are all living in the Matrix just like in that movie and the idea that we all actually exist in one single objective reality. *** {{Objection}} Simply because a scenario makes "the most sense" does not mean that it is correct (see refutations to "Occam's Razor" above). Similarly, just because a scenario is unlikely/unproven, does not mean that it is false (just look to the numerous unproven conjectures in mathematics as an example - both the ones that are currently unproven but seem to be true, and the ones that used to seem true but are now proven to be false). **{{Objection}} As a matter of fact, probability shouldn't really be assigned here; this is to prove whether or not an objective reality exists, not if we can ever find such an objective reality. The two arguments are different (though one directly proves the other). * {{Argument against}} Reality nowadays is just too ridiculous to be true. Too many practically unbelievable things keep being reported as factual. ** {{Objection}} This reflects a failure of imagination on the part of the human mind to understand and accept objective reality. It does not disprove objective reality. As the saying goes, "Truth is stranger than fiction." This would seem to indicate that, if reality seems incredibly ridiculous and weird, it probably really is real, since it is many times weirder than anything the limited capacity of the human imagination could possibly think up using only the creativity of a single finite mortal human mind. *** {{Objection}} Multiple people can work together to create a very intricate and detailed narrative that runs counter to reality. Who is to say that the most popular narrative used to describe what people call "objective reality" is not similarly false, with its elaborate complexity and absurdity the result of the cooperation of many people? **** {{Objection}} There is a major qualitative difference between double-blind peer-reviewed scientific studies with reproducible results and conspiracy theories or tall tales that are simply made up. Your argument deliberately ignores this in order to equate the two and dismiss both of them. * {{Argument against}} We all live in our own subjective realities. The human mind is not capable of being truly objective. Therefore, the entire idea of a single objective reality is purely speculative, an assumption that, while popular, is not necessary. ** {{Objection}} Why must there exist only one objective reality? This has not been proven (though it is not particularly relevant to the point of the argument). **{{Objection}} This isn't about the necessity of believing whether or not an objective reality exists, this is about whether or not an objective reality exists. * {{Argument against}} {{W|Postmodernism}} is one of many philosophies that does not believe in any objective reality and successfully undermines the idea of objective reality. ** {{Objection}} Postmodernism is rather absurd, either sheer nihilism or some form of relativism, disbelieving in things that are obvious facts. *** {{Objection}} "Obvious" is subjective - though saying that doesn't take much away from the argument. I will preface my objection by stating that I have not personally researched postmodernism in depth, so I do not know what facts are exactly disbelieved by postmodernism. That said, postmodernism is built upon skepticism of facts in general, which is understandable, considering that our facts come from what we observe, and our perspectives are subjective. Thus, disbelief of facts in general is not particularly absurd. As an example, flat-earth was assumed to be fact for an incredibly long time (and by some people, it still is assumed to be true), but was eventually disproved. There's no reason that any of our present facts won't have the same thing happen to them either. ** {{Objection}} Postmodernism has not been proven. *** {{Objection}} Postmodernism doesn't have to be 'proven'. **** {{Objection}} In the context of this debate, postmodernism, as a matter of fact, does need to be proven. The argument is assuming that postmodernism is true, but does not prove that the ideas and beliefs expressed by postmodernism are factual. As of right now, there is no particular reason for me to believe that postmodernism is "true" - that is, the concepts that form postmodernism are objectively true. * {{Argument against}} Again one must consider using which parameters is objective reality to be understood. How is it perceived or measured. To be truly objective, objective reality must be perceived and measured without any limits or filters. Because any such limitations arise from a subjective viewpoint. For example, humans perceive 0.0035% of the entire electromagnetic spectrum. But the entire electromagnetic spectrum must be included in an understanding of objective reality. Any scientific instrument is likewise circumscribed in its ability to measure everything. Therefore what is objective must be unlimited - everything that is possible. And that means the whole universe without limits. All of it including dark matter, dark energy etc. And such an understanding is only theoretically possible for the whole universe itself to have. We humans are permanently in subjective reality, as are all conscious life forms. ** {{Objection}} Objective reality must exist independent of subjective reality. Just because we do not or cannot perceive it, does not mean it does not exist. * {{Argument against}} As it is impossible to see the world, that is the eyes are like cameras only receiving light, they supposedly get two images, somehow these images get merged as one and then projected as "the world around us". The same can be said of all the senses. And, this sense of world includes the body which is external to that which sees it. ** {{Objection}} What does this say about the existence of an objective reality? Is an objective reality necessarily perceived? It certainly isn't by the definition above, and it certainly isn't by my definition. * {{Argument against}} The fact that objective reality is a debatable concept makes objective reality subjective. ** {{Objection}} Debating its existence does not make its existence subjective, it just means that we do not have enough evidence (and may never have the evidence) to prove that it exists. For instance, debating if the twin prime conjecture is true does not disprove it, only shows that we do not have enough information to currently prove it. ** {{Objection}} Debatability does not imply subjectivity. A debate could be caused by a mistaken interlocutor, with the debated subject nevertheless being objective. * {{Argument against}} Recent physics experiments suggest objective reality isn't real but directly relative to individual perceptions, and is basically a subjectively manifested, consciously perceived, and somewhat mutually agreed upon collective manifestation.<ref>{{Cite web |url=https://www.technologyreview.com/2019/03/12/136684/a-quantum-experiment-suggests-theres-no-such-thing-as-objective-reality/ |title=A quantum experiment suggests there's no such thing as objective reality |website=MIT Technology Review |language=en |access-date=2021-10-03}}</ref><ref>{{Cite web |url=https://interestingengineering.com/new-physics-experiment-indicates-no-objective-reality |title=New Physics Experiment Indicates There's No Objective Reality |date=2021-08-30 |website=interestingengineering.com |language=en-US |access-date=2021-10-03}}</ref> ** {{Objection}} The experiment that the article refers to has tested the Wigner's friend thought experiment where the friend's "measurement" seems to not be a proper observation (which would collapse the wavefunction) and instead is a unitary action on the state being measured, and the results of the experiment (unsurprisingly) reflect that. The result simply means that it doesn't matter which observer collapses the wavefunction as long as they are all entangled, but that was never debated in the first place. In day-to-day scenarios we deal with situations where (we assume) the quantum system has already collapsed, which is why we don't see most quantum effects in daily life and why things like the Schrodinger's cat never happen. This "subjectivity" in the experiment only holds for unobserved quantum systems. == See also == * [[Does God exist?]] * [[Do humans have free will?]] == Notes and references == {{Reflist}} [[Category:Philosophy]] mndwbfe18cnfdxpot2djm23o1h3jx1n 0 249885 2410809 2410026 2022-08-01T18:59:37Z Parodda 2936296 Added more detailed descriptions for the Miller and Humphreys talks wikitext text/x-wiki == '''Addresses''' == === '''''Future Directions Address 1: "Future Directions in Adversity and Mental Health"''''' === '''Presented by Dr. Kate McLaughlin, Ph.D.''' ==== Description ==== Research has shown that experiences of childhood adversity are more common than previously thought among children in the US. Furthermore, children who have experienced more than six instances of childhood adversity are five times more likely than their peers to develop at least one clinical manifestation of psychopathology by adulthood. Yet, many current approaches to child psychopathology have a tendency to lump vastly different experiences of adversity together - a practice that has likely served to further complicate efforts to identify an underlying neurological mechanism linking childhood adversity and trauma. In her address, McLaughlin identifies level of threat and level of deprivation as two distinct contextual axes of childhood adversity, influencing underlying neurological mechanisms and eventual manifestation of psychopathology in distinctly different ways. The threat exposure axis pertains to the harm or threat of harm that a child may experience during their developmental years. As McLaughlin discusses, threat exposure influences neural systems involved in fear learning and salience processing. This adaptive response to growing up in a dangerous environment can be essential for the child’s survival in the short term, but as children grow up these changes can result in increased emotional reactivity and poor emotion regulation. McLaughlin details how her lab used brain maps to identify and track possible neurological mechanisms that explain a significant portion of the relationship between childhood trauma and the general psychopathology (p) factor. Pulling in additional lab data, McLaughlin explains that sustained childhood exposure to threat results in children having a harder time distinguishing between what is safe and what is dangerous, and likely perceiving a broader set of stimuli to be potentially dangerous compared to children who have never experienced trauma. McLaughlin states that a key additional implication of this data is that the poor cross talk between the amygdala and hippocampus that occurs as a result of trauma serves as a direct predictor of higher levels of psychopathology across the board. McLaughlin then shifts the focus of her discussion to discuss possible mechanisms linking the p factor and deprivation during key developmental years. McLaughlin begins by stating that children with inadequate exposure to caregivers are not exposed to many of the stimuli necessary for optimal learning and development, including language, nurturing, and complex interpersonal stimulation. It is hypothesized that the result of this deprivation is an exaggerated experience of synaptic pruning, wherein unused neural circuits are destroyed. While synaptic pruning is an important component of neural development, exaggerated synaptic pruning can lead to a dramatic reduction in the density of areas of the brain relating to language processing, social cognition, attention, executive function, and working memory, describing how these data apply to children growing up in poverty in the US. Watch the YouTube video recording of the address [https://www.youtube.com/watch?v=ceZUPrRHxlo here]. === '''''Future Directions Address 2: "Future Directions in Mediators of Treatment"''''' === '''Presented by [[wikipedia:Philip_C._Kendall|Dr. Philip Kendall, Ph.D]].''' ==== Description ==== How do psychological therapies work? How can we enhance treatment to improve outcomes? Questions of mediation lie at the heart of these inquiries. In this address, Dr. Philip Kendall delineates some of the issues confronting tests of treatment mediation in youth mental health and suggests future directions in research on addressing these issues.   Kendall begins his address by defining predictors, moderators, and mediators within the context of clinical psychology. In contrast to both predictors and moderators, a mediator is a variable that has implications for possible mechanisms. A change in a mediating variable will tell you something about the possible mechanism of change in outcome. Importantly, mediators have temporal precedence and can help explain observed outcomes within a clinical or experimental setting. As Kendall points out, there are very few instances of mediation in child and adolescent outcome studies. In order to identify a mediating variable, one has to test each possible variable in order to understand all of the potential mediating factors. Additionally, there is a need for full temporal precedence in measurement and identification of a mediator. In identification and analysis of mediating variables, weekly study is ideal. However, surveying participants at such a high frequency over the course of a trial can be challenging, since responses tend to decrease in complexity and frequency over time if a significant incentive is not offered. Because of this tendency, psychologists must balance the intrusive quality of such measurements against the minimum amount of inquiries necessary to properly identify and examine a mediating variable. As Kendall mentions, burst-style surveying presents a possible solution to upholding this balance. Kendall discusses in detail the limitations of prior group-based approaches, pointing out the dangers of assuming that change over time is equivalent across participants, that change occurs as a result of the same mechanism, and that change is linear. Kendall emphasizes the need for more frequent measurement in mediator analysis, highlighting the use of dynamical systems approaches to test potential mechanisms of mediation. Kendall concludes his discussion by touching on several potential novel study designs to best facilitate medication, emphasizing Bayesian approaches. To summarize his discussion, Kendall leaves the audience with several mechanisms of incorporating mediation analysis into their research: implementing high-frequency data collection methods, establishing temporal precedence, and considering the use of person-centered approaches. Watch the YouTube video recording of the address [https://www.youtube.com/watch?v=0w0ul3YH17o here]. === '''''Future Directions Address 3: "Future Directions in Immunology and Mental Health"''''' === '''Presented by Dr. Greg Miller, Ph.D.''' ==== Description ==== An established body of research has identified significant disparities in health outcomes between racial and ethnic groups, patterning linearly by socioeconomic status (SES). These disparities start early in life, continue throughout the life course, and cannot be explained by genetics, lifestyle choices, or lack of access to healthcare alone. In this address, Miller seeks to identify both the biological mechanisms linking childhood disadvantage with risk for health problems across the lifespan, as well as potential protective/vulnerability factors that could be used to assess health risk, with the end goal of using these results to inform better policy and practice. Miller begins by conceptualizing the immune system as a key interface between the social environment and physical/mental health. In his discussion of the immune interface, Miller places special focus on non-resolving, low-grade inflammation as an identified underlying factor in many chronic health conditions that pattern by SES. As Miller explains, childhood socioeconomic disadvantages put kids at greater risk of negative social and physical exposures. These experiences profoundly shape our our brains develop, and how our brain/body system learns to engage with stressors. Kids who live in low SES environments display an exaggerated immune response compared with their peers. This exaggerated immune response hangs around in stem and progenitor cells, perpetuating a cycle of negative health outcomes throughout the lifespan. However, as Miller notes, having a nurturing mother has been shown to almost completely negate the pro-inflammatory effects of SES, pointing toward a great potential for parenting interventions to stop this pro-inflammatory cycle. Starting as early as pregnancy, Miller discusses multiple novel horizons for parenting and child development interventions focused on decreasing the gap in health outcomes by racial/ethnic groups and SES. Watch the YouTube video recording of the address [https://www.youtube.com/watch?v=kgu3lnyZNmA&feature=youtu.be here]. === '''''Future Directions Address 4: "Future Directions in Parent-Child Separation"''''' === ''' Presented by Dr. Kathryn Humphreys, Ph.D.''' ==== Description ==== Modern societies have long since struggled to understand how to best provide care for orphaned/neglected children. As issues involving child/caregiver separation become increasingly present in the political and cultural spotlight amidst immigration debates, the relationship between early childhood adversity by way of neglect and later manifestation of psychopathology has become especially relevant. As Humphreys emphasizes, the presence of a nurturing relationship between a caregiver and a child is absolutely essential for normal development. In almost every domain of psychopathology there exists a correlation between early live separation/neglect and later risk of psychiatric illness, suggesting that there is something about the caregiver/child separation itself that is traumatizing. Throughout her discussion, Humphreys incorporates data to back up her claim that children who are separated from their caregivers (such as immigrants at the US/Mexico border) are hit with a "double whammy" of trauma, both from losing the presence of a committed caregiver, and from having to navigate adverse conditions without the presence of that caregiver. Pre- and post-separation adverse experiences can compound the effect of this double trauma, leading to increased rates of attachment disorders, internalizing/externalizing disorders, and callous-unemotional traits in these children. Given the importance of attachment processes in early life, Humphreys discusses how timing and sensitive developmental periods may be particularly relevant in the study of parent-child separations. In addition to discussing multiple neurological mechanisms of stress stemming from neglect, Humphreys frames adolescence as a second sensitive period in development, in addition to the first five years of life. As Humphreys explains, the plasticity associated with adolescence, taken in conjunction with the face that development does not end with infancy, points to the potential for interventions focused on building healthy caregiver/child relationships during adolescence and late childhood to impact the later physical and mental health of children who experienced early-live neglect in a positive manner. Watch the YouTube video recording of the address [https://www.youtube.com/watch?v=fq_OHfGyF0o&feature=youtu.be here]. == '''Workshops''' == === '''''Strategies for Improving Writing Clarity''''' === '''Presented by Dr. Andres De Los Reyes, Ph.D.''' ===== Description ===== People tend to be drawn to and understand information best when it is communicated to them in the form of a narrative or “story” rather than a list of facts. However, researchers rarely receive formal training on leveraging narrative tools when writing about their academic work. In this workshop, Dr. Andres De Los Reyes describes evidence-based strategies for consistently applying narrative structure to academic work, with a focus on preparing manuscripts for submission to peer-reviewed academic journals. This includes his description of the and-but-therefore approach to writing and how this narrative structure can be utilized in academic writing as a way to make scientific information more interesting to consume and memorable to the audience. === '''''Job Options in Academia''''' === '''Presented by Dr. Susan White, Ph.D and Dr. Matthew Lerner, Ph.D. ''' ===== Description ===== Graduate training in fields relevant to child and adolescent mental health (e.g., Education, Psychiatry, Psychology, and Social Work) prepares trainees for careers in a variety of policy, research, and practice settings. While there are many options one can take with a career in mental health, academic jobs are among the most common and include traditional academic settings such as R1 research universities and research positions in a medical school, the government, or a research center. Another big consideration is whether or not to go for a job with tenure track (which is like a probationary period that typically lasts about 6 years) especially because these positions are very difficult to come by. There are also considerations to be made on whether to take a position that requires/allows teaching and service work like serving on an editorial board. Research finds that the best predictor of success in academia is: department reputation. Drs. White and Lerner give advice on the timeline of when to apply and resources to use on your applications such as your advisor, organization, and practice. They also provide brief discussion on job interviews which mostly consist of job talks in academia. Lastly, the give advice on negotiating a job offer with the most important piece of advice being to '''get everything in writing''' when negotiating a position. === '''''Preparing a Training Grant''''' === '''Presented by Dr. Deborah A.G. Drabick, Ph.D. and Dr. Tara Peris, Ph.D. ''' ===== Description ===== Submitting a training grant involves considering multiple factors that focus on not only a proposed study but also a concrete plan for developing the skills needed to execute this study. By construction, these applications carry many expectations, requirements, and complicated forms. In this workshop, Drs. Deborah Drabick and Tara Peris leverage their years of experience with extramural funding to clarify the process of submitting a training grant, and provide attendees with concrete tools for submitting successful training grant applications. Training grants are important because they offer opportunities to graduate students and post-docs that may not otherwise be available and provide additional mentorship and consultation. There are several types of training grants that typically fall into either F grant class or K grant class. Based on the type of grant you are applying for, the information needed to apply will vary; however, picking your topic and telling your story will always be required. In doing this, you should describe what you are interested in researching, the current state of the field, and how your research will address gaps you have identified in the current research. Grants are typically reviewed on a scale of 1-9 in 5 areas which are then compiled into an overall impact score. Reviewers then meet and discuss the top 50% of the grant applications and make a decision on who will receive funding. === '''''Responding to Peer Review''''' === '''Presented by Dr. Andres De Los Reyes, Ph.D.''' ===== Description ===== Publishing academic work often involves submitting scholarly manuscripts to peer-reviewed journals. A key component of the publishing process involves receiving commentary about your work from peers in your field and satisfactorily responding to such commentary. Despite it being a core feature of the publishing process, researchers rarely receive formal training on responding to peer review commentary. In this workshop, Dr. Andres De Los Reyes describes evidence-based strategies for responding to peer review commentary, including strategies for how to compose cover letters for responding to such commentary. Along these lines, Dr. De Los Reyes details how to respond to peer reviews line by line to ensure all comments have been addressed. Additionally, he outlines how to respectfully respond to and address comments from a reviewer that you may not agree with. === '''''Job Search and Negotiation''''' === ''' Presented by Dr. Deborah A.G. Drabick, Ph.D. and Dr. Tara Peris, Ph.D. ''' ===== Description ===== In this workshop Drs. Deborah Drabick and Tara Peris provide advice when it comes to searching for jobs and negotiating job offers. They give practical advice for determining when to go on the job market, where are the best places to search for job openings, how to write a good cover letter that will catch an employer's attention, and where to find resources that will assist in the preparation for a job interview. It is always good to monitor list-servs from APA, ABCT, and other relevant groups for job postings as well as checking APA psyc careers and following up via word of mouth. When applying for jobs, you should have your CV up to date and a cover letter ready that details your experiences and qualifications. They also detail how to advocate for yourself, negotiate salary and benefits, and how to get a lab startup package when given a job offer. When negotiating you can negotiate on salary, title, space, start-up funds, time between moving from your current place to your new job, protected time, parking, childcare, and housing. You should always ask for more than what you want since you are negotiating and will not get everything you ask for. Lastly, the remind you to always be gracious when interviewing and negotiating job offers as well as to send thank you notes to those you interact with during the process. === '''''Preparing a Grant Post-Ph.D''''' === ''' Presented by Dr. Joshua Langberg, Ph.D. and Dr. Susan White, Ph.D. ''' ===== Description ===== Submitting your first grant as a Ph.D. can appear on the surface to be a daunting task, with many expectations, requirements, and complicated forms. In this workshop, we leverage years of experience with extramural funding to explain the grant submission process, and provide attendees with concrete tools for submitting successful applications via multiple post-Ph.D. mechanisms, including project grants and K Series, F Series, and T Series applications. K Series grants are most often used to fund early career research, F Series are individual fellowships, and T Series are institutional training grants. Each grant will have different elements specific to that grant; however, each will require picking a topic/telling your story, showing preliminary data, forming a team, and budgeting. It is important to establish a timeline for writing the grant and submitting materials to ensure that everything is submitted on time. You should ideally have multiple grants under review at one time to maximize the chance of having a successfully funded grant. Once you have written one, it is easy to tweak it to meet the requirements of other grant applications. Lastly, ask others in the field for examples of grants they have written and ask them to review yours before submitting. === '''''Strategies for Developing a Research Program''''' === ''' Presented by Dr. Andres De Los Reyes, Ph.D. ''' ===== Description ===== Our first two writing workshops dealt with applying narrative tools to academic work and responding to peer review commentary, with the key goal of publishing a single journal article. How might you use these tools to connect separate articles together into a larger story? In research, our larger stories are the “research programs” we build from years of work and multiple articles. These are the stories we take with us “on the road” when interviewing for jobs and applying for grants. In this workshop, we discover how narrative devices commonly used in filmmaking actually help us weave related but distinct articles together into the “story” of an entire body of work. === '''''Networking at Conferences''''' === ''' Presented by Dr. Deborah A.G. Drabick, Ph.D. and Dr. Matthew Lerner, Ph.D. ''' ===== Description ===== To an early career scientist, attending professional meetings can be an overwhelming experience, with many opportunities to not only learn new things but also connect with like-minded scholars in the field. In this workshop, Drs. Deborah Drabick and Matthew Lerner demystify the process of networking at conferences, and provide attendees with concrete tools for developing and maintaining professional relationships with conference attendees. The presenters discuss how to approach a researcher who you are interested in building a connection with and how to maximize the time you have to speak with them and to make yourself stand out. Drs. Drabick and Lerner also discuss how to have a meaningful discussion that can carry into a long-term professional relationship such as collaborations on research projects. It is also important to be mindful of how you are carrying yourself and what your interactions look like as you never know who is around and watching. Having a go to question can be a good way to start the conversation such as asking what the current state of that person's research is or discussing the current state of the literature in their field of study. Always make sure you know about the person's research before approaching them and consider how their work can inform yours. It is also important to attend their session if they are speaking at the conference or event so you can be informed on their current work and research goals. === '''''Work-Life Balance''''' === '''Presented by Dr. Joshua Langberg, Ph.D. and Dr. Sarah Racz, Ph.D.''' ===== Description ===== Sometimes it feels like everyone in our field is “always on task” and unable to “unplug”. But is that a realistic view of how we balance our work lives with our lives outside of work? In this workshop, Drs. Joshua Langberg and Sarah Racz discuss the competing demands placed on us across our various work, family, and social spheres; and strategies to manage these demands in the necessary pursuit of healthy, balanced lives. They begin with practical advice such as knowing what your priorities are and how you operationalize them. Then map your schedule so you have a clear picture of your time and your commitments. From there you will need to make choices out of the time items you have left on your priorities and what time you have left in the day. By having routine and structure for each day, you are able to maximize your time and make conscious choices of what you are spending your day doing. They also recommend being flexible, but organized when creating a work-life balance as sometimes things in your schedule may shift and you need to quickly adjust. Being organized may look like keeping a calendar and/or to do list as well as using family management apps. It is also important to communicate with others in your household about your schedule and set up a plan that works for all of you. Lastly, it is important to advocate for yourself and to know your rights and responsibilities if you are a working parent when it comes to parental leave and time off. == '''Ceremony for the ''Future Directions Launch Award''''' == === John L. Cooley === * Received Ph.D. from the University of Kansas ====== About the award recipient ====== John is a recipient of the 2019 Future Directions Launch Award in Adversity.  After receiving his Ph.D. in Clinical Child Psychology from University of Kansas in 2018, he worked as a NIMH T32 postdoctoral fellow in the Department of Psychiatry at the University of Colorado Anschutz Medical Campus. He is currently a tenure-track Assistant Professor in the Department of Psychological Sciences at Texas Tech University, where his lab is guided by two overarching questions: “Why are some children and adolescents more impacted by peer victimization/bullying than others?” and “How can we address the mental health needs of peer-victimized/bullied youth?" More specifically, John’s lab is focused on investigating a) factors that influence risk for peer victimization/bullying and their associated negative outcomes, b) methods for identifying victims of peer aggression in need of intervention, and c) prevention and intervention approaches. Learn more about John's lab here: [https://www.peerrelationslab.com www.peerrelationslab.com] Watch the YouTube video recording of the remarks [https://www.youtube.com/watch?v=Jq6k0URJ1es here]. === Erin Kang === * Ph.D. Candidate at Stony Brook University at the time of the award ====== About the award recipient ====== Erin received the 2019 Future Directions Launch Award in Treatment. She earned her Ph.D. in Clinical Psychology in 2020 at Stony Brook University under the mentorship of Dr. Matthew Lerner. She is currently a tenure-track Assistant Professor in the Psychology Department at Montclair State University, where her lab focuses on understanding how the processing of social information shapes, and is shaped by, social experience in autistic youth and those with related neurodevelopmental disorders. This focus on social plasticity, or capacity to learn from and adapt to their complex social environments, includes the role of social experiences, affective processing, and neural plasticity that underlie this capacity. Learn more about Erin's lab here: [https://www.erinkanglab.com www.erinkanglab.com] Watch the YouTube video recording of the remarks [https://www.youtube.com/watch?v=oaK14hoO3UI here]. === Nicole Lorenzo === * Ph.D. candidate at Florida International University at the time of the award ====== About the award recipient ====== Nicole received the 2019 Future Directions Launch Award in Treatment. After receiving her Ph.D. in Clinical Science at Florida International University in 2019, she worked as a post-doctoral fellow at the University of Maryland, College Park. Currently, she is a tenure-track Assistant Professor in the psychology department at American University. Her research focuses on the transactional processes involved in parent-child interactions, examining how these processes develop and the impact of factors like temperament and parent mental health. This developmental work informs her intervention work which seeks to understand how we can refine and individualize treatment targets to develop early intervention programs that are accessible and scalable for families and providers, particularly those from underserved and underrepresented backgrounds. Learn more about Nicole's work here: [https://www.researchgate.net/profile/Nicole&#x20;Lorenzo www.researchgate.net/profile/Nicole_Lorenzo] Watch the YouTube video recording of the remarks [https://www.youtube.com/watch?v=3T53yHtdcqY here]. hdn8oo1txhlmpk6bkgpgeyrjy6mgj4a 3 250786 2410828 2409290 2022-08-01T21:21:53Z MediaWiki message delivery 983498 /* Tech News: 2022-31 */ new section wikitext text/x-wiki == First Message Posting == Update Talk on Wikiversity [[User:Bnhassin|Bnhassin]] ([[User talk:Bnhassin|discuss]] • [[Special:Contributions/Bnhassin|contribs]]) 21:04, 29 June 2019 (UTC) == Update Sandbox User == == Posting to sandbox == Update Sandbox on Wikiversity [[User:Bnhassin/sandbox]] ([[User talk:Bnhassin|discuss]] • [[Special:Contributions/Bnhassin|contribs]])[[User:Bnhassin|Bnhassin]] ([[User talk:Bnhassin|discuss]] • [[Special:Contributions/Bnhassin|contribs]]) 11:04, 25 October 2020 (UTC) == [[m:Special:MyLanguage/Tech/News/2020/50|Tech News: 2020-50]] == <section begin="technews-2020-W50"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2020/50|Translations]] are available. '''Recent changes''' * You can now put pages on your watchlist for a limited period of time. Some wikis already had this function. [https://meta.wikimedia.org/wiki/Community_Tech/Watchlist_Expiry][https://www.mediawiki.org/wiki/Help:Watchlist_expiry] '''Changes later this week''' * Information from Wikidata that is used on a wiki page can be shown in recent changes and watchlists on a Wikimedia wiki. To see this you need to turn on showing Wikidata edits in your watchlist in the preferences. Changes to the Wikidata description in the language of a Wikimedia wiki will then be shown in recent changes and watchlists. This will not show edits to languages that are not relevant to your wiki. [https://lists.wikimedia.org/pipermail/wikidata/2020-November/014402.html][https://phabricator.wikimedia.org/T191831] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2020-12-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2020-12-09|en}}. It will be on all wikis from {{#time:j xg|2020-12-10|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''Future changes''' * You can vote on proposals in the [[m:Special:MyLanguage/Community Wishlist Survey 2021|Community Wishlist Survey]] between 8 December and 21 December. The survey decides what the [[m:Community Tech|Community Tech team]] will work on. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2020/50|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2020-W50"/> 16:15, 7 December 2020 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=20754641 --> == [[m:Special:MyLanguage/Tech/News/2020/51|Tech News: 2020-51]] == <section begin="technews-2020-W51"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2020/51|Translations]] are available. '''Recent changes''' * There is a [[mw:Wikipedia for KaiOS|Wikipedia app]] for [[:w:en:KaiOS|KaiOS]] phones. It was released in India in September. It can now be downloaded in other countries too. [https://diff.wikimedia.org/2020/12/10/growing-wikipedias-reach-with-an-app-for-kaios-feature-phones/] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2020-12-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2020-12-16|en}}. It will be on all wikis from {{#time:j xg|2020-12-17|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2020/51|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2020-W51"/> 21:34, 14 December 2020 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=20803489 --> == [[m:Special:MyLanguage/Tech/News/2020/52|Tech News: 2020-52]] == <section begin="technews-2020-W52"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2020/52|Translations]] are available. '''Tech News''' * Because of the [[w:en:Christmas and holiday season|holidays]] the next issue of Tech News will be sent out on 11 January 2021. '''Recent changes''' * The <code><nowiki>{{citation needed}}</nowiki></code> template shows when a statement in a Wikipedia article needs a source. If you click on it when you edit with the visual editor there is a popup that explains this. Now it can also show the reason and when it was added. [https://phabricator.wikimedia.org/T270107] '''Changes later this week''' * There is no new MediaWiki version this week or next week. '''Future changes''' * You can [[m:WMDE Technical Wishes/Geoinformation/Ideas|propose and discuss]] what technical improvements should be done for geographic information. This could be coordinates, maps or other related things. * Some wikis use [[mw:Writing systems/LanguageConverter|LanguageConverter]] to switch between writing systems or variants of a language. This can only be done for the entire page. There will be a <code><nowiki><langconvert></nowiki></code> tag that can convert a piece of text on a page. [https://phabricator.wikimedia.org/T263082] * Oversighters and stewards can hide entries in [[Special:AbuseLog|Special:AbuseLog]]. They can soon hide multiple entries at once using checkboxes. This works like hiding normal edits. It will happen in early January. [https://phabricator.wikimedia.org/T260904] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2020/52|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2020-W52"/> 20:54, 21 December 2020 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=20833836 --> == [[m:Special:MyLanguage/Tech/News/2021/02|Tech News: 2021-02]] == <section begin="technews-2021-W02"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/02|Translations]] are available. '''Recent changes''' * You can choose to be reminded when you have not added an edit summary. This can be done in your preferences. This could conflict with the [[:w:en:CAPTCHA|CAPTCHA]]. This has now been fixed. [https://phabricator.wikimedia.org/T12729] * You can link to specific log entries. You can get these links for example by clicking the timestamps in the log. Until now, such links to private log entries showed no entry even if you had permission to view private log entries. The links now show the entry. [https://phabricator.wikimedia.org/T269761] * Admins can use the [[:mw:Special:MyLanguage/Extension:AbuseFilter|abuse filter tool]] to automatically prevent bad edits. Three changes happened last week: ** The filter editing interface now shows syntax errors while you type. This is similar to JavaScript pages. It also shows a warning for regular expressions that match the empty string. New warnings will be added later. [https://phabricator.wikimedia.org/T187686] ** [[m:Special:MyLanguage/Meta:Oversighters|Oversighters]] can now hide multiple filter log entries at once using checkboxes on [[Special:AbuseLog]]. This is how the usual revision deletion works. [https://phabricator.wikimedia.org/T260904] ** When a filter matches too many actions after it has been changed it is "throttled". The most powerful actions are disabled. This is to avoid many editors getting blocked when an administrator made a mistake. The administrator will now get a notification about this "throttle". * [[File:Octicons-tools.svg|15px|link=|Advanced item]] There is a new tool to [https://skins.wmflabs.org/?#/add build new skins]. You can also [https://skins.wmflabs.org/?#/ see] existing [[mw:Special:MyLanguage/Manual:Skins|skins]]. You can [[mw:User talk:Jdlrobson|give feedback]]. [https://lists.wikimedia.org/pipermail/wikitech-l/2020-December/094130.html] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Bots using the API no longer watch pages automatically based on account preferences. Setting the <code>watchlist</code> to <code>watch</code> will still work. This is to reduce the size of the watchlist data in the database. [https://phabricator.wikimedia.org/T258108] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[mw:Special:MyLanguage/Extension:Scribunto|Scribunto's]] [[:mw:Extension:Scribunto/Lua reference manual#File metadata|file metadata]] now includes length. [https://phabricator.wikimedia.org/T209679] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:w:en:CSS|CSS]] and [[:w:en:JavaScript|JavaScript]] code pages now have link anchors to [https://patchdemo.wmflabs.org/wikis/40e4795d4448b55a6d8c46ff414bcf78/w/index.php/MediaWiki:En.js#L-125 line numbers]. You can use wikilinks like [[:w:en:MediaWiki:Common.js#L-50]]. [https://phabricator.wikimedia.org/T29531] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] There was a [[mw:MediaWiki 1.36/wmf.25|new version]] of MediaWiki last week. You can read [[mw:MediaWiki 1.36/wmf.25/Changelog|a detailed log]] of all 763 changes. Most of them are very small and will not affect you. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-01-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-01-13|en}}. It will be on all wikis from {{#time:j xg|2021-01-14|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/02|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W02"/> 15:42, 11 January 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=20950047 --> == [[m:Special:MyLanguage/Tech/News/2021/03|Tech News: 2021-03]] == <section begin="technews-2021-W03"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/03|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-01-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-01-20|en}}. It will be on all wikis from {{#time:j xg|2021-01-21|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''Future changes''' * The [[mw:Special:MyLanguage/Growth|Growth team]] plans to add features to [[mw:Special:MyLanguage/Growth/Personalized first day/Newcomer tasks/Experiment analysis, November 2020|get more visitors to edit]] to more Wikipedias. You can help [https://translatewiki.net/w/i.php?title=Special:Translate&group=ext-growthexperiments&language=&filter=&action=translate translating the interface]. * You will be able to read but not to edit Wikimedia Commons for a short time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210126T07 {{#time:j xg|2021-01-26|en}} at 07:00 (UTC)]. [https://phabricator.wikimedia.org/T271791] * [[m:Special:MyLanguage/MassMessage|MassMessage]] posts could be automatically timestamped in the future. This is because MassMessage senders can now send pages using MassMessage. Pages are more difficult to sign. If there are times when a MassMessage post should not be timestamped you can [[phab:T270435|let the developers know]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/03|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W03"/> 16:10, 18 January 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=20974628 --> == [[m:Special:MyLanguage/Tech/News/2021/04|Tech News: 2021-04]] == <section begin="technews-2021-W04"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/04|Translations]] are available. '''Problems''' * You will be able to read but not to edit Wikimedia Commons for a short time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210126T07 {{#time:j xg|2021-01-26|en}} at 07:00 (UTC)]. You will not be able to read or edit [[:wikitech:Main Page|Wikitech]] for a short time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210128T09 {{#time:j xg|2021-01-28|en}} at 09:00 (UTC)]. [https://phabricator.wikimedia.org/T271791][https://phabricator.wikimedia.org/T272388] '''Changes later this week''' * [[m:WMDE Technical Wishes/Bracket Matching|Bracket matching]] will be added to the [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] syntax highlighter on the first wikis. The first wikis are German and Catalan Wikipedia and maybe other Wikimedia wikis. This will happen on 27 January. [https://phabricator.wikimedia.org/T270238] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.28|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-01-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-01-27|en}}. It will be on all wikis from {{#time:j xg|2021-01-28|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/04|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W04"/> 18:31, 25 January 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21007423 --> == [[m:Special:MyLanguage/Tech/News/2021/05|Tech News: 2021-05]] == <section begin="technews-2021-W05"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/05|Translations]] are available. '''Problems''' * [[:w:en:IPv6|IPv6 addresses]] were written in lowercase letters in diffs. This caused dead links since [[Special:Contributions|Special:Contributions]] only accepted uppercase letters for the IPs. This has been fixed. [https://phabricator.wikimedia.org/T272225] '''Changes later this week''' * You can soon use Wikidata to link to pages on the multilingual Wikisource. [https://phabricator.wikimedia.org/T138332] * Often editors use a "non-breaking space" to make a gap between two items when reading but still show them together. This can be used to avoid a line break. You will now be able to add new ones via the special character tool in the 2010, 2017, and visual editors. The character will be shown in the visual editor as a space with a grey background. [https://phabricator.wikimedia.org/T70429][https://phabricator.wikimedia.org/T96666] * [[File:Octicons-tools.svg|15px|link=| Advanced item]] Wikis use [[mw:Special:MyLanguage/Extension:AbuseFilter|abuse filters]] to stop bad edits being made. Filter maintainers can now use syntax like <code>1.2.3.4 - 1.2.3.55</code> as well as the <code>1.2.3.4/27</code> syntax for IP ranges. [https://phabricator.wikimedia.org/T218074] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.29|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-03|en}}. It will be on all wikis from {{#time:j xg|2021-02-04|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''Future changes''' * [[mw:Skin:Minerva Neue|Minerva]] is the skin Wikimedia wikis use for mobile traffic. When a page is protected and you can't edit it you can normally read the source wikicode. This doesn't work on Minerva on mobile devices. This is being fixed. Some text might overlap. This is because your community needs to update [[MediaWiki:Protectedpagetext|MediaWiki:Protectedpagetext]] to work on mobile. You can [[phab:T208827|read more]]. [https://www.mediawiki.org/wiki/Recommendations_for_mobile_friendly_articles_on_Wikimedia_wikis#Inline_styles_should_not_use_properties_that_impact_sizing_and_positioning][https://www.mediawiki.org/wiki/Recommendations_for_mobile_friendly_articles_on_Wikimedia_wikis#Avoid_tables_for_anything_except_data] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:wikitech:Portal:Cloud VPS|Cloud VPS]] and [[:wikitech:Portal:Toolforge|Toolforge]] will change the IP address they use to contact the wikis. The new IP address will be <code>185.15.56.1</code>. This will happen on February 8. You can [[:wikitech:News/CloudVPS NAT wikis|read more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/05|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W05"/> 22:38, 1 February 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21033195 --> == [[m:Special:MyLanguage/Tech/News/2021/06|Tech News: 2021-06]] == <section begin="technews-2021-W06"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/06|Translations]] are available. '''Recent changes''' * The [[mw:Special:MyLanguage/Wikimedia Apps|Wikipedia app]] for Android now has watchlists and talk pages in the app. [https://play.google.com/store/apps/details?id=org.wikipedia] '''Changes later this week''' * You can see edits to chosen pages on [[Special:Watchlist|Special:Watchlist]]. You can add pages to your watchlist on every wiki you like. The [[:mw:Special:MyLanguage/Extension:GlobalWatchlist|GlobalWatchlist]] extension will come to Meta on 11 February. There you can see entries on watched pages on different wikis on the same page. The new watchlist will be found on [[m:Special:GlobalWatchlist|Special:GlobalWatchlist]] on Meta. You can choose which wikis to watch and other preferences on [[m:Special:GlobalWatchlistSettings|Special:GlobalWatchlistSettings]] on Meta. You can watch up to five wikis. [https://phabricator.wikimedia.org/T260862] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.30|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-10|en}}. It will be on all wikis from {{#time:j xg|2021-02-11|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''Future changes''' * When admins [[mw:Special:MyLanguage/Help:Protecting and unprotecting pages|protect]] pages the form will use the [[mw:UX standardization|OOUI look]]. [[Special:Import|Special:Import]] will also get the new look. This will make them easier to use on mobile phones. [https://phabricator.wikimedia.org/T235424][https://phabricator.wikimedia.org/T108792] * Some services will not work for a short period of time from 07:00 UTC on 17 February. There might be problems with new [[m:Special:MyLanguage/Wikimedia URL Shortener|short links]], new translations, new notifications, adding new items to your [[mw:Reading/Reading Lists|reading lists]] or recording [[:w:en:Email#Tracking of sent mail|email bounces]]. This is because of database maintenance. [https://phabricator.wikimedia.org/T273758] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[m:Tech/News/2021/05|Last week]] Tech News reported that the IP address [[:wikitech:Portal:Cloud VPS|Cloud VPS]] and [[:wikitech:Portal:Toolforge|Toolforge]] use to contact the wikis will change on 8 February. This is delayed. It will happen later instead. [https://wikitech.wikimedia.org/wiki/News/CloudVPS_NAT_wikis] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/06|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W06"/> 17:42, 8 February 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21082948 --> == [[m:Special:MyLanguage/Tech/News/2021/07|Tech News: 2021-07]] == <section begin="technews-2021-W07"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/07|Translations]] are available. '''Problems''' * There were problems with recent versions of MediaWiki. Because the updates caused problems the developers rolled back to an earlier version. Some updates and new functions will come later than planned. [https://lists.wikimedia.org/pipermail/wikitech-l/2021-February/094255.html][https://lists.wikimedia.org/pipermail/wikitech-l/2021-February/094271.html] * Some services will not work for a short period of time from 07:00 UTC on 17 February. There might be problems with new [[m:Special:MyLanguage/Wikimedia URL Shortener|short links]], new translations, new notifications, adding new items to your [[mw:Reading/Reading Lists|reading lists]] or recording [[:w:en:Email#Tracking of sent mail|email bounces]]. This is because of database maintenance. [https://phabricator.wikimedia.org/T273758] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.31|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-17|en}}. It will be on all wikis from {{#time:j xg|2021-02-18|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/07|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W07"/> 17:56, 15 February 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21105437 --> == [[m:Special:MyLanguage/Tech/News/2021/08|Tech News: 2021-08]] == <div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/08|Translations]] are available. '''Recent changes''' * The visual editor will now use [[:c:Commons:Structured data/Media search|MediaSearch]] to find images. You can search for images on Commons in the visual editor when you are looking for illustrations. This is to help editors find better images. [https://phabricator.wikimedia.org/T259896] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[mw:Special:MyLanguage/Extension:SyntaxHighlight|syntax highlighter]] now works with more languages: [[:w:en:Futhark (programming language)|Futhark]], [[:w:en:Graphviz|Graphviz]]/[[:w:en:DOT (graph description language)|DOT]], CDDL and AMDGPU. [https://phabricator.wikimedia.org/T274741] '''Problems''' * Editing a [[mw:Special:MyLanguage/Extension:EasyTimeline|timeline]] might have removed all text from it. This was because of a bug and has been fixed. You might need to edit the timeline again for it to show properly. [https://phabricator.wikimedia.org/T274822] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.32|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-24|en}}. It will be on all wikis from {{#time:j xg|2021-02-25|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] There is a [[:m:Wikimedia Rust developers user group|user group]] for developers and users interested in working on Wikimedia wikis with the [[:w:en:Rust (programming language)|Rust programming language]]. You can join or tell others who want to make your wiki better in the future. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/08|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div> ---- 00:17, 23 February 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21134058 --> == [[m:Special:MyLanguage/Tech/News/2021/09|Tech News: 2021-09]] == <div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/09|Translations]] are available. '''Recent changes''' * Wikis using the [[mw:Special:MyLanguage/Growth/Feature summary|Growth team tools]] can now show the name of a newcomer's mentor anywhere [[mw:Special:MyLanguage/Help:Growth/Mentorship/Integrating_mentorship|through a magic word]]. This can be used for welcome messages or userboxes. * A new version of the [[c:Special:MyLanguage/Commons:VideoCutTool|VideoCutTool]] is now available. It enables cropping, trimming, audio disabling, and rotating video content. It is being created as part of the developer outreach programs. '''Problems''' * There was a problem with the [[mw:Special:MyLanguage/Manual:Job queue|job queue]]. This meant some functions did not save changes and mass messages were delayed. This did not affect wiki edits. [https://phabricator.wikimedia.org/T275437] * Some editors may not be logged in to their accounts automatically in the latest versions of Firefox and Safari. [https://phabricator.wikimedia.org/T226797] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.33|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-03|en}}. It will be on all wikis from {{#time:j xg|2021-03-04|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/09|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div> ---- 19:08, 1 March 2021 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21161722 --> == [[m:Special:MyLanguage/Tech/News/2021/10|Tech News: 2021-10]] == <section begin="technews-2021-W10"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/10|Translations]] are available. '''Recent changes''' * [[mw:Special:MyLanguage/Content translation/Section translation|Section translation]] now works on Bengali Wikipedia. It helps mobile editors translate sections of articles. It will come to more wikis later. The first focus is active wikis with a smaller number of articles. You can [https://sx.wmflabs.org/index.php/Main_Page test it] and [[mw:Talk:Content translation/Section translation|leave feedback]]. * [[mw:Special:MyLanguage/Help:Extension:FlaggedRevs|Flagged revisions]] now give admins the review right. [https://phabricator.wikimedia.org/T275293] * When someone links to a Wikipedia article on Twitter this will now show a preview of the article. [https://phabricator.wikimedia.org/T276185] '''Problems''' * Many graphs have [[:w:en:JavaScript|JavaScript]] errors. Graph editors can check their graphs in their browser's developer console after editing. [https://phabricator.wikimedia.org/T275833] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.34|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-10|en}}. It will be on all wikis from {{#time:j xg|2021-03-11|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). * The [[mw:Talk pages project/New discussion|New Discussion]] tool will soon be a new [[mw:Special:MyLanguage/Extension:DiscussionTools|discussion tools]] beta feature for on most Wikipedias. The goal is to make it easier to start new discussions. [https://phabricator.wikimedia.org/T275257] '''Future changes''' * There will be a number of changes to make it easier to work with templates. Some will come to the first wikis in March. Other changes will come to the first wikis in June. This is both for those who use templates and those who create or maintain them. You can [[:m:WMDE Technical Wishes/Templates|read more]]. * [[m:WMDE Technical Wishes/ReferencePreviews|Reference Previews]] will become a default feature on some wikis on 17 March. They will share a setting with [[mw:Page Previews|Page Previews]]. If you prefer the Reference Tooltips or Navigation-Popups gadget you can keep using them. If so Reference Previews won't be shown. [https://phabricator.wikimedia.org/T271206][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/ReferencePreviews] * New JavaScript-based functions will not work in [[:w:en:Internet Explorer 11|Internet Explorer 11]]. This is because Internet Explorer is an old browser that doesn't work with how JavaScript is written today. Everything that works in Internet Explorer 11 today will continue working in Internet Explorer for now. You can [[mw:Compatibility/IE11|read more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/10|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W10"/> 17:51, 8 March 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21175593 --> == [[m:Special:MyLanguage/Tech/News/2021/11|Tech News: 2021-11]] == <section begin="technews-2021-W11"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/11|Translations]] are available. '''Recent changes''' * Wikis that are part of the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|desktop improvements]] project can now use a new [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Search|search function]]. The desktop improvements and the new search will come to more wikis later. You can also [[mw:Reading/Web/Desktop Improvements#Deployment plan and timeline|test it early]]. * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Editors who put up banners or change site-wide [[:w:en:JavaScript|JavaScript]] code should use the [https://grafana.wikimedia.org/d/000000566/overview?viewPanel=16&orgId=1 client error graph] to see that their changes has not caused problems. You can [https://diff.wikimedia.org/2021/03/08/sailing-steady%e2%80%8a-%e2%80%8ahow-you-can-help-keep-wikimedia-sites-error-free read more]. [https://phabricator.wikimedia.org/T276296] '''Problems''' * Due to [[phab:T276968|database issues]] the [https://meta.wikimedia.beta.wmflabs.org Wikimedia Beta Cluster] was read-only for over a day. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.34|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-17|en}}. It will be on all wikis from {{#time:j xg|2021-03-18|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''Future changes''' * You can add a [[:w:en:Newline|newline]] or [[:w:en:Carriage return|carriage return]] character to a custom signature if you use a template. There is a proposal to not allow them in the future. This is because they can cause formatting problems. [https://www.mediawiki.org/wiki/New_requirements_for_user_signatures#Additional_proposal_(2021)][https://phabricator.wikimedia.org/T272322] * You will be able to read but not edit [[phab:T276899|12 wikis]] for a short period of time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210323T06 {{#time:j xg|2021-03-23|en}} at 06:00 (UTC)]. This could take 30 minutes but will probably be much faster. * [[File:Octicons-tools.svg|15px|link=|Advanced item]] You can use [https://quarry.wmflabs.org/ Quarry] for [[:w:en:SQL|SQL]] queries to the [[wikitech:Wiki replicas|Wiki Replicas]]. Cross-database <code>JOINS</code> will no longer work from 23 March. There will be a new field to specify the database to connect to. If you think this affects you and you need help you can [[phab:T268498|post on Phabricator]] or on [[wikitech:Talk:News/Wiki Replicas 2020 Redesign|Wikitech]]. [https://wikitech.wikimedia.org/wiki/PAWS PAWS] and other ways to do [[:w:en:SQL|SQL]] queries to the Wiki Replicas will be affected later. [https://wikitech.wikimedia.org/wiki/News/Wiki_Replicas_2020_Redesign] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/11|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W11"/> 23:22, 15 March 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21226057 --> == [[m:Special:MyLanguage/Tech/News/2021/12|Tech News: 2021-12]] == <section begin="technews-2021-W12"/><div class="plainlinks mw-content-ltr" lang="en" dir="ltr"><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/12|Translations]] are available. '''Recent changes''' * There is a [[mw:Wikipedia for KaiOS|Wikipedia app]] for [[:w:en:KaiOS|KaiOS]] phones. They don't have a touch screen so readers navigate with the phone keys. There is now a [https://wikimedia.github.io/wikipedia-kaios/sim.html simulator] so you can see what it looks like. * The [[mw:Special:MyLanguage/Talk pages project/Replying|reply tool]] and [[mw:Special:MyLanguage/Talk pages project/New discussion|new discussion tool]] are now available as the "{{int:discussiontools-preference-label}}" [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] in almost all wikis except German Wikipedia. '''Problems''' * You will be able to read but not edit [[phab:T276899|twelve wikis]] for a short period of time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210323T06 {{#time:j xg|2021-03-23|{{PAGELANGUAGE}}}} at 06:00 (UTC)]. This can also affect password changes, logging in to new wikis, global renames and changing or confirming emails. This could take 30 minutes but will probably be much faster. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.36|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-24|en}}. It will be on all wikis from {{#time:j xg|2021-03-25|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). * [[:w:en:Syntax highlighting|Syntax highlighting]] colours will change to be easier to read. This will soon come to the [[phab:T276346|first wikis]]. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Improved_Color_Scheme_of_Syntax_Highlighting] '''Future changes''' * [[mw:Special:MyLanguage/Extension:FlaggedRevs|Flagged revisions]] will no longer have multiple tags like "tone" or "depth". It will also only have one tier. This was changed because very few wikis used these features and they make the tool difficult to maintain. [https://phabricator.wikimedia.org/T185664][https://phabricator.wikimedia.org/T277883] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets and user scripts can access variables about the current page in JavaScript. In 2015 this was moved from <code dir=ltr>wg*</code> to <code dir=ltr>mw.config</code>. <code dir=ltr>wg*</code> will soon no longer work. [https://phabricator.wikimedia.org/T72470] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/12|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div></div> <section end="technews-2021-W12"/> 16:53, 22 March 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21244806 --> == [[m:Special:MyLanguage/Tech/News/2021/13|Tech News: 2021-13]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/13|Translations]] are available. '''Recent changes''' * Some very old [[:w:en:Web browser|web browsers]] [[:mw:Special:MyLanguage/Compatibility|don’t work]] well with the Wikimedia wikis. Some old code for browsers that used to be supported is being removed. This could cause issues in those browsers. [https://phabricator.wikimedia.org/T277803] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:m:IRC/Channels#Raw_feeds|IRC recent changes feeds]] have been moved to a new server. Make sure all tools automatically reconnect to <code>irc.wikimedia.org</code> and not to the name of any specific server. Users should also consider switching to the more modern [[:wikitech:Event Platform/EventStreams|EventStreams]]. [https://phabricator.wikimedia.org/T224579] '''Problems''' * When you move a page that many editors have on their watchlist the history can be split. It might also not be possible to move it again for a while. This is because of a [[:w:en:Job queue|job queue]] problem. [https://phabricator.wikimedia.org/T278350] * Some translatable pages on Meta could not be edited. This was because of a bug in the translation tool. The new MediaWiki version was delayed because of problems like this. [https://phabricator.wikimedia.org/T278429][https://phabricator.wikimedia.org/T274940] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.37|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-31|en}}. It will be on all wikis from {{#time:j xg|2021-04-01|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/13|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 17:30, 29 March 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21267131 --> == [[m:Special:MyLanguage/Tech/News/2021/14|Tech News: 2021-14]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/14|Translations]] are available. '''Recent changes''' * Editors can collapse part of an article so you have to click on it to see it. When you click a link to a section inside collapsed content it will now expand to show the section. The browser will scroll down to the section. Previously such links didn't work unless you manually expanded the content first. [https://phabricator.wikimedia.org/T276741] '''Changes later this week''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[mw:Special:MyLanguage/Citoid|citoid]] [[:w:en:API|API]] will use for example <code>2010-12-XX</code> instead of <code>2010-12</code> for dates with a month but no days. This is because <code>2010-12</code> could be confused with <code>2010-2012</code> instead of <code>December 2010</code>. This is called level 1 instead of level 0 in the [https://www.loc.gov/standards/datetime/ Extended Date/Time Format]. [https://phabricator.wikimedia.org/T132308] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.38|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-04-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-04-07|en}}. It will be on all wikis from {{#time:j xg|2021-04-08|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]). '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:wikitech:PAWS|PAWS]] can now connect to the new [[:wikitech:Wiki Replicas|Wiki Replicas]]. Cross-database <code>JOINS</code> will no longer work from 28 April. There is [[:wikitech:News/Wiki Replicas 2020 Redesign#How should I connect to databases in PAWS?|a new way to connect]] to the databases. Until 28 April both ways to connect to the databases will work. If you think this affects you and you need help you can post [[phab:T268498|on Phabricator]] or on [[wikitech:Talk:News/Wiki Replicas 2020 Redesign|Wikitech]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/14|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 19:41, 5 April 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21287348 --> == [[m:Special:MyLanguage/Tech/News/2021/16|Tech News: 2021-16]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/16|Translations]] are available. '''Recent changes''' * Email to the Wikimedia wikis are handled by groups of Wikimedia editors. These volunteer response teams now use [https://github.com/znuny/Znuny Znuny] instead of [[m:Special:MyLanguage/OTRS|OTRS]]. The functions and interface remain the same. The volunteer administrators will give more details about the next steps soon. [https://phabricator.wikimedia.org/T279303][https://phabricator.wikimedia.org/T275294] * If you use [[Mw:Special:MyLanguage/Extension:CodeMirror|syntax highlighting]], you can see line numbers in the 2010 and 2017 wikitext editors when editing templates. This is to make it easier to see line breaks or talk about specific lines. Line numbers will soon come to all namespaces. [https://phabricator.wikimedia.org/T267911][https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Line_Numbering][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/Line_Numbering] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Because of a technical change there could be problems with gadgets and scripts that have an edit summary area that looks [https://phab.wmfusercontent.org/file/data/llvdqqnb5zpsfzylbqcg/PHID-FILE-25vs4qowibmtysl7cbml/Screen_Shot_2021-04-06_at_2.34.04_PM.png similar to this one]. If they look strange they should use <code>mw.loader.using('mediawiki.action.edit.styles')</code> to go back to how they looked before. [https://phabricator.wikimedia.org/T278898] * The [[mw:MediaWiki 1.37/wmf.1|latest version]] of MediaWiki came to the Wikimedia wikis last week. There was no Tech News issue last week. '''Changes later this week''' * There is no new MediaWiki version this week. '''Future changes''' * The user group <code>oversight</code> will be renamed <code>suppress</code>. This is for [[phab:T109327|technical reasons]]. This is the technical name. It doesn't affect what you call the editors with this user right on your wiki. This is planned to happen in two weeks. You can comment [[phab:T112147|in Phabricator]] if you have objections. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/16|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 16:48, 19 April 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21356080 --> == [[m:Special:MyLanguage/Tech/News/2021/17|Tech News: 2021-17]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/17|Translations]] are available. '''Recent changes''' * Templates have parameters that can have specific values. It is possible to suggest values for editors with [[mw:Special:MyLanguage/Extension:TemplateData|TemplateData]]. You can soon see them as a drop-down list in the visual editor. This is to help template users find the right values faster. [https://phabricator.wikimedia.org/T273857][https://meta.wikimedia.org/wiki/Special:MyLanguage/WMDE_Technical_Wishes/Suggested_values_for_template_parameters][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/Suggested_values_for_template_parameters] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-04-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-04-28|en}}. It will be on all wikis from {{#time:j xg|2021-04-29|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/17|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 21:24, 26 April 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21391118 --> == [[m:Special:MyLanguage/Tech/News/2021/18|Tech News: 2021-18]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/18|Translations]] are available. '''Recent changes''' * [[w:en:Wikipedia:Twinkle|Twinkle]] is a gadget on English Wikipedia. It can help with maintenance and patrolling. It can [[m:Grants:Project/Rapid/SD0001/Twinkle localisation/Report|now be used on other wikis]]. You can get Twinkle on your wiki using the [https://github.com/wikimedia-gadgets/twinkle-starter twinkle-starter] GitHub repository. '''Problems''' * The [[mw:Special:MyLanguage/Content translation|content translation tool]] did not work for many articles for a little while. This was because of a bug. [https://phabricator.wikimedia.org/T281346] * Some things will not work for about a minute on 5 May. This will happen [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210505T0600 around 06:00 UTC]. This will affect the content translation tool and notifications among other things. This is because of an upgrade to avoid crashes. [https://phabricator.wikimedia.org/T281212] '''Changes later this week''' * [[mw:Special:MyLanguage/Help:Reference Previews|Reference Previews]] will become a default feature on a number of wikis on 5 May. This is later than planned because of some changes. You can use it without using [[mw:Special:MyLanguage/Page Previews|Page Previews]] if you want to. The earlier plan was to have the preference to use both or none. [https://phabricator.wikimedia.org/T271206][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/ReferencePreviews] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-05|en}}. It will be on all wikis from {{#time:j xg|2021-05-06|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[:w:en:CSS|CSS]] classes <code dir=ltr>.error</code>, <code dir=ltr>.warning</code> and <code dir=ltr>.success</code> do not work for mobile readers if they have not been specifically defined on your wiki. From June they will not work for desktop readers. This can affect gadgets and templates. The classes can be defined in [[MediaWiki:Common.css]] or template styles instead. [https://phabricator.wikimedia.org/T280766] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/18|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 15:43, 3 May 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21418010 --> == [[m:Special:MyLanguage/Tech/News/2021/19|Tech News: 2021-19]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/19|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-12|en}}. It will be on all wikis from {{#time:j xg|2021-05-13|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * You can see what participants plan to work on at the online [[mw:Wikimedia Hackathon 2021|Wikimedia hackathon]] 22–23 May. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/19|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 15:10, 10 May 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21428676 --> == [[m:Special:MyLanguage/Tech/News/2021/20|Tech News: 2021-20]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/20|Translations]] are available. '''Recent changes''' * There is a new toolbar in [[mw:Talk pages project/Replying|the Reply tool]]. It works in the wikitext source mode. You can enable it in [[Special:Preferences#mw-htmlform-discussion|your preferences]]. [https://phabricator.wikimedia.org/T276608] [https://www.mediawiki.org/wiki/Talk_pages_project/Replying#13_May_2021] [https://www.mediawiki.org/wiki/Talk_pages_project/New_discussion#13_May_2021] * Wikimedia [https://lists.wikimedia.org/mailman/listinfo mailing lists] are being moved to [[:w:en:GNU Mailman|Mailman 3]]. This is a newer version. For the [[:w:en:Character encoding|character encoding]] to work it will change from <code>[[:w:en:UTF-8|UTF-8]]</code> to <code>utf8mb3</code>. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IEYQ2HS3LZF2P3DAYMNZYQDGHWPVMTPY/][https://phabricator.wikimedia.org/T282621] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] An [[m:Special:MyLanguage/Tech/News/2021/14|earlier issue]] of Tech News said that the [[mw:Special:MyLanguage/Citoid|citoid]] [[:w:en:API|API]] would handle dates with a month but no days in a new way. This has been reverted for now. There needs to be more discussion of how it affects different wikis first. [https://phabricator.wikimedia.org/T132308] '''Changes later this week''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] <code>MediaWiki:Pageimages-blacklist</code> will be renamed <code>MediaWiki:Pageimages-denylist</code>. The list can be copied to the new name. It will happen on 19 May for some wikis and 20 May for some wikis. Most wikis don't use it. It lists images that should never be used as thumbnails for articles. [https://phabricator.wikimedia.org/T282626] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-19|en}}. It will be on all wikis from {{#time:j xg|2021-05-20|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/20|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 13:49, 17 May 2021 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21464279 --> == [[m:Special:MyLanguage/Tech/News/2021/21|Tech News: 2021-21]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/21|Translations]] are available. '''Recent changes''' * The Wikimedia movement has been using [[:m:Special:MyLanguage/IRC|IRC]] on a network called [[:w:en:Freenode|Freenode]]. There have been changes around who is in control of the network. The [[m:Special:MyLanguage/IRC/Group_Contacts|Wikimedia IRC Group Contacts]] have [[m:Special:Diff/21476411|decided]] to move to the new [[:w:en:Libera Chat|Libera Chat]] network instead. This is not a formal decision for the movement to move all channels but most Wikimedia IRC channels will probably leave Freenode. There is a [[:m:IRC/Migrating_to_Libera_Chat|migration guide]] and ongoing Wikimedia [[m:Wikimedia Forum#Freenode (IRC)|discussions about this]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-26|en}}. It will be on all wikis from {{#time:j xg|2021-05-27|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/21|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 17:07, 24 May 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21477606 --> == [[m:Special:MyLanguage/Tech/News/2021/22|Tech News: 2021-22]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/22|Translations]] are available. '''Problems''' * There was an issue on the Vector skin with the text size of categories and notices under the page title. It was fixed last Monday. [https://phabricator.wikimedia.org/T283206] '''Changes later this week''' * There is no new MediaWiki version this week. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/22|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 17:05, 31 May 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21516076 --> == [[m:Special:MyLanguage/Tech/News/2021/23|Tech News: 2021-23]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/23|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-06-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-06-09|en}}. It will be on all wikis from {{#time:j xg|2021-06-10|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * The Wikimedia movement uses [[:mw:Special:MyLanguage/Phabricator|Phabricator]] for technical tasks. This is where we collect technical suggestions, bugs and what developers are working on. The company behind Phabricator will stop working on it. This will not change anything for the Wikimedia movement now. It could lead to changes in the future. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/message/YAXOD46INJLAODYYIJUVQWOZFIV54VUI/][https://admin.phacility.com/phame/post/view/11/phacility_is_winding_down_operations/][https://phabricator.wikimedia.org/T283980] * Searching on Wikipedia will find more results in some languages. This is mainly true for when those who search do not use the correct [[:w:en:Diacritic|diacritics]] because they are not seen as necessary in that language. For example searching for <code>Bedusz</code> doesn't find <code>Będusz</code> on German Wikipedia. The character <code>ę</code> isn't used in German so many would write <code>e</code> instead. This will work better in the future in some languages. [https://phabricator.wikimedia.org/T219550] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[:w:en:Cross-site request forgery|CSRF token parameters]] in the [[:mw:Special:MyLanguage/API:Main page|action API]] were changed in 2014. The old parameters from before 2014 will stop working soon. This can affect bots, gadgets and user scripts that still use the old parameters. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IMP43BNCI32C524O5YCUWMQYP4WVBQ2B/][https://phabricator.wikimedia.org/T280806] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/23|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 20:02, 7 June 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21551759 --> == [[m:Special:MyLanguage/Tech/News/2021/24|Tech News: 2021-24]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/24|Translations]] are available. '''Recent changes''' * Logged-in users on the mobile web can choose to use the [[:mw:Special:MyLanguage/Reading/Web/Advanced mobile contributions|advanced mobile mode]]. They now see categories in a similar way as users on desktop do. This means that some gadgets that have just been for desktop users could work for users of the mobile site too. If your wiki has such gadgets you could decide to turn them on for the mobile site too. Some gadgets probably need to be fixed to look good on mobile. [https://phabricator.wikimedia.org/T284763] * Language links on Wikidata now works for [[:oldwikisource:Main Page|multilingual Wikisource]]. [https://phabricator.wikimedia.org/T275958] '''Changes later this week''' * There is no new MediaWiki version this week. '''Future changes''' * In the future we [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation|can't show the IP]] of unregistered editors to everyone. This is because privacy regulations and norms have changed. There is now a rough draft of how [[m:IP Editing: Privacy Enhancement and Abuse Mitigation#Updates|showing the IP to those who need to see it]] could work. * German Wikipedia, English Wikivoyage and 29 smaller wikis will be read-only for a few minutes on 22 June. This is planned between 5:00 and 5:30 UTC. [https://phabricator.wikimedia.org/T284530] * All wikis will be read-only for a few minutes in the week of 28 June. More information will be published in Tech News later. It will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T281515][https://phabricator.wikimedia.org/T281209] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/24|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 20:26, 14 June 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21587625 --> == [[m:Special:MyLanguage/Tech/News/2021/25|Tech News: 2021-25]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/25|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The <code>otrs-member</code> group name is now <code>vrt-permissions</code>. This could affect abuse filters. [https://phabricator.wikimedia.org/T280615] '''Problems''' * You will be able to read but not edit German Wikipedia, English Wikivoyage and 29 smaller wikis for a few minutes on 22 June. This is planned between [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210623T0500 5:00 and 5:30 UTC]. [https://phabricator.wikimedia.org/T284530] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-06-22|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-06-23|en}}. It will be on all wikis from {{#time:j xg|2021-06-24|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/25|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 15:49, 21 June 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21593987 --> == [[m:Special:MyLanguage/Tech/News/2021/26|Tech News: 2021-26]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/26|Translations]] are available. '''Recent changes''' * Wikis with the [[mw:Special:MyLanguage/Growth|Growth features]] now can [[mw:Special:MyLanguage/Growth/Community configuration|configure Growth features directly on their wiki]]. This uses the new special page <code>Special:EditGrowthConfig</code>. [https://phabricator.wikimedia.org/T285423] * Wikisources have a new [[m:Special:MyLanguage/Community Tech/OCR Improvements|OCR tool]]. If you don't want to see the "extract text" button on Wikisource you can add <code>.ext-wikisource-ExtractTextWidget { display: none; }</code> to your [[Special:MyPage/common.css|common.css page]]. [https://phabricator.wikimedia.org/T285311] '''Problems''' *You will be able to read but not edit the Wikimedia wikis for a few minutes on 29 June. This is planned at [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210629T1400 14:00 UTC]. [https://phabricator.wikimedia.org/T281515][https://phabricator.wikimedia.org/T281209] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-06-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-06-30|en}}. It will be on all wikis from {{#time:j xg|2021-07-01|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * <code>Threshold for stub link formatting</code>, <code>thumbnail size</code> and <code>auto-number headings</code> can be set in preferences. They are expensive to maintain and few editors use them. The developers are planning to remove them. Removing them will make pages load faster. You can [[mw:Special:MyLanguage/User:SKim (WMF)/Performance Dependent User Preferences|read more and give feedback]]. * A toolbar will be added to the [[mw:Talk pages project/Replying|Reply tool]]'s wikitext source mode. This will make it easier to link to pages and to ping other users. [https://phabricator.wikimedia.org/T276609][https://www.mediawiki.org/wiki/Talk_pages_project/Replying#Status_updates] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/26|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 16:32, 28 June 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21653312 --> == [[m:Special:MyLanguage/Tech/News/2021/27|Tech News: 2021-27]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/27|Translations]] are available. '''Tech News''' * The next issue of Tech News will be sent out on 19 July. '''Recent changes''' * [[:wikidata:Q4063270|AutoWikiBrowser]] is a tool to make repetitive tasks easier. It now uses [[:w:en:JSON|JSON]]. <code>Wikipedia:AutoWikiBrowser/CheckPage</code> has moved to <code>Wikipedia:AutoWikiBrowser/CheckPageJSON</code> and <code>Wikipedia:AutoWikiBrowser/Config</code>. <code>Wikipedia:AutoWikiBrowser/CheckPage/Version</code> has moved to <code>Wikipedia:AutoWikiBrowser/CheckPage/VersionJSON</code>. The tool will eventually be configured on the wiki so that you don't have to wait until the new version to add templates or regular expression fixes. [https://phabricator.wikimedia.org/T241196] '''Problems''' * [[m:Special:MyLanguage/InternetArchiveBot|InternetArchiveBot]] helps saving online sources on some wikis. It adds them to [[:w:en:Wayback Machine|Wayback Machine]] and links to them there. This is so they don't disappear if the page that was linked to is removed. It currently has a problem with linking to the wrong date when it moves pages from <code>archive.is</code> to <code>web.archive.org</code>. [https://phabricator.wikimedia.org/T283432] '''Changes later this week''' * The tool to [[m:WMDE Technical Wishes/Finding and inserting templates|find, add and remove templates]] will be updated. This is to make it easier to find and use the right templates. It will come to the first wikis on 7 July. It will come to more wikis later this year. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Removing_a_template_from_a_page_using_the_VisualEditor][https://phabricator.wikimedia.org/T284553] * There is no new MediaWiki version this week. '''Future changes''' * Some Wikimedia wikis use [[m:Special:MyLanguage/Flagged Revisions|Flagged Revisions]] or pending changes. It hides edits from new and unregistered accounts for readers until they have been patrolled. The auto review action in Flagged Revisions will no longer be logged. All old logs of auto-review will be removed. This is because it creates a lot of logs that are not very useful. [https://phabricator.wikimedia.org/T285608] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/27|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 17:33, 5 July 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21694636 --> == [[m:Special:MyLanguage/Tech/News/2021/29|Tech News: 2021-29]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/29|Translations]] are available. '''Recent changes''' * The tool to [[m:WMDE Technical Wishes/Finding and inserting templates|find, add and remove templates]] was updated. This is to make it easier to find and use the right templates. It was supposed to come to the first wikis on 7 July. It was delayed to 12 July instead. It will come to more wikis later this year. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Removing_a_template_from_a_page_using_the_VisualEditor][https://phabricator.wikimedia.org/T284553] * [[Special:UnconnectedPages|Special:UnconnectedPages]] lists pages that are not connected to Wikidata. This helps you find pages that can be connected to Wikidata items. Some pages should not be connected to Wikidata. You can use the magic word <code><nowiki>__EXPECTED_UNCONNECTED_PAGE__</nowiki></code> on pages that should not be listed on the special page. [https://phabricator.wikimedia.org/T97577] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-07-20|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-07-21|en}}. It will be on all wikis from {{#time:j xg|2021-07-22|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] How media is structured in the [[:w:en:Parsing|parser's]] HTML output will soon change. This can affect bots, gadgets, user scripts and extensions. You can [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/L2UQJRHTFK5YG3IOZEC7JSLH2ZQNZRVU/ read more]. You can test it on [[:testwiki:Main Page|Testwiki]] or [[:test2wiki:Main Page|Testwiki 2]]. * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The parameters for how you obtain [[mw:API:Tokens|tokens]] in the MediaWiki API were changed in 2014. The old way will no longer work from 1 September. Scripts, bots and tools that use the parameters from before the 2014 change need to be updated. You can [[phab:T280806#7215377|read more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/29|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 15:31, 19 July 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21755027 --> == [[m:Special:MyLanguage/Tech/News/2021/30|Tech News: 2021-30]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/30|Translations]] are available. '''Recent changes''' * A [[mw:MediaWiki 1.37/wmf.14|new version]] of MediaWiki came to the Wikimedia wikis the week before last week. This was not in Tech News because there was no newsletter that week. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-07-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-07-28|en}}. It will be on all wikis from {{#time:j xg|2021-07-29|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * If you use the [[mw:Special:MyLanguage/Skin:MonoBook|Monobook skin]] you can choose to switch off [[:w:en:Responsive web design|responsive design]] on mobile. This will now work for more skins. If <code>{{int:monobook-responsive-label}}</code> is unticked you need to also untick the new [[Special:Preferences#mw-prefsection-rendering|preference]] <code>{{int:prefs-skin-responsive}}</code>. Otherwise it will stop working. Interface admins can automate this process on your wiki. You can [[phab:T285991|read more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/30|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 21:11, 26 July 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21771634 --> == [[m:Special:MyLanguage/Tech/News/2021/31|Tech News: 2021-31]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/31|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] If your wiki uses markup like <bdi lang="zxx" dir="ltr"><code><nowiki><div class="mw-content-ltr"></nowiki></code></bdi> or <bdi lang="zxx" dir="ltr"><code><nowiki><div class="mw-content-rtl"></nowiki></code></bdi> without the required <bdi lang="zxx" dir="ltr"><code>dir</code></bdi> attribute, then these will no longer work in 2 weeks. There is a short-term fix that can be added to your local wiki's Common.css page, which is explained at [[phab:T287701|T287701]]. From now on, all usages should include the full attributes, for example: <bdi lang="zxx" dir="ltr"><code><nowiki><div class="mw-content-ltr" dir="ltr" lang="en"></nowiki></code></bdi> or <bdi lang="zxx" dir="ltr"><code><nowiki><div class="mw-content-rtl" dir="rtl" lang="he"></nowiki></code></bdi>. This also applies to some other HTML tags, such as <code>span</code> or <code>code</code>. You can find existing examples on your wiki that need to be updated, using the instructions at [[phab:T287701|T287701]]. * Reminder: Wikimedia has [[m:Special:MyLanguage/IRC/Migrating to Libera Chat|migrated to the Libera Chat IRC network]], from the old Freenode network. Local documentation should be updated. '''Problems''' * Last week, all wikis had slow access or no access for 30 minutes. There was a problem with generating dynamic lists of articles on the Russian Wikinews, due to the bulk import of 200,000+ new articles over 3 days, which led to database problems. The problematic feature has been disabled on that wiki and developers are discussing if it can be fixed properly. [https://phabricator.wikimedia.org/T287380][https://wikitech.wikimedia.org/wiki/Incident_documentation/2021-07-26_ruwikinews_DynamicPageList] '''Changes later this week''' * When adding links to a page using [[mw:VisualEditor|VisualEditor]] or the [[mw:Special:MyLanguage/2017 wikitext editor|2017 wikitext editor]], [[mw:Special:MyLanguage/Extension:Disambiguator|disambiguation pages]] will now only appear at the bottom of search results. This is because users do not often want to link to disambiguation pages. [https://phabricator.wikimedia.org/T285510] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-03|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-04|en}}. It will be on all wikis from {{#time:j xg|2021-08-05|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * The [[mw:Wikimedia Apps/Team/Android|team of the Wikipedia app for Android]] is working on communication in the app. The developers are working on how to talk to other editors and get notifications. You can [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Communication|read more]]. They are looking for users who want to [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Communication/UsertestingJuly2021|test the plans]]. Any editor who has an Android phone and is willing to download the app can do this. * The [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature]] for {{int:discussiontools-preference-label}} will be updated in the coming weeks. You will be able to [[mw:Talk pages project/Notifications|subscribe to individual sections]] on a talk page at more wikis. You can test this now by adding <code>?dtenable=1</code> to the end of the talk page's URL ([https://meta.wikimedia.org/wiki/Meta_talk:Sandbox?dtenable=1 example]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/31|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 20:47, 2 August 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21818289 --> == [[m:Special:MyLanguage/Tech/News/2021/32|Tech News: 2021-32]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/32|Translations]] are available. '''Problems''' * You can read but not edit 17 wikis for a few minutes on 10 August. This is planned at [https://zonestamp.toolforge.org/1628571650 05:00 UTC]. This is because of work on the database. [https://phabricator.wikimedia.org/T287449] '''Changes later this week''' * The [[wmania:Special:MyLanguage/2021:Hackathon|Wikimania Hackathon]] will take place remotely on 13 August, starting at 5:00 UTC, for 24 hours. You can participate in many ways. You can still propose projects and sessions. * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-10|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-11|en}}. It will be on all wikis from {{#time:j xg|2021-08-12|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The old CSS <bdi lang="zxx" dir="ltr"><code><nowiki><div class="visualClear"></div></nowiki></code></bdi> will not be supported after 12 August. Instead, templates and pages should use <bdi lang="zxx" dir="ltr"><code><nowiki><div style="clear:both;"></div></nowiki></code></bdi>. Please help to replace any existing uses on your wiki. There are global-search links available at [[phab:T287962|T287962]]. '''Future changes''' * [[m:Special:MyLanguage/The Wikipedia Library|The Wikipedia Library]] is a place for Wikipedia editors to get access to sources. There is an [[mw:Special:MyLanguage/Extension:TheWikipediaLibrary|extension]] which has a new function to tell users when they can take part in it. It will use notifications. It will start pinging the first users in September. It will ping more users later. [https://phabricator.wikimedia.org/T288070] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[w:en:Vue.js|Vue.js]] will be the [[w:en:JavaScript|JavaScript]] framework for MediaWiki in the future. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/SOZREBYR36PUNFZXMIUBVAIOQI4N7PDU/] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/32|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 16:21, 9 August 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21856726 --> == [[m:Special:MyLanguage/Tech/News/2021/33|Tech News: 2021-33]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/33|Translations]] are available. '''Recent changes''' * You can add language links in the sidebar in the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|new Vector skin]] again. You do this by connecting the page to a Wikidata item. The new Vector skin has moved the language links but the new language selector cannot add language links yet. [https://phabricator.wikimedia.org/T287206] '''Problems''' * There was a problem on wikis which use the Translate extension. Translations were not updated or were replaced with the English text. The problems have been fixed. [https://phabricator.wikimedia.org/T288700][https://phabricator.wikimedia.org/T288683][https://phabricator.wikimedia.org/T288719] '''Changes later this week''' * A [[mw:Help:Tags|revision tag]] will soon be added to edits that add links to [[mw:Special:MyLanguage/Extension:Disambiguator|disambiguation pages]]. This is because these links are usually added by accident. The tag will allow editors to easily find the broken links and fix them. If your wiki does not like this feature, it can be [[mw:Help:Tags#Deleting a tag added by the software|hidden]]. [https://phabricator.wikimedia.org/T287549] *Would you like to help improve the information about tools? Would you like to attend or help organize a small virtual meetup for your community to discuss the list of tools? Please get in touch on the [[m:Toolhub/The Quality Signal Sessions|Toolhub Quality Signal Sessions]] talk page. We are also looking for feedback [[m:Talk:Toolhub/The Quality Signal Sessions#Discussion topic for "Quality Signal Sessions: The Tool Maintainers edition"|from tool maintainers]] on some specific questions. * In the past, edits to any page in your user talk space ignored your [[mw:Special:MyLanguage/Help:Notifications#mute|mute list]], e.g. sub-pages. Starting this week, this is only true for edits to your talk page. [https://phabricator.wikimedia.org/T288112] * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-18|en}}. It will be on all wikis from {{#time:j xg|2021-08-19|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/33|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 19:27, 16 August 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21889213 --> == [[m:Special:MyLanguage/Tech/News/2021/34|Tech News: 2021-34]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/34|Translations]] are available. '''Recent changes''' * The [[mw:Special:MyLanguage/Extension:Score|Score]] extension (<bdi lang="zxx" dir="ltr"><code><nowiki><score></nowiki></code></bdi> notation) has been re-enabled on public wikis and upgraded to a newer version. Some musical score functionality may no longer work because the extension is only enabled in "safe mode". The security issue has been fixed and an [[mw:Special:MyLanguage/Extension:Score/2021 security advisory|advisory published]]. '''Problems''' * You will be able to read but not edit [[phab:T289130|some wikis]] for a few minutes on {{#time:j xg|2021-08-25|en}}. This will happen around [https://zonestamp.toolforge.org/1629871217 06:00 UTC]. This is for database maintenance. During this time, operations on the CentralAuth will also not be possible. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-25|en}}. It will be on all wikis from {{#time:j xg|2021-08-26|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/34|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 21:58, 23 August 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21923254 --> == Read-only reminder == <section begin="MassMessage"/> A maintenance operation will be performed on [https://zonestamp.toolforge.org/1629871231 {{#time: l F d H:i e|2021-08-25T06:00|en}}]. It should only last for a few minutes. This will affect your wiki as well as 11 other wikis. During this time, publishing edits will not be possible. Also during this time, operations on the CentralAuth will not be possible (GlobalRenames, changing/confirming e-mail addresses, logging into new wikis, password changes). For more details about the operation and on all impacted services, please check [[phab:T289130|on Phabricator]]. A banner will be displayed 30 minutes before the operation. Please help your community to be aware of this maintenance operation. {{Int:Feedback-thanks-title}}<section end="MassMessage"/> 20:35, 24 August 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21927201 --> == [[m:Special:MyLanguage/Tech/News/2021/35|Tech News: 2021-35]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/35|Translations]] are available. '''Recent changes''' * Some musical score syntax no longer works and may needed to be updated, you can check [[:Category:{{MediaWiki:score-error-category}}]] on your wiki for a list of pages with errors. '''Problems''' * Musical scores were unable to render lyrics in some languages because of missing fonts. This has been fixed now. If your language would prefer a different font, please file a request in Phabricator. [https://phabricator.wikimedia.org/T289554] '''Changes later this week''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The parameters for how you obtain [[mw:API:Tokens|tokens]] in the MediaWiki API were changed in 2014. The old way will no longer work from 1 September. Scripts, bots and tools that use the parameters from before the 2014 change need to be updated. You can [[phab:T280806#7215377|read more]] about this. * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-01|en}}. It will be on all wikis from {{#time:j xg|2021-09-02|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). '''Future changes''' * You will be able to read but not edit [[phab:T289660|Commons]] for a few minutes on {{#time:j xg|2021-09-06|en}}. This will happen around [https://zonestamp.toolforge.org/1630818058 05:00 UTC]. This is for database maintenance. * All wikis will be read-only for a few minutes in the week of 13 September. More information will be published in Tech News later. It will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T287539] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/35|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 16:01, 30 August 2021 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21954810 --> == [[m:Special:MyLanguage/Tech/News/2021/36|Tech News: 2021-36]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/36|Translations]] are available. '''Recent changes''' * The wikis that have [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] deployed have been part of A/B testing since deployment, in which some newcomers did not receive the new features. Now, all of the newcomers on 21 of the smallest of those wikis will be receiving the features. [https://phabricator.wikimedia.org/T289786] '''Changes later this week''' * There is no new MediaWiki version this week. '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] In 2017, the provided jQuery library was upgraded from version 1 to 3, with a compatibility layer. The migration will soon finish, to make the site load faster for everyone. If you maintain a gadget or user script, check if you have any JQMIGRATE errors and fix them, or they will break. [https://phabricator.wikimedia.org/T280944][https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/6Z2BVLOBBEC2QP4VV4KOOVQVE52P3HOP/] * Last year, the Portuguese Wikipedia community embarked on an experiment to make log-in compulsory for editing.  The [[m:IP Editing: Privacy Enhancement and Abuse Mitigation/Impact report for Login Required Experiment on Portuguese Wikipedia|impact report of this trial]] is ready. Moving forward, the Anti-Harassment Tools team is looking for projects that are willing to experiment with restricting IP editing on their wiki for a short-term experiment. [[m:IP Editing: Privacy Enhancement and Abuse Mitigation/Login Required Experiment|Learn more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/36|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 15:20, 6 September 2021 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=21981010 --> == [[m:Special:MyLanguage/Tech/News/2021/37|Tech News: 2021-37]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/37|Translations]] are available. '''Recent changes''' * 45 new Wikipedias now have access to the [[mw:Special:MyLanguage/Growth/Feature summary|Growth features]]. [https://phabricator.wikimedia.org/T289680] * [[mw:Special:MyLanguage/Growth/Deployment table|A majority of Wikipedias]] now have access to the Growth features. The Growth team [[mw:Special:MyLanguage/Growth/FAQ|has published an FAQ page]] about the features. This translatable FAQ covers the description of the features, how to use them, how to change the configuration, and more. '''Problems''' * [[m:Special:MyLanguage/Tech/Server switch|All wikis will be read-only]] for a few minutes on 14 September. This is planned at [https://zonestamp.toolforge.org/1631628002 14:00 UTC]. [https://phabricator.wikimedia.org/T287539] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-09-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-15|en}}. It will be on all wikis from {{#time:j xg|2021-09-16|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]). * Starting this week, Wikipedia in Italian will receive weekly software updates on Wednesdays. It used to receive the updates on Thursdays. Due to this change, bugs will be noticed and fixed sooner. [https://phabricator.wikimedia.org/T286664] * You can add language links in the sidebar in [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|the new Vector skin]] again. You do this by connecting the page to a Wikidata item. The new Vector skin has moved the language links but the new language selector cannot add language links yet. [https://phabricator.wikimedia.org/T287206] * The [[mw:Special:MyLanguage/Extension:SyntaxHighlight|syntax highlight]] tool marks up code with different colours. It now can highlight 23 new code languages. Additionally, <bdi lang="zxx" dir="ltr"><code>golang</code></bdi> can now be used as an alias for the [[d:Q37227|Go programming language]], and a special <bdi lang="zxx" dir="ltr"><code>output</code></bdi> mode has been added to show a program's output. [https://phabricator.wikimedia.org/T280117][https://gerrit.wikimedia.org/r/c/mediawiki/extensions/SyntaxHighlight_GeSHi/+/715277/] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/37|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 15:35, 13 September 2021 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22009517 --> == [[m:Special:MyLanguage/Tech/News/2021/38|Tech News: 2021-38]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/38|Translations]] are available. '''Recent changes''' * Growth features are now deployed to almost all Wikipedias. [[phab:T290582|For the majority of small Wikipedias]], the features are only available for experienced users, to [[mw:Special:MyLanguage/Growth/FAQ#enable|test the features]] and [[mw:Special:MyLanguage/Growth/FAQ#config|configure them]]. Features will be available for newcomers starting on 20 September 2021. * MediaWiki had a feature that would highlight local links to short articles in a different style. Each user could pick the size at which "stubs" would be highlighted. This feature was very bad for performance, and following a consultation, has been removed. [https://phabricator.wikimedia.org/T284917] * A technical change was made to the MonoBook skin to allow for easier maintenance and upkeep. This has resulted in some minor changes to HTML that make MonoBook's HTML consistent with other skins. Efforts have been made to minimize the impact on editors, but please ping [[m:User:Jon (WMF)|Jon (WMF)]] on wiki or in [[phab:T290888|phabricator]] if any problems are reported. '''Problems''' * There was a problem with search last week. Many search requests did not work for 2 hours because of an accidental restart of the search servers. [https://wikitech.wikimedia.org/wiki/Incident_documentation/2021-09-13_cirrussearch_restart] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-09-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-22|en}}. It will be on all wikis from {{#time:j xg|2021-09-23|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[s:Special:ApiHelp/query+proofreadinfo|meta=proofreadpage API]] has changed. The <bdi lang="zxx" dir="ltr"><code><nowiki>piprop</nowiki></code></bdi> parameter has been renamed to <bdi lang="zxx" dir="ltr"><code><nowiki>prpiprop</nowiki></code></bdi>. API users should update their code to avoid unrecognized parameter warnings. Pywikibot users should upgrade to 6.6.0. [https://phabricator.wikimedia.org/T290585] '''Future changes''' * The [[mw:Special:MyLanguage/Help:DiscussionTools#Replying|Reply tool]] will be deployed to the remaining wikis in the coming weeks. It is currently part of "{{int:discussiontools-preference-label}}" in [[Special:Preferences#mw-prefsection-betafeatures|Beta features]] at most wikis. You will be able to turn it off in [[Special:Preferences#mw-prefsection-editing-discussion|Editing Preferences]]. [https://phabricator.wikimedia.org/T262331] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[mw:MediaWiki_1.37/Deprecation_of_legacy_API_token_parameters|previously announced]] change to how you obtain tokens from the API has been delayed to September 21 because of an incompatibility with Pywikibot. Bot operators using Pywikibot can follow [[phab:T291202|T291202]] for progress on a fix, and should plan to upgrade to 6.6.1 when it is released. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/38|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 18:32, 20 September 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22043415 --> == [[m:Special:MyLanguage/Tech/News/2021/39|Tech News: 2021-39]] == <section begin="technews-2021-W39"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/39|Translations]] are available. '''Recent changes''' * [[w:en:IOS|iOS 15]] has a new function called [https://support.apple.com/en-us/HT212614 Private Relay] (Apple website). This can hide the user's IP when they use [[w:en:Safari (software)|Safari]] browser. This is like using a [[w:en:Virtual private network|VPN]] in that we see another IP address instead. It is opt-in and only for those who pay extra for [[w:en:ICloud|iCloud]]. It will come to Safari users on [[:w:en:OSX|OSX]] later. There is a [[phab:T289795|technical discussion]] about what this means for the Wikimedia wikis. '''Problems''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Some gadgets and user-scripts add items to the [[m:Customization:Explaining_skins#Portlets|portlets]] (article tools) part of the skin. A recent change to the HTML may have made those links a different font-size. This can be fixed by adding the CSS class <bdi lang="zxx" dir="ltr"><code>.vector-menu-dropdown-noicon</code></bdi>. [https://phabricator.wikimedia.org/T291438] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-09-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-29|en}}. It will be on all wikis from {{#time:j xg|2021-09-30|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * The [[mw:Special:MyLanguage/Onboarding_new_Wikipedians#New_experience|GettingStarted extension]] was built in 2013, and provides an onboarding process for new account holders in a few versions of Wikipedia. However, the recently developed [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] provide a better onboarding experience. Since the vast majority of Wikipedias now have access to the Growth features, GettingStarted will be deactivated starting on 4 October. [https://phabricator.wikimedia.org/T235752] * A small number of users will not be able to connect to the Wikimedia wikis after 30 September. This is because an old [[:w:en:root certificate|root certificate]] will no longer work. They will also have problems with many other websites. Users who have updated their software in the last five years are unlikely to have problems. Users in Europe, Africa and Asia are less likely to have immediate problems even if their software is too old. You can [[m:Special:MyLanguage/HTTPS/2021 Let's Encrypt root expiry|read more]]. * You can [[mw:Special:MyLanguage/Help:Notifications|receive notifications]] when someone leaves a comment on user talk page or mentions you in a talk page comment. Clicking the notification link will now bring you to the comment and highlight it. Previously, doing so brought you to the top of the section that contained the comment. You can find [[phab:T282029|more information in T282029.]] '''Future changes''' * The [[mw:Special:MyLanguage/Help:DiscussionTools#Replying|Reply tool]] will be deployed to the remaining wikis in the coming weeks. It is currently part of "{{int:discussiontools-preference-label}}" in [[Special:Preferences#mw-prefsection-betafeatures|Beta features]] at most wikis. You will be able to turn it off in [[Special:Preferences#mw-prefsection-editing-discussion|Editing Preferences]]. [[phab:T288485|See the list of wikis.]] [https://phabricator.wikimedia.org/T262331] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/39|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W39"/> 22:23, 27 September 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22077885 --> == [[m:Special:MyLanguage/Tech/News/2021/40|Tech News: 2021-40]] == <section begin="tech-newsletter-content"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/40|Translations]] are available. '''Recent changes''' * A more efficient way of sending changes from Wikidata to Wikimedia wikis that show them has been enabled for the following 10 wikis: mediawiki.org, the Italian, Catalan, Hebrew and Vietnamese Wikipedias, French Wikisource, and English Wikivoygage, Wikibooks, Wiktionary and Wikinews. If you notice anything strange about how changes from Wikidata appear in recent changes or your watchlist on those wikis you can [[phab:T48643|let the developers know]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-06|en}}. It will be on all wikis from {{#time:j xg|2021-10-07|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Some gadgets and bots that use the API to read the AbuseFilter log might break. The <bdi lang="zxx" dir="ltr"><code>hidden</code></bdi> property will no longer say an entry is <bdi lang="zxx" dir="ltr"><code>implicit</code></bdi> for unsuppressed log entries about suppressed edits. If your bot needs to know this, do a separate revision query. Additionally, the property will have the value <bdi lang="zxx" dir="ltr"><code>false</code></bdi> for visible entries; previously, it wasn't included in the response. [https://phabricator.wikimedia.org/T291718] * A more efficient way of sending changes from Wikidata to Wikimedia wikis that show them will be enabled for ''all production wikis''. If you notice anything strange about how changes from Wikidata appear in recent changes or your watchlist you can [[phab:T48643|let the developers know]]. '''Future changes''' * You can soon get cross-wiki notifications in the [[mw:Wikimedia Apps/Team/iOS|iOS Wikipedia app]]. You can also get notifications as push notifications. More notification updates will follow in later versions. [https://www.mediawiki.org/wiki/Wikimedia_Apps/Team/iOS/Notifications#September_2021_update] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The JavaScript variables <bdi lang="zxx" dir="ltr"><code>wgExtraSignatureNamespaces</code></bdi>, <bdi lang="zxx" dir="ltr"><code>wgLegalTitleChars</code></bdi>, and <bdi lang="zxx" dir="ltr"><code>wgIllegalFileChars</code></bdi> will soon be removed from <bdi lang="zxx" dir="ltr"><code>[[mw:Special:MyLanguage/Manual:Interface/JavaScript#mw.config|mw.config]]</code></bdi>. These are not part of the "stable" variables available for use in wiki JavaScript. [https://phabricator.wikimedia.org/T292011] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The JavaScript variables <bdi lang="zxx" dir="ltr"><code>wgCookiePrefix</code></bdi>, <bdi lang="zxx" dir="ltr"><code>wgCookieDomain</code></bdi>, <bdi lang="zxx" dir="ltr"><code>wgCookiePath</code></bdi>, and <bdi lang="zxx" dir="ltr"><code>wgCookieExpiration</code></bdi> will soon be removed from mw.config. Scripts should instead use <bdi lang="zxx" dir="ltr"><code>mw.cookie</code></bdi> from the "<bdi lang="zxx" dir="ltr">[[mw:ResourceLoader/Core_modules#mediawiki.cookie|mediawiki.cookie]]</bdi>" module. [https://phabricator.wikimedia.org/T291760] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/40|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="tech-newsletter-content"/> 16:32, 4 October 2021 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22101208 --> == [[m:Special:MyLanguage/Tech/News/2021/41|Tech News: 2021-41]] == <section begin="technews-2021-W41"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/41|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-13|en}}. It will be on all wikis from {{#time:j xg|2021-10-14|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * The [[mw:Manual:Table_of_contents#Auto-numbering|"auto-number headings" preference]] is being removed. You can read [[phab:T284921]] for the reasons and discussion. This change was [[m:Tech/News/2021/26|previously]] announced. [[mw:Snippets/Auto-number_headings|A JavaScript snippet]] is available which can be used to create a Gadget on wikis that still want to support auto-numbering. '''Meetings''' * You can join a meeting about the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Desktop Improvements]]. A demonstration version of the [[mw:Reading/Web/Desktop Improvements/Features/Sticky Header|newest feature]] will be shown. The event will take place on Tuesday, 12 October at 16:00 UTC. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web/12-10-2021|See how to join]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/41|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W41"/> 15:30, 11 October 2021 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22152137 --> == [[m:Special:MyLanguage/Tech/News/2021/42|Tech News: 2021-42]] == <section begin="technews-2021-W42"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/42|Translations]] are available. '''Recent changes''' *[[m:Toolhub|Toolhub]] is a catalogue to make it easier to find software tools that can be used for working on the Wikimedia projects. You can [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/LF4SSR4QRCKV6NPRFGUAQWUFQISVIPTS/ read more]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-20|en}}. It will be on all wikis from {{#time:j xg|2021-10-21|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Future changes''' * The developers of the [[mw:Wikimedia Apps/Team/Android|Wikipedia Android app]] are working on [[mw:Wikimedia Apps/Team/Android/Communication|communication in the app]]. You can now answer questions in [[mw:Wikimedia Apps/Team/Android/Communication/UsertestingOctober2021|survey]] to help the development. * 3–5% of editors may be blocked in the next few months. This is because of a new service in Safari, which is similar to a [[w:en:Proxy server|proxy]] or a [[w:en:VPN|VPN]]. It is called iCloud Private Relay. There is a [[m:Special:MyLanguage/Apple iCloud Private Relay|discussion about this]] on Meta. The goal is to learn what iCloud Private Relay could mean for the communities. * [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] is a new [[w:en:API|API]] for those who use a lot of information from the Wikimedia projects on other sites. It is a way to get big commercial users to pay for the data. There will soon be a copy of the Wikimedia Enterprise dataset. You can [https://lists.wikimedia.org/hyperkitty/list/wikitech-ambassadors@lists.wikimedia.org/message/B2AX6PWH5MBKB4L63NFZY3ADBQG7MSBA/ read more]. You can also ask the team questions [https://wikimedia.zoom.us/j/88994018553 on Zoom] on [https://www.timeanddate.com/worldclock/fixedtime.html?hour=15&min=00&sec=0&day=22&month=10&year=2021 22 October 15:00 UTC]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/42|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W42"/> 20:53, 18 October 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22176877 --> == [[m:Special:MyLanguage/Tech/News/2021/43|Tech News: 2021-43]] == <section begin="technews-2021-W43"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/43|Translations]] are available. '''Recent changes''' * The [[m:Special:MyLanguage/Coolest_Tool_Award|Coolest Tool Award 2021]] is looking for nominations. You can recommend tools until 27 October. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-27|en}}. It will be on all wikis from {{#time:j xg|2021-10-28|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Future changes''' *[[m:Special:MyLanguage/Help:Diff|Diff pages]] will have an improved copy and pasting experience. [[m:Special:MyLanguage/Community Wishlist Survey 2021/Copy paste diffs|The changes]] will allow the text in the diff for before and after to be treated as separate columns and will remove any unwanted syntax. [https://phabricator.wikimedia.org/T192526] * The version of the [[w:en:Liberation fonts|Liberation fonts]] used in SVG files will be upgraded. Only new thumbnails will be affected. Liberation Sans Narrow will not change. [https://phabricator.wikimedia.org/T253600] '''Meetings''' * You can join a meeting about the [[m:Special:MyLanguage/Community Wishlist Survey|Community Wishlist Survey]]. News about the [[m:Special:MyLanguage/Community Wishlist Survey 2021/Warn when linking to disambiguation pages|disambiguation]] and the [[m:Special:MyLanguage/Community Wishlist Survey 2021/Real Time Preview for Wikitext|real-time preview]] wishes will be shown. The event will take place on Wednesday, 27 October at 14:30 UTC. [[m:Special:MyLanguage/Community Wishlist Survey/Updates/Talk to Us|See how to join]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/43|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W43"/> 20:08, 25 October 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22232718 --> == [[m:Special:MyLanguage/Tech/News/2021/44|Tech News: 2021-44]] == <section begin="technews-2021-W44"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/44|Translations]] are available. '''Recent changes''' * There is a limit on the amount of emails a user can send each day. This limit is now global instead of per-wiki. This change is to prevent abuse. [https://phabricator.wikimedia.org/T293866] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-11-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-11-03|en}}. It will be on all wikis from {{#time:j xg|2021-11-04|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/44|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W44"/> 20:28, 1 November 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22269406 --> == [[m:Special:MyLanguage/Tech/News/2021/45|Tech News: 2021-45]] == <section begin="technews-2021-W45"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/45|Translations]] are available. '''Recent changes''' * Mobile IP editors are now able to receive warning notices indicating they have a talk page message on the mobile website (similar to the orange banners available on desktop). These notices will be displayed on every page outside of the main namespace and every time the user attempts to edit. The notice on desktop now has a slightly different colour. [https://phabricator.wikimedia.org/T284642][https://phabricator.wikimedia.org/T278105] '''Changes later this week''' * [[phab:T294321|Wikidata will be read-only]] for a few minutes on 11 November. This will happen around [https://zonestamp.toolforge.org/1636610400 06:00 UTC]. This is for database maintenance. [https://phabricator.wikimedia.org/T294321] * There is no new MediaWiki version this week. '''Future changes''' * In the future, unregistered editors will be given an identity that is not their [[:w:en:IP address|IP address]]. This is for legal reasons. A new user right will let editors who need to know the IPs of unregistered accounts to fight vandalism, spam, and harassment, see the IP. You can read the [[m:IP Editing: Privacy Enhancement and Abuse Mitigation#IP Masking Implementation Approaches (FAQ)|suggestions for how that identity could work]] and [[m:Talk:IP Editing: Privacy Enhancement and Abuse Mitigation|discuss on the talk page]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/45|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W45"/> 20:36, 8 November 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22311003 --> == [[m:Special:MyLanguage/Tech/News/2021/46|Tech News: 2021-46]] == <section begin="technews-2021-W46"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/46|Translations]] are available. '''Recent changes''' * Most [[c:Special:MyLanguage/Commons:Maximum_file_size#MAXTHUMB|large file uploads]] errors that had messages like "<bdi lang="zxx" dir="ltr"><code>stashfailed</code></bdi>" or "<bdi lang="zxx" dir="ltr"><code>DBQueryError</code></bdi>" have now been fixed. An [[wikitech:Incident documentation/2021-11-04 large file upload timeouts|incident report]] is available. '''Problems''' * Sometimes, edits made on iOS using the visual editor save groups of numbers as telephone number links, because of a feature in the operating system. This problem is under investigation. [https://phabricator.wikimedia.org/T116525] * There was a problem with search last week. Many search requests did not work for 2 hours because of a configuration error. [https://wikitech.wikimedia.org/wiki/Incident_documentation/2021-11-10_cirrussearch_commonsfile_outage] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-11-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-11-17|en}}. It will be on all wikis from {{#time:j xg|2021-11-18|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/46|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W46"/> 22:06, 15 November 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22338097 --> == [[m:Special:MyLanguage/Tech/News/2021/47|Tech News: 2021-47]] == <section begin="technews-2021-W47"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/47|Translations]] are available. '''Changes later this week''' * There is no new MediaWiki version this week. *The template dialog in VisualEditor and in the [[Special:Preferences#mw-prefsection-betafeatures|new wikitext mode]] Beta feature will be [[m:WMDE Technical Wishes/VisualEditor template dialog improvements|heavily improved]] on [[phab:T286992|a few wikis]]. Your [[m:Talk:WMDE Technical Wishes/VisualEditor template dialog improvements|feedback is welcome]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/47|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W47"/> 20:02, 22 November 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22366010 --> == [[m:Special:MyLanguage/Tech/News/2021/48|Tech News: 2021-48]] == <section begin="technews-2021-W48"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/48|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-11-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-12-01|en}}. It will be on all wikis from {{#time:j xg|2021-12-02|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/48|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W48"/> 21:15, 29 November 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22375666 --> == [[m:Special:MyLanguage/Tech/News/2021/49|Tech News: 2021-49]] == <section begin="technews-2021-W49"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/49|Translations]] are available. '''Problems''' * MediaWiki 1.38-wmf.11 was scheduled to be deployed on some wikis last week. The deployment was delayed because of unexpected problems. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-12-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-12-08|en}}. It will be on all wikis from {{#time:j xg|2021-12-09|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * At all Wikipedias, a Mentor Dashboard is now available at <bdi lang="zxx" dir="ltr"><code><nowiki>Special:MentorDashboard</nowiki></code></bdi>. It allows registered mentors, who take care of newcomers' first steps, to monitor their assigned newcomers' activity. It is part of the [[mw:Special:MyLanguage/Growth/Feature summary|Growth features]]. You can learn more about [[mw:Special:MyLanguage/Growth/Communities/How_to_configure_the_mentors%27_list|activating the mentor list]] on your wiki and about [[mw:Special:MyLanguage/Growth/Mentor dashboard|the mentor dashboard project]]. * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The predecessor to the current [[mw:API|MediaWiki Action API]] (which was created in 2008), <bdi lang="zxx" dir="ltr"><code><nowiki>action=ajax</nowiki></code></bdi>, will be removed this week. Any scripts or bots using it will need to switch to the corresponding API module. [https://phabricator.wikimedia.org/T42786] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] An old ResourceLoader module, <bdi lang="zxx" dir="ltr"><code><nowiki>jquery.jStorage</nowiki></code></bdi>, which was deprecated in 2016, will be removed this week. Any scripts or bots using it will need to switch to <bdi lang="zxx" dir="ltr"><code><nowiki>mediawiki.storage</nowiki></code></bdi> instead. [https://phabricator.wikimedia.org/T143034] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/49|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W49"/> 21:59, 6 December 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22413926 --> == [[m:Special:MyLanguage/Tech/News/2021/50|Tech News: 2021-50]] == <section begin="technews-2021-W50"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/50|Translations]] are available. '''Recent changes''' * There are now default [[m:Special:MyLanguage/Help:Namespace#Other_namespace_aliases|short aliases]] for the "Project:" namespace on most wikis. E.g. On Wikibooks wikis, <bdi lang="zxx" dir="ltr"><code><nowiki>[[WB:]]</nowiki></code></bdi> will go to the local language default for the <bdi lang="zxx" dir="ltr"><code><nowiki>[[Project:]]</nowiki></code></bdi> namespace. This change is intended to help the smaller communities have easy access to this feature. Additional local aliases can still be requested via [[m:Special:MyLanguage/Requesting wiki configuration changes|the usual process]]. [https://phabricator.wikimedia.org/T293839] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-12-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-12-15|en}}. It will be on all wikis from {{#time:j xg|2021-12-16|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/50|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W50"/> 22:27, 13 December 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22441074 --> == [[m:Special:MyLanguage/Tech/News/2021/51|Tech News: 2021-51]] == <section begin="technews-2021-W51"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/51|Translations]] are available. '''Tech News''' * Because of the [[w:en:Christmas and holiday season|holidays]] the next issue of Tech News will be sent out on 10 January 2022. '''Recent changes''' * Queries made by the DynamicPageList extension (<bdi lang="zxx" dir="ltr"><code><nowiki><DynamicPageList></nowiki></code></bdi>) are now only allowed to run for 10 seconds and error if they take longer. This is in response to multiple outages where long-running queries caused an outage on all wikis. [https://phabricator.wikimedia.org/T287380#7575719] '''Changes later this week''' * There is no new MediaWiki version this week or next week. '''Future changes''' * The developers of the Wikipedia iOS app are looking for testers who edit in multiple languages. You can [[mw:Wikimedia Apps/Team/iOS/202112 testing|read more and let them know if you are interested]]. * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The Wikimedia [[wikitech:Portal:Cloud VPS|Cloud VPS]] hosts technical projects for the Wikimedia movement. Developers need to [[wikitech:News/Cloud VPS 2021 Purge|claim projects]] they use. This is because old and unused projects are removed once a year. Unclaimed projects can be shut down from February. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/2B7KYL5VLQNHGQQHMYLW7KTUKXKAYY3T/] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2021/51|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2021-W51"/> 22:05, 20 December 2021 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22465395 --> == [[m:Special:MyLanguage/Tech/News/2022/02|Tech News: 2022-02]] == <section begin="technews-2022-W02"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/02|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] A <bdi lang="zxx" dir="ltr"><code>oauth_consumer</code></bdi> variable has been added to the [[mw:Special:MyLanguage/AbuseFilter|AbuseFilter]] to enable identifying changes made by specific tools. [https://phabricator.wikimedia.org/T298281] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets are [[mw:Special:MyLanguage/ResourceLoader/Migration_guide_(users)#Package_Gadgets|now able to directly include JSON pages]]. This means some gadgets can now be configured by administrators without needing the interface administrator permission, such as with the Geonotice gadget. [https://phabricator.wikimedia.org/T198758] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets [[mw:Extension:Gadgets#Options|can now specify page actions]] on which they are available. For example, <bdi lang="zxx" dir="ltr"><code>|actions=edit,history</code></bdi> will load a gadget only while editing and on history pages. [https://phabricator.wikimedia.org/T63007] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets can now be loaded on demand with the <bdi lang="zxx" dir="ltr"><code>withgadget</code></bdi> URL parameter. This can be used to replace [[mw:Special:MyLanguage/Snippets/Load JS and CSS by URL|an earlier snippet]] that typically looks like <bdi lang="zxx" dir="ltr"><code>withJS</code></bdi> or <bdi lang="zxx" dir="ltr"><code>withCSS</code></bdi>. [https://phabricator.wikimedia.org/T29766] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] At wikis where [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list|the Mentorship system is configured]], you can now use the Action API to get a list of a [[mw:Special:MyLanguage/Growth/Mentor_dashboard|mentor's]] mentees. [https://phabricator.wikimedia.org/T291966] * The heading on the main page can now be configured using <span class="mw-content-ltr" lang="en" dir="ltr">[[MediaWiki:Mainpage-title-loggedin]]</span> for logged-in users and <span class="mw-content-ltr" lang="en" dir="ltr">[[MediaWiki:Mainpage-title]]</span> for logged-out users. Any CSS that was previously used to hide the heading should be removed. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Small_wiki_toolkits/Starter_kit/Main_page_customization#hide-heading] [https://phabricator.wikimedia.org/T298715] * Four special pages (and their API counterparts) now have a maximum database query execution time of 30 seconds. These special pages are: RecentChanges, Watchlist, Contributions, and Log. This change will help with site performance and stability. You can read [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IPJNO75HYAQWIGTHI5LJHTDVLVOC4LJP/ more details about this change] including some possible solutions if this affects your workflows. [https://phabricator.wikimedia.org/T297708] * The [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Sticky Header|sticky header]] has been deployed for 50% of logged-in users on [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Frequently asked questions#pilot-wikis|more than 10 wikis]]. This is part of the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Desktop Improvements]]. See [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Participate|how to take part in the project]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-12|en}}. It will be on all wikis from {{#time:j xg|2022-01-13|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Events''' * [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey 2022]] begins. All contributors to the Wikimedia projects can propose for tools and platform improvements. The proposal phase takes place from {{#time:j xg|2022-01-10|en}} 18:00 UTC to {{#time:j xg|2022-01-23|en}} 18:00 UTC. [[m:Special:MyLanguage/Community_Wishlist_Survey/FAQ|Learn more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/02|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W02"/> 01:23, 11 January 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22562156 --> == [[m:Special:MyLanguage/Tech/News/2022/03|Tech News: 2022-03]] == <section begin="technews-2022-W03"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/03|Translations]] are available. '''Recent changes''' * When using [[mw:Special:MyLanguage/Extension:WikiEditor|WikiEditor]] (also known as the 2010 wikitext editor), people will now see a warning if they link to disambiguation pages. If you click "{{int:Disambiguator-review-link}}" in the warning, it will ask you to correct the link to a more specific term. You can [[m:Community Wishlist Survey 2021/Warn when linking to disambiguation pages#Jan 12, 2021: Turning on the changes for all Wikis|read more information]] about this completed 2021 Community Wishlist item. * You can [[mw:Special:MyLanguage/Help:DiscussionTools#subscribe|automatically subscribe to all of the talk page discussions]] that you start or comment in using [[mw:Special:MyLanguage/Talk pages project/Feature summary|DiscussionTools]]. You will receive [[mw:Special:MyLanguage/Notifications|notifications]] when another editor replies. This is available at most wikis. Go to your [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]] and turn on "{{int:discussiontools-preference-autotopicsub}}". [https://phabricator.wikimedia.org/T263819] * When asked to create a new page or talk page section, input fields can be [[mw:Special:MyLanguage/Manual:Creating_pages_with_preloaded_text|"preloaded" with some text]]. This feature is now limited to wikitext pages. This is so users can't be tricked into making malicious edits. There is a discussion about [[phab:T297725|if this feature should be re-enabled]] for some content types. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-19|en}}. It will be on all wikis from {{#time:j xg|2022-01-20|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Events''' * [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey 2022]] continues. All contributors to the Wikimedia projects can propose for tools and platform improvements. The proposal phase takes place from {{#time:j xg|2022-01-10|en}} 18:00 UTC to {{#time:j xg|2022-01-23|en}} 18:00 UTC. [[m:Special:MyLanguage/Community_Wishlist_Survey/FAQ|Learn more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/03|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W03"/> 19:55, 17 January 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22620285 --> == [[m:Special:MyLanguage/Tech/News/2022/04|Tech News: 2022-04]] == <section begin="technews-2022-W04"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/04|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-26|en}}. It will be on all wikis from {{#time:j xg|2022-01-27|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * The following languages can now be used with [[mw:Special:MyLanguage/Extension:SyntaxHighlight|syntax highlighting]]: BDD, Elpi, LilyPond, Maxima, Rita, Savi, Sed, Sophia, Spice, .SRCINFO. * You can now access your watchlist from outside of the user menu in the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|new Vector skin]]. The watchlist link appears next to the notification icons if you are at the top of the page. [https://phabricator.wikimedia.org/T289619] '''Events''' * You can see the results of the [[m:Special:MyLanguage/Coolest Tool Award|Coolest Tool Award 2021]] and learn more about 14 tools which were selected this year. * You can [[m:Special:MyLanguage/Community_Wishlist_Survey/Help_us|translate, promote]], or comment on [[m:Special:MyLanguage/Community Wishlist Survey 2022/Proposals|the proposals]] in the Community Wishlist Survey. Voting will begin on {{#time:j xg|2022-01-28|en}}. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/04|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W04"/> 21:38, 24 January 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22644148 --> == [[m:Special:MyLanguage/Tech/News/2022/05|Tech News: 2022-05]] == <section begin="technews-2022-W05"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/05|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] If a gadget should support the new <bdi lang="zxx" dir="ltr"><code>?withgadget</code></bdi> URL parameter that was [[m:Special:MyLanguage/Tech/News/2022/02|announced]] 3 weeks ago, then it must now also specify <bdi lang="zxx" dir="ltr"><code>supportsUrlLoad</code></bdi> in the gadget definition ([[mw:Special:MyLanguage/Extension:Gadgets#supportsUrlLoad|documentation]]). [https://phabricator.wikimedia.org/T29766] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-02|en}}. It will be on all wikis from {{#time:j xg|2022-02-03|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Future changes''' * A change that was [[m:Special:MyLanguage/Tech/News/2021/16|announced]] last year was delayed. It is now ready to move ahead: ** The user group <code>oversight</code> will be renamed <code>suppress</code>. This is for [[phab:T109327|technical reasons]]. This is the technical name. It doesn't affect what you call the editors with this user right on your wiki. This is planned to happen in three weeks. You can comment [[phab:T112147|in Phabricator]] if you have objections. As usual, these labels can be translated on translatewiki ([[phab:T112147|direct links are available]]) or by administrators on your wiki. '''Events''' * You can vote on proposals in the [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey]] between 28 January and 11 February. The survey decides what the [[m:Special:MyLanguage/Community Tech|Community Tech team]] will work on. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/05|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W05"/> 17:42, 31 January 2022 (UTC) <!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22721804 --> == [[m:Special:MyLanguage/Tech/News/2022/06|Tech News: 2022-06]] == <section begin="technews-2022-W06"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/06|Translations]] are available. '''Recent changes''' * English Wikipedia recently set up a gadget for dark mode. You can enable it there, or request help from an [[m:Special:MyLanguage/Interface administrators|interface administrator]] to set it up on your wiki ([[w:en:Wikipedia:Dark mode (gadget)|instructions and screenshot]]). * Category counts are sometimes wrong. They will now be completely recounted at the beginning of every month. [https://phabricator.wikimedia.org/T299823] '''Problems''' * A code-change last week to fix a bug with [[mw:Special:MyLanguage/Manual:Live preview|Live Preview]] may have caused problems with some local gadgets and user-scripts. Any code with skin-specific behaviour for <bdi lang="zxx" dir="ltr"><code>vector</code></bdi> should be updated to also check for <bdi lang="zxx" dir="ltr"><code>vector-2022</code></bdi>. [[phab:T300987|A code-snippet, global search, and example are available]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-09|en}}. It will be on all wikis from {{#time:j xg|2022-02-10|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/06|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W06"/> 21:15, 7 February 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22765948 --> == [[m:Special:MyLanguage/Tech/News/2022/07|Tech News: 2022-07]] == <section begin="technews-2022-W07"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/07|Translations]] are available. '''Recent changes''' * [[mw:Special:MyLanguage/Manual:Purge|Purging]] a category page with fewer than 5,000 members will now recount it completely. This will allow editors to fix incorrect counts when it is wrong. [https://phabricator.wikimedia.org/T85696] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-16|en}}. It will be on all wikis from {{#time:j xg|2022-02-17|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * [[File:Octicons-tools.svg|15px|link=|Advanced item]] In the [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] extension, the <code dir=ltr>rmspecials()</code> function has been updated so that it does not remove the "space" character. Wikis are advised to wrap all the uses of <code dir=ltr>rmspecials()</code> with <code dir=ltr>rmwhitespace()</code> wherever necessary to keep filters' behavior unchanged. You can use the search function on [[Special:AbuseFilter]] to locate its usage. [https://phabricator.wikimedia.org/T263024] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/07|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W07"/> 19:18, 14 February 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22821788 --> == [[m:Special:MyLanguage/Tech/News/2022/08|Tech News: 2022-08]] == <section begin="technews-2022-W08"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/08|Translations]] are available. '''Recent changes''' * [[Special:Nuke|Special:Nuke]] will now provide the standard deletion reasons (editable at <bdi lang="en" dir="ltr">[[MediaWiki:Deletereason-dropdown]]</bdi>) to use when mass-deleting pages. This was [[m:Community Wishlist Survey 2022/Admins and patrollers/Mass-delete to offer drop-down of standard reasons, or templated reasons.|a request in the 2022 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T25020] * At Wikipedias, all new accounts now get the [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] by default when creating an account. Communities are encouraged to [[mw:Special:MyLanguage/Help:Growth/Tools/Account_creation|update their help resources]]. Previously, only 80% of new accounts would get the Growth features. A few Wikipedias remain unaffected by this change. [https://phabricator.wikimedia.org/T301820] * You can now prevent specific images that are used in a page from appearing in other locations, such as within PagePreviews or Search results. This is done with the markup <bdi lang="zxx" dir="ltr"><code><nowiki>class=notpageimage</nowiki></code></bdi>. For example, <code><nowiki>[[File:Example.png|class=notpageimage]]</nowiki></code>. [https://phabricator.wikimedia.org/T301588] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] There has been a change to the HTML of Special:Contributions, Special:MergeHistory, and History pages, to support the grouping of changes by date in [[mw:Special:MyLanguage/Skin:Minerva_Neue|the mobile skin]]. While unlikely, this may affect gadgets and user scripts. A [[phab:T298638|list of all the HTML changes]] is on Phabricator. '''Events''' * [[m:Special:MyLanguage/Community Wishlist Survey 2022/Results|Community Wishlist Survey results]] have been published. The [[m:Special:MyLanguage/Community Wishlist Survey/Updates/2022 results#leaderboard|ranking of prioritized proposals]] is also available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-22|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-23|en}}. It will be on all wikis from {{#time:j xg|2022-02-24|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Future changes''' * The software to play videos and audio files on pages will change soon on all wikis. The old player will be removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Toolforge's underlying operating system is being updated. If you maintain any tools there, there are two options for migrating your tools into the new system. There are [[wikitech:News/Toolforge Stretch deprecation|details, deadlines, and instructions]] on Wikitech. [https://lists.wikimedia.org/hyperkitty/list/cloud-announce@lists.wikimedia.org/thread/EPJFISC52T7OOEFH5YYMZNL57O4VGSPR/] * Administrators will soon have [[m:Special:MyLanguage/Community Wishlist Survey 2021/(Un)delete associated talk page|the option to delete/undelete]] the associated "talk" page when they are deleting a given page. An API endpoint with this option will also be available. This was [[m:Community Wishlist Survey 2021/Admins and patrollers/(Un)delete associated talk page|a request from the 2021 Wishlist Survey]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/08|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W08"/> 19:12, 21 February 2022 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22847768 --> == [[m:Special:MyLanguage/Tech/News/2022/09|Tech News: 2022-09]] == <section begin="technews-2022-W09"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/09|Translations]] are available. '''Recent changes''' * When searching for edits by [[mw:Special:MyLanguage/Help:Tags|change tags]], e.g. in page history or user contributions, there is now a dropdown list of possible tags. This was [[m:Community Wishlist Survey 2022/Miscellaneous/Improve plain-text change tag selector|a request in the 2022 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T27909] * Mentors using the [[mw:Special:MyLanguage/Growth/Mentor_dashboard|Growth Mentor dashboard]] will now see newcomers assigned to them who have made at least one edit, up to 200 edits. Previously, all newcomers assigned to the mentor were visible on the dashboard, even ones without any edit or ones who made hundred of edits. Mentors can still change these values using the filters on their dashboard. Also, the last choice of filters will now be saved. [https://phabricator.wikimedia.org/T301268][https://phabricator.wikimedia.org/T294460] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The user group <code>oversight</code> was renamed <code>suppress</code>. This is for [[phab:T109327|technical reasons]]. You may need to update any local references to the old name, e.g. gadgets, links to Special:Listusers, or uses of [[mw:Special:MyLanguage/Help:Magic_words|NUMBERINGROUP]]. '''Problems''' * The recent change to the HTML of [[mw:Special:MyLanguage/Help:Tracking changes|tracking changes]] pages caused some problems for screenreaders. This is being fixed. [https://phabricator.wikimedia.org/T298638] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.24|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-02|en}}. It will be on all wikis from {{#time:j xg|2022-03-03|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''Future changes''' * Working with templates will become easier. [[m:WMDE_Technical_Wishes/Templates|Several improvements]] are planned for March 9 on most wikis and on March 16 on English Wikipedia. The improvements include: Bracket matching, syntax highlighting colors, finding and inserting templates, and related visual editor features. * If you are a template developer or an interface administrator, and you are intentionally overriding or using the default CSS styles of user feedback boxes (the classes: <code dir=ltr>successbox, messagebox, errorbox, warningbox</code>), please note that these classes and associated CSS will soon be removed from MediaWiki core. This is to prevent problems when the same class-names are also used on a wiki. Please let us know by commenting at [[phab:T300314]] if you think you might be affected. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/09|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W09"/> 22:59, 28 February 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22902593 --> == [[m:Special:MyLanguage/Tech/News/2022/10|Tech News: 2022-10]] == <section begin="technews-2022-W10"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/10|Translations]] are available. '''Problems''' * There was a problem with some interface labels last week. It will be fixed this week. This change was part of ongoing work to simplify the support for skins which do not have active maintainers. [https://phabricator.wikimedia.org/T301203] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-09|en}}. It will be on all wikis from {{#time:j xg|2022-03-10|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/10|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W10"/> 21:16, 7 March 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22958074 --> == [[m:Special:MyLanguage/Tech/News/2022/11|Tech News: 2022-11]] == <section begin="technews-2022-W11"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/11|Translations]] are available. '''Recent changes''' * In the Wikipedia Android app [[mw:Special:MyLanguage/Wikimedia_Apps/Team/Android/Communication#Updates|it is now possible]] to change the toolbar at the bottom so the tools you use more often are easier to click on. The app now also has a focused reading mode. [https://phabricator.wikimedia.org/T296753][https://phabricator.wikimedia.org/T254771] '''Problems''' * There was a problem with the collection of some page-view data from June 2021 to January 2022 on all wikis. This means the statistics are incomplete. To help calculate which projects and regions were most affected, relevant datasets are being retained for 30 extra days. You can [[m:Talk:Data_retention_guidelines#Added_exception_for_page_views_investigation|read more on Meta-wiki]]. * There was a problem with the databases on March 10. All wikis were unreachable for logged-in users for 12 minutes. Logged-out users could read pages but could not edit or access uncached content then. [https://wikitech.wikimedia.org/wiki/Incident_documentation/2022-03-10_MediaWiki_availability] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-16|en}}. It will be on all wikis from {{#time:j xg|2022-03-17|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]). * When [[mw:Special:MyLanguage/Help:System_message#Finding_messages_and_documentation|using <bdi lang="zxx" dir="ltr"><code>uselang=qqx</code></bdi> to find localisation messages]], it will now show all possible message keys for navigation tabs such as "{{int:vector-view-history}}". [https://phabricator.wikimedia.org/T300069] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Access to [[{{#special:RevisionDelete}}]] has been expanded to include users who have <code dir=ltr>deletelogentry</code> and <code dir=ltr>deletedhistory</code> rights through their group memberships. Before, only those with the <code dir=ltr>deleterevision</code> right could access this special page. [https://phabricator.wikimedia.org/T301928] * On the [[{{#special:Undelete}}]] pages for diffs and revisions, there will be a link back to the main Undelete page with the list of revisions. [https://phabricator.wikimedia.org/T284114] '''Future changes''' * The Wikimedia Foundation has announced the IP Masking implementation strategy and next steps. The [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation#feb25|announcement can be read here]]. * The [[mw:Special:MyLanguage/Wikimedia Apps/Android FAQ|Wikipedia Android app]] developers are working on [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Communication|new functions]] for user talk pages and article talk pages. [https://phabricator.wikimedia.org/T297617] '''Events''' * The [[mw:Wikimedia Hackathon 2022|Wikimedia Hackathon 2022]] will take place as a hybrid event on 20-22 May 2022. The Hackathon will be held online and there are grants available to support local in-person meetups around the world. Grants can be requested until 20 March. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/11|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W11"/> 22:07, 14 March 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=22993074 --> == [[m:Special:MyLanguage/Tech/News/2022/12|Tech News: 2022-12]] == <section begin="technews-2022-W12"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/12|Translations]] are available. '''New code release schedule for this week''' * There will be four MediaWiki releases this week, instead of just one. This is an experiment which should lead to fewer problems and to faster feature updates. The releases will be on all wikis, at different times, on Monday, Tuesday, and Wednesday. You can [[mw:Special:MyLanguage/Wikimedia Release Engineering Team/Trainsperiment week|read more about this project]]. '''Recent changes''' * You can now set how many search results to show by default in [[Special:Preferences#mw-prefsection-searchoptions|your Preferences]]. This was the 12th most popular wish in the [[m:Special:MyLanguage/Community Wishlist Survey 2022/Results|Community Wishlist Survey 2022]]. [https://phabricator.wikimedia.org/T215716] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The Jupyter notebooks tool [[wikitech:PAWS|PAWS]] has been updated to a new interface. [https://phabricator.wikimedia.org/T295043] '''Future changes''' * Interactive maps via [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] will soon work on wikis using the [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevisions]] extension. [https://wikimedia.sslsurvey.de/Kartographer-Workflows-EN/ Please tell us] which improvements you want to see in Kartographer. You can take this survey in simple English. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/12|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W12"/> 16:01, 21 March 2022 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23034693 --> == [[m:Special:MyLanguage/Tech/News/2022/13|Tech News: 2022-13]] == <section begin="technews-2022-W13"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/13|Translations]] are available. '''Recent changes''' * There is a simple new Wikimedia Commons upload tool available for macOS users, [[c:Commons:Sunflower|Sunflower]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-30|en}}. It will be on all wikis from {{#time:j xg|2022-03-31|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * Some wikis will be in read-only for a few minutes because of regular database maintenance. It will be performed on {{#time:j xg|2022-03-29|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]) and on {{#time:j xg|2022-03-31|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]). [https://phabricator.wikimedia.org/T301850][https://phabricator.wikimedia.org/T303798] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/13|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W13"/> 19:54, 28 March 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23073711 --> == [[m:Special:MyLanguage/Tech/News/2022/14|Tech News: 2022-14]] == <section begin="technews-2022-W14"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/14|Translations]] are available. '''Problems''' * For a few days last week, edits that were suggested to newcomers were not tagged in the [[{{#special:recentchanges}}]] feed. This bug has been fixed. [https://phabricator.wikimedia.org/T304747] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-06|en}}. It will be on all wikis from {{#time:j xg|2022-04-07|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-07|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]). '''Future changes''' * Starting next week, Tech News' title will be translatable. When the newsletter is distributed, its title may not be <code dir=ltr>Tech News: 2022-14</code> anymore. It may affect some filters that have been set up by some communities. [https://phabricator.wikimedia.org/T302920] * Over the next few months, the "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" Growth feature [[phab:T304110|will become available to more Wikipedias]]. Each week, a few wikis will get the feature. You can test this tool at [[mw:Special:MyLanguage/Growth#deploymentstable|a few wikis where "Link recommendation" is already available]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/14|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W14"/> 21:01, 4 April 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23097604 --> == Tech News: 2022-15 == <section begin="technews-2022-W15"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/15|Translations]] are available. '''Recent changes''' * There is a new public status page at <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wikimediastatus.net/ www.wikimediastatus.net]</span>. This site shows five automated high-level metrics where you can see the overall health and performance of our wikis' technical environment. It also contains manually-written updates for widespread incidents, which are written as quickly as the engineers are able to do so while also fixing the actual problem. The site is separated from our production infrastructure and hosted by an external service, so that it can be accessed even if the wikis are briefly unavailable. You can [https://diff.wikimedia.org/2022/03/31/announcing-www-wikimediastatus-net/ read more about this project]. * On Wiktionary wikis, the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-13|en}}. It will be on all wikis from {{#time:j xg|2022-04-14|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/15|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W15"/> 19:44, 11 April 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23124108 --> == Tech News: 2022-16 == <section begin="technews-2022-W16"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/16|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-20|en}}. It will be on all wikis from {{#time:j xg|2022-04-21|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-19|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]) and on {{#time:j xg|2022-04-21|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s8.dblist targeted wikis]). * Administrators will now have [[m:Community Wishlist Survey 2021/(Un)delete associated talk page|the option to delete/undelete the associated "Talk" page]] when they are deleting a given page. An API endpoint with this option is also available. This concludes the [[m:Community Wishlist Survey 2021/Admins and patrollers/(Un)delete associated talk page|11th wish of the 2021 Community Wishlist Survey]]. * On [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements#test-wikis|selected wikis]], 50% of logged-in users will see the new [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Table of contents|table of contents]]. When scrolling up and down the page, the table of contents will stay in the same place on the screen. This is part of the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Desktop Improvements]] project. [https://phabricator.wikimedia.org/T304169] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Message boxes produced by MediaWiki code will no longer have these CSS classes: <code dir=ltr>successbox</code>, <code dir=ltr>errorbox</code>, <code dir=ltr>warningbox</code>. The styles for those classes and <code dir=ltr>messagebox</code> will be removed from MediaWiki core. This only affects wikis that use these classes in wikitext, or change their appearance within site-wide CSS. Please review any local usage and definitions for these classes you may have. This was previously announced in the [[m:Special:MyLanguage/Tech/News/2022/09|28 February issue of Tech News]]. '''Future changes''' * [[mw:Special:MyLanguage/Extension:Kartographer|Kartographer]] will become compatible with [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevisions page stabilization]]. Kartographer maps will also work on pages with [[mw:Special:MyLanguage/Help:Pending changes|pending changes]]. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation#Project_descriptions] The Kartographer documentation has been thoroughly updated. [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Extension:Kartographer/Getting_started] [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:VisualEditor/Maps] [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Extension:Kartographer] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/16|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W16"/> 23:11, 18 April 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23167004 --> == Tech News: 2022-17 == <section begin="technews-2022-W17"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/17|Translations]] are available. '''Recent changes''' * On [https://noc.wikimedia.org/conf/dblists/group1.dblist many wikis] (group 1), the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-27|en}}. It will be on all wikis from {{#time:j xg|2022-04-28|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-26|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s2.dblist targeted wikis]). * Some very old browsers and operating systems are no longer supported. Some things on the wikis might look weird or not work in very old browsers like Internet Explorer 9 or 10, Android 4, or Firefox 38 or older. [https://phabricator.wikimedia.org/T306486] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/17|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W17"/> 22:56, 25 April 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23187115 --> == Tech News: 2022-18 == <section begin="technews-2022-W18"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/18|Translations]] are available. '''Recent changes''' * On [https://noc.wikimedia.org/conf/dblists/group2.dblist all remaining wikis] (group 2), the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-03|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-04|en}}. It will be on all wikis from {{#time:j xg|2022-05-05|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''Future changes''' * The developers are working on talk pages in the [[mw:Wikimedia Apps/Team/iOS|Wikipedia app for iOS]]. You can [https://wikimedia.qualtrics.com/jfe/form/SV_9GBcHczQGLbQWTY give feedback]. You can take the survey in English, German, Hebrew or Chinese. * [[m:WMDE_Technical_Wishes/VisualEditor_template_dialog_improvements#Status_and_next_steps|Most wikis]] will receive an [[m:WMDE_Technical_Wishes/VisualEditor_template_dialog_improvements|improved template dialog]] in VisualEditor and New Wikitext mode. [https://phabricator.wikimedia.org/T296759] [https://phabricator.wikimedia.org/T306967] * If you use syntax highlighting while editing wikitext, you can soon activate a [[m:WMDE_Technical_Wishes/Improved_Color_Scheme_of_Syntax_Highlighting#Color-blind_mode|colorblind-friendly color scheme]]. [https://phabricator.wikimedia.org/T306867] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Several CSS IDs related to MediaWiki interface messages will be removed. Technical editors should please [[phab:T304363|review the list of IDs and links to their existing uses]]. These include <code dir=ltr>#mw-anon-edit-warning</code>, <code dir=ltr>#mw-undelete-revision</code> and 3 others. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/18|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W18"/> 19:33, 2 May 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23232924 --> == Tech News: 2022-19 == <section begin="technews-2022-W19"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/19|Translations]] are available. '''Recent changes''' * You can now see categories in the [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android|Wikipedia app for Android]]. [https://phabricator.wikimedia.org/T73966] '''Problems''' * Last week, there was a problem with Wikidata's search autocomplete. This has now been fixed. [https://phabricator.wikimedia.org/T307586] * Last week, all wikis had slow access or no access for 20 minutes, for logged-in users and non-cached pages. This was caused by a problem with a database change. [https://phabricator.wikimedia.org/T307647] '''Changes later this week''' * There is no new MediaWiki version this week. [https://phabricator.wikimedia.org/T305217#7894966] * [[m:WMDE Technical Wishes/Geoinformation#Current issues|Incompatibility issues]] with [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] and the [[mw:Special:MyLanguage/Help:Extension:FlaggedRevs|FlaggedRevs extension]] will be fixed: Deployment is planned for May 10 on all wikis. Kartographer will then be enabled on the [[phab:T307348|five wikis which have not yet enabled the extension]] on May 24. * The [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector (2022)]] skin will be set as the default on several more wikis, including Arabic and Catalan Wikipedias. Logged-in users will be able to switch back to the old Vector (2010). See the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/2022-04 for the largest wikis|latest update]] about Vector (2022). '''Future meetings''' * The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place on 17 May. The following meetings are currently planned for: 7 June, 21 June, 5 July, 19 July. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/19|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W19"/> 15:22, 9 May 2022 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23256717 --> == Tech News: 2022-20 == <section begin="technews-2022-W20"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/20|Translations]] are available. '''Changes later this week''' * Some wikis can soon use the [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|add a link]] feature. This will start on Wednesday. The wikis are {{int:project-localized-name-cawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ptwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-simplewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-svwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ukwiki/en}}. This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304542] * The [[mw:Special:MyLanguage/Wikimedia Hackathon 2022|Wikimedia Hackathon 2022]] will take place online on May 20–22. It will be in English. There are also local [[mw:Special:MyLanguage/Wikimedia Hackathon 2022/Meetups|hackathon meetups]] in Germany, Ghana, Greece, India, Nigeria and the United States. Technically interested Wikimedians can work on software projects and learn new skills. You can also host a session or post a project you want to work on. * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-18|en}}. It will be on all wikis from {{#time:j xg|2022-05-19|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''Future changes''' * You can soon edit translatable pages in the visual editor. Translatable pages exist on for examples Meta and Commons. [https://diff.wikimedia.org/2022/05/12/mediawiki-1-38-brings-support-for-editing-translatable-pages-with-the-visual-editor/] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/20|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W20"/> 18:58, 16 May 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23291515 --> == Tech News: 2022-21 == <section begin="technews-2022-W21"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/21|Translations]] are available. '''Recent changes''' * Administrators using the mobile web interface can now access Special:Block directly from user pages. [https://phabricator.wikimedia.org/T307341] * The <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wiktionary.org/ www.wiktionary.org]</span> portal page now uses an automated update system. Other [[m:Project_portals|project portals]] will be updated over the next few months. [https://phabricator.wikimedia.org/T304629] '''Problems''' * The Growth team maintains a mentorship program for newcomers. Previously, newcomers weren't able to opt out from the program. Starting May 19, 2022, newcomers are able to fully opt out from Growth mentorship, in case they do not wish to have any mentor at all. [https://phabricator.wikimedia.org/T287915] * Some editors cannot access the content translation tool if they load it by clicking from the contributions menu. This problem is being worked on. It should still work properly if accessed directly via Special:ContentTranslation. [https://phabricator.wikimedia.org/T308802] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-25|en}}. It will be on all wikis from {{#time:j xg|2022-05-26|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''Future changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Gadget and user scripts developers are invited to give feedback on a [[mw:User:Jdlrobson/Extension:Gadget/Policy|proposed technical policy]] aiming to improve support from MediaWiki developers. [https://phabricator.wikimedia.org/T308686] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/21|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W21"/> 00:21, 24 May 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23317250 --> == Tech News: 2022-22 == <section begin="technews-2022-W22"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/22|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|Advanced item]] In the [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] extension, an <code dir=ltr>ip_in_ranges()</code> function has been introduced to check if an IP is in any of the ranges. Wikis are advised to combine multiple <code dir=ltr>ip_in_range()</code> expressions joined by <code>|</code> into a single expression for better performance. You can use the search function on [[Special:AbuseFilter|Special:AbuseFilter]] to locate its usage. [https://phabricator.wikimedia.org/T305017] * The [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature|IP Info feature]] which helps abuse fighters access information about IPs, [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#May 24, 2022|has been deployed]] to all wikis as a beta feature. This comes after weeks of beta testing on test.wikipedia.org. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.14|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-01|en}}. It will be on all wikis from {{#time:j xg|2022-06-02|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-05-31|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]). * The [[mw:Special:MyLanguage/Help:DiscussionTools#New topic tool|New Topic Tool]] will be deployed for all editors at most wikis soon. You will be able to opt out from within the tool and in [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Talk_pages_project/New_discussion][https://phabricator.wikimedia.org/T287804] * [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[:mw:Special:ApiHelp/query+usercontribs|list=usercontribs API]] will support fetching contributions from an [[mw:Special:MyLanguage/Help:Range blocks#Non-technical explanation|IP range]] soon. API users can set the <code>uciprange</code> parameter to get contributions from any IP range within [[:mw:Manual:$wgRangeContributionsCIDRLimit|the limit]]. [https://phabricator.wikimedia.org/T177150] * A new parser function will be introduced: <bdi lang="zxx" dir="ltr"><code><nowiki>{{=}}</nowiki></code></bdi>. It will replace existing templates named "=". It will insert an [[w:en:Equals sign|equal sign]]. This can be used to escape the equal sign in the parameter values of templates. [https://phabricator.wikimedia.org/T91154] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/22|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W22"/> 20:28, 30 May 2022 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23340178 --> == Tech News: 2022-23 == <section begin="technews-2022-W23"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/23|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-08|en}}. It will be on all wikis from {{#time:j xg|2022-06-09|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] A new <bdi lang="zxx" dir="ltr"><code>str_replace_regexp()</code></bdi> function can be used in [[Special:AbuseFilter|abuse filters]] to replace parts of text using a [[w:en:Regular expression|regular expression]]. [https://phabricator.wikimedia.org/T285468] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/23|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W23"/> 02:46, 7 June 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23366979 --> == Tech News: 2022-24 == <section begin="technews-2022-W24"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/24|Translations]] are available. '''Recent changes''' * All wikis can now use [[mw:Special:MyLanguage/Extension:Kartographer|Kartographer]] maps. Kartographer maps now also work on pages with [[mw:Special:MyLanguage/Help:Pending changes|pending changes]]. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation#Project_descriptions][https://phabricator.wikimedia.org/T307348] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-15|en}}. It will be on all wikis from {{#time:j xg|2022-06-16|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-14|en}} at 06:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]). [https://phabricator.wikimedia.org/T300471] * Starting on Wednesday, a new set of Wikipedias will get "[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]" ({{int:project-localized-name-abwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-acewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-adywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-afwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-akwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-alswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-amwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-anwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-angwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-arcwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-arzwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-astwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-atjwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-avwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-aywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-azwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-azbwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304548] * The [[mw:Special:MyLanguage/Help:DiscussionTools#New topic tool|New Topic Tool]] will be deployed for all editors at Commons, Wikidata, and some other wikis soon. You will be able to opt out from within the tool and in [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Talk_pages_project/New_discussion][https://phabricator.wikimedia.org/T287804] '''Future meetings''' * The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place today (13 June). The following meetings will take place on: 28 June, 12 July, 26 July. '''Future changes''' * By the end of July, the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector 2022]] skin should be ready to become the default across all wikis. Discussions on how to adjust it to the communities' needs will begin in the next weeks. It will always be possible to revert to the previous version on an individual basis. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/2022-04 for the largest wikis|Learn more]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/24|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W24"/> 16:58, 13 June 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23389956 --> == Tech News: 2022-25 == <section begin="technews-2022-W25"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/25|Translations]] are available. '''Recent changes''' * The [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android|Wikipedia App for Android]] now has an option for editing the whole page at once, located in the overflow menu (three-dots menu [[File:Ic more vert 36px.svg|15px|link=|alt=]]). [https://phabricator.wikimedia.org/T103622] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Some recent database changes may affect queries using the [[m:Research:Quarry|Quarry tool]]. Queries for <bdi lang="zxx" dir="ltr"><code>site_stats</code></bdi> at English Wikipedia, Commons, and Wikidata will need to be updated. [[phab:T306589|Read more]]. * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] A new <bdi lang="zxx" dir="ltr"><code>user_global_editcount</code></bdi> variable can be used in [[Special:AbuseFilter|abuse filters]] to avoid affecting globally active users. [https://phabricator.wikimedia.org/T130439] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-22|en}}. It will be on all wikis from {{#time:j xg|2022-06-23|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * Users of non-responsive skins (e.g. MonoBook or Vector) on mobile devices may notice a slight change in the default zoom level. This is intended to optimize zooming and ensure all interface elements are present on the page (for example the table of contents on Vector 2022). In the unlikely event this causes any problems with how you use the site, we'd love to understand better, please ping <span class="mw-content-ltr" lang="en" dir="ltr">[[m:User:Jon (WMF)|Jon (WMF)]]</span> to any on-wiki conversations. [https://phabricator.wikimedia.org/T306910] '''Future changes''' * The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July. Discussions will look different. You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Parsoid's HTML output will soon stop annotating file links with different <bdi lang="zxx" dir="ltr"><code>typeof</code></bdi> attribute values, and instead use <bdi lang="zxx" dir="ltr"><code>mw:File</code></bdi> for all types. Tool authors should adjust any code that expects: <bdi lang="zxx" dir="ltr"><code>mw:Image</code></bdi>, <bdi lang="zxx" dir="ltr"><code>mw:Audio</code></bdi>, or <bdi lang="zxx" dir="ltr"><code>mw:Video</code></bdi>. [https://phabricator.wikimedia.org/T273505] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/25|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W25"/> 20:18, 20 June 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23425855 --> == Tech News: 2022-26 == <section begin="technews-2022-W26"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/26|Translations]] are available. '''Recent changes''' * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] API service now has self-service accounts with free on-demand requests and monthly snapshots ([https://enterprise.wikimedia.com/docs/ API documentation]). Community access [[m:Special:MyLanguage/Wikimedia Enterprise/FAQ#community-access|via database dumps & Wikimedia Cloud Services]] continues. * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] [[d:Special:MyLanguage/Wikidata:Wiktionary#lua|All Wikimedia wikis can now use Wikidata Lexemes in Lua]] after creating local modules and templates. Discussions are welcome [[d:Wikidata_talk:Lexicographical_data#You_can_now_reuse_Wikidata_Lexemes_on_all_wikis|on the project talk page]]. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-29|en}}. It will be on all wikis from {{#time:j xg|2022-06-30|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-28|en}} at 06:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]). [https://phabricator.wikimedia.org/T311033] * Some global and cross-wiki services will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-30|en}} at 06:00 UTC. This will impact ContentTranslation, Echo, StructuredDiscussions, Growth experiments and a few more services. [https://phabricator.wikimedia.org/T300472] * Users will be able to sort columns within sortable tables in the mobile skin. [https://phabricator.wikimedia.org/T233340] '''Future meetings''' * The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place tomorrow (28 June). The following meetings will take place on 12 July and 26 July. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/26|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W26"/> 20:02, 27 June 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23453785 --> == Tech News: 2022-27 == <section begin="technews-2022-W27"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/27|Translations]] are available. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-06|en}}. It will be on all wikis from {{#time:j xg|2022-07-07|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-07-05|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]) and on {{#time:j xg|2022-07-07|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]). * The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July. Discussions will look different. You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. * [[File:Octicons-tools.svg|15px|link=|alt=| Advanced item]] This change only affects pages in the main namespace in Wikisource. The Javascript config variable <bdi lang="zxx" dir="ltr"><code>proofreadpage_source_href</code></bdi> will be removed from <bdi lang="zxx" dir="ltr"><code>[[mw:Special:MyLanguage/Manual:Interface/JavaScript#mw.config|mw.config]]</code></bdi> and be replaced with the variable <bdi lang="zxx" dir="ltr"><code>prpSourceIndexPage</code></bdi>. [https://phabricator.wikimedia.org/T309490] '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/27|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W27"/> 19:32, 4 July 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23466250 --> == Tech News: 2022-28 == <section begin="technews-2022-W28"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/28|Translations]] are available. '''Recent changes''' * In the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector 2022 skin]], the page title is now displayed above the tabs such as Discussion, Read, Edit, View history, or More. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates#Page title/tabs switch|Learn more]]. [https://phabricator.wikimedia.org/T303549] * [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] It is now possible to easily view most of the configuration settings that apply to just one wiki, and to compare settings between two wikis if those settings are different. For example: [https://noc.wikimedia.org/wiki.php?wiki=jawiktionary Japanese Wiktionary settings], or [https://noc.wikimedia.org/wiki.php?wiki=eswiki&compare=eowiki settings that are different between the Spanish and Esperanto Wikipedias]. Local communities may want to [[m:Special:MyLanguage/Requesting_wiki_configuration_changes|discuss and propose changes]] to their local settings. Details about each of the named settings can be found by [[mw:Special:Search|searching MediaWiki.org]]. [https://phabricator.wikimedia.org/T308932] *The Anti-Harassment Tools team [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#May|recently deployed]] the IP Info Feature as a [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature at all wikis]]. This feature allows abuse fighters to access information about IP addresses. Please check our update on [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#April|how to find and use the tool]]. Please share your feedback using a link you will be given within the tool itself. '''Changes later this week''' * There is no new MediaWiki version this week. * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-07-12|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]). '''Future changes''' * The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July. Discussions will look different. You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/28|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W28"/> 19:24, 11 July 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23502519 --> == Tech News: 2022-29 == <section begin="technews-2022-W29"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/29|Translations]] are available. '''Problems''' * The feature on mobile web for [[mw:Special:MyLanguage/Extension:NearbyPages|Nearby Pages]] was missing last week. It will be fixed this week. [https://phabricator.wikimedia.org/T312864] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-20|en}}. It will be on all wikis from {{#time:j xg|2022-07-21|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). '''Future changes''' * The [[mw:Technical_decision_making/Forum|Technical Decision Forum]] is seeking [[mw:Technical_decision_making/Community_representation|community representatives]]. You can apply on wiki or by emailing <span class="mw-content-ltr" lang="en" dir="ltr">TDFSupport@wikimedia.org</span> before 12 August. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/29|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W29"/> 22:59, 18 July 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23517957 --> == Tech News: 2022-30 == <section begin="technews-2022-W30"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/30|Translations]] are available. '''Recent changes''' * The <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wikibooks.org/ www.wikibooks.org]</span> and <span class="mw-content-ltr" lang="en" dir="ltr">[https://www.wikiquote.org/ www.wikiquote.org]</span> portal pages now use an automated update system. Other [[m:Project_portals|project portals]] will be updated over the next few months. [https://phabricator.wikimedia.org/T273179] '''Problems''' * Last week, some wikis were in read-only mode for a few minutes because of an emergency switch of their main database ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]). [https://phabricator.wikimedia.org/T313383] '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-27|en}}. It will be on all wikis from {{#time:j xg|2022-07-28|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * The external link icon will change slightly in the skins Vector legacy and Vector 2022. The new icon uses simpler shapes to be more recognizable on low-fidelity screens. [https://phabricator.wikimedia.org/T261391] * Administrators will now see buttons on user pages for "{{int:changeblockip}}" and "{{int:unblockip}}" instead of just "{{int:blockip}}" if the user is already blocked. [https://phabricator.wikimedia.org/T308570] '''Future meetings''' * The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place tomorrow (26 July). '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/30|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W30"/> 19:27, 25 July 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23545370 --> == Tech News: 2022-31 == <section begin="technews-2022-W31"/><div class="plainlinks"> Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/31|Translations]] are available. '''Recent changes''' * Improved [[m:Special:MyLanguage/Help:Displaying_a_formula#Phantom|LaTeX capabilities for math rendering]] are now available in the wikis thanks to supporting <bdi lang="zxx" dir="ltr"><code>Phantom</code></bdi> tags. This completes part of [[m:Community_Wishlist_Survey_2022/Editing/Missing_LaTeX_capabilities_for_math_rendering|the #59 wish]] of the 2022 Community Wishlist Survey. '''Changes later this week''' * [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-08-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-08-03|en}}. It will be on all wikis from {{#time:j xg|2022-08-04|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]). * The [[mw:Special:MyLanguage/Help:Extension:WikiEditor/Realtime_Preview|Realtime Preview]] will be available as a Beta Feature on wikis in [https://noc.wikimedia.org/conf/highlight.php?file=dblists%2Fgroup0.dblist Group 0]. This feature was built in order to fulfill [[m:Special:MyLanguage/Community_Wishlist_Survey_2021/Real_Time_Preview_for_Wikitext|one of the Community Wishlist Survey proposals]]. '''Future changes''' * The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout August. Discussions will look different. You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. '''Future meetings''' * This week, three meetings about [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector (2022)]] with live interpretation will take place. On Tuesday, interpretation in Russian will be provided. On Thursday, meetings for Arabic and Spanish speakers will take place. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|See how to join]]. '''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&nbsp;• [[m:Special:MyLanguage/Tech/News/2022/31|Translate]]&nbsp;• [[m:Tech|Get help]]&nbsp;• [[m:Talk:Tech/News|Give feedback]]&nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' </div><section end="technews-2022-W31"/> 21:21, 1 August 2022 (UTC) <!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&oldid=23615613 --> 93hm7mgzqry73krm7yhuuubcbtcdpzg 0 263444 2410833 2410467 2022-08-01T22:40:46Z Scogdill 1331941 wikitext text/x-wiki == Also Known As == *Louise, Duchess of Devonshire *Louisa, Duchess of Manchester *Luise Friederike August Gräfin von Alten *Louisa Montagu *Louise Cavendish *The Double Duchess == Acquaintances, Friends and Enemies == === Friends === *[[Social Victorians/People/Albert Edward, Prince of Wales | Albert Edward, Prince of Wales]] (beginning about 1852) *[[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish]], Lord Hartington (later 8th Duke of Devonshire) *Daisy, Lady Warwick *Lady Mayoress, Mrs. Benjamin Samuel Faudel-Phillips, 2nd Baronet,<ref>{{Cite journal|date=2020-08-25|title=Faudel-Phillips baronets|url=https://en.wikipedia.org/w/index.php?title=Faudel-Phillips_baronets&oldid=974879290|journal=Wikipedia|language=en}}</ref> presented to Victoria by Louisa Cavendish at a Queen's Drawing-room on Wednesday, 24 February 1897 at Buckingham Palace.<ref name=":4">"The Queen's Drawing Room" ''Morning Post'' 25 February 1897 Thursday: 5 [of 10], Col. 5a–7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970225/047/0005.</ref>{{rp|p. 5, Col. 6c}} *Mrs. J. E. Mellor, presented to Victoria by Louisa Cavendish at a Queen's Drawing-room on Wednesday, 24 February 1897 at Buckingham Palace.<ref name=":4" />{{rp|p. 5, Col. 6c}} === Enemies === * Consuelo, Duchess of Marlborough (at least, in 1901)<ref name=":1">Murphy, Sophia. ''The Duchess of Devonshire's Ball''. London: Sidgwick & Jackson, 1984.</ref>{{rp|pp. 31–32}} == Organizations == == Timeline == '''1852 July 22''', Luise Friederike Auguste Gräfin von Alten and William Drogo Montagu married.<ref name=":2">"Luise Friederike Auguste Gräfin von Alten." {{Cite web|url=http://www.thepeerage.com/p10947.htm#i109469|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1863, early, or late 1862''', Louise and Spencer Compton Cavendish began a relationship.<ref name=":1" />{{rp|p. 26}} '''1873 December 10''', Mary Louise Elizabeth Montagu (daughter) and William Douglas-Hamilton married. '''1876 May 22''', Consuelo Iznaga y Clement and George Victor Drogo Montagu (son) married in Grace Church, New York City.<ref>{{Cite journal|date=2020-08-24|title=George Montagu, 8th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=George_Montagu,_8th_Duke_of_Manchester&oldid=974659520|journal=Wikipedia|language=en}}</ref><ref>{{Cite journal|date=2020-07-27|title=Consuelo Montagu, Duchess of Manchester|url=https://en.wikipedia.org/w/index.php?title=Consuelo_Montagu,_Duchess_of_Manchester&oldid=969888488|journal=Wikipedia|language=en}}</ref> '''1876 August 10''', Louisa Augusta Beatrice Montagu (daughter) and Archibald Acheson married. '''1889 January 5''', Alice Maude Olivia Montagu (daughter) and Edward Stanley married. '''1890 March 22''', William Drogo Montagu (7th Duke) died.<ref name=":3">"William Drogo Montagu, 7th Duke of Manchester." {{Cite web|url=http://www.thepeerage.com/p10128.htm#i101274|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1890 November 14''', William Angus Drogo Montagu (grandson) and Helena Zimmerman married secretly, in London.<ref>"Helena Zimmerman." {{Cite web|url=http://www.thepeerage.com/p34555.htm#i345545|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1892 August 16''', Louise Friederike Auguste Gräfin von Alten Montagu and Spencer Compton Cavendish, her second husband, married.<ref name=":2" /> '''1897 July 2, Friday''', Louise Cavendish (#18 on the list of attendees) hosted her famous [[Social Victorians/1897 Fancy Dress Ball| fancy-dress ball]] at Devonshire House in London. '''1897 July 20''', Mary Louise Elizabeth Montagu Douglas-Hamilton and Robert Carnaby Foster married. '''1900 November 14''', William Angus Drogo Montagu and Helena Zimmerman married.<ref>{{Cite journal|date=2020-07-17|title=Helena, Countess of Kintore|url=https://en.wikipedia.org/w/index.php?title=Helena,_Countess_of_Kintore&oldid=968067371|journal=Wikipedia|language=en}}</ref> '''1901 Spring''', Paris, Consuelo Spencer-Churchill, Duchess of Marlborough, describes a meeting with Louise Cavendish in the spring following Queen Victoria's death at the horse racetrack, Longchamps:<blockquote>A renowned character and virtually dictator of what was known as the fast set as opposed to the Victorian, Her Grace was a German aristocrat by birth. She had first been married to the impoverished Duke of Manchester, and when he died had improved her status by marriage to the rich Duke of Devonshire, who waged an undisputed influence in politics. Rumour had her beautiful, but when I knew her she was a raddled old woman, covering her wrinkles with paint and her pate with a brown wig. Her mouth was a red gash and from it, when she saw me, issued a stream of abuse. How could I, she complained, pointing to my white gloves, show so little respect to the memory of a great Queen? What a carefree world we must have lived in, that etiquette even in such small matters could assume so much importance?<ref>Balsan, Consuelo Vanderbilt. ''The Glitter and the Gold: The American Duchess — In Her Own Words''. New York: St. Martin's, 1953.</ref>{{rp|p. 115}}</blockquote> === Annual Events === Every year, as Duchess of Devonshire, Louise held a dance on the night after the Derby at Epsom Downs, which at this point was held on Wednesdays after Easter. == Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball == [[File:Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester) as Zenobia, Queen of Palmyra.jpg|thumb|Louise, Duchess of Devonshire as Zenobia, Queen of Palmyra|alt=Louise, Duchess of Devonshire in costume as Zenobia, Queen of Palmyra]] At their fancy-dress ball, Louisa, Duchess of Devonshire sat at Table 1 during the first seating for supper, escorted in to the table by the Prince of Wales.<ref name=":7">"Fancy Dress Ball at Devonshire House." ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.</ref>{{rp|p. 7, Col. 4c}} Her costume was designed by M. Comelli (Attillo Giuseppe Comelli, 1858–1925, artist and costumier for opera, ballet and theatre in London as well as Europe and the U.S.<ref name=":13">{{Cite book|url=https://books.google.com/books?id=SZh2DwAAQBAJ&pg=PT207&lpg=PT207&dq=Attilio+Comelli&source=bl&ots=lFB0If7CwV&sig=ACfU3U1_Ost_lhmMvzMMs6NvuhK5SlRhJw&hl=en&sa=X&ved=2ahUKEwjKlsTw2sH3AhXYAp0JHVIxDWA4KBDoAXoECBAQAw#v=onepage&q=Attilio%20Comelli&f=false|title=Forgotten Designers Costume Designers of American Broadway Revues and Musicals From 1900-1930|last=Unruh|first=Delbert|date=2018-11-06|publisher=Page Publishing Inc|isbn=978-1-64082-758-5|language=en}} N.P.</ref>)<ref name=":5">“The Devonshire House Ball.” The ''Man of Ross'' 10 July 1897, Saturday: 2 [of 8], Col. 4b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0001463/18970710/033/0002.</ref> <ref name=":8">"The Duchess of Devonshire's Fancy Dress Ball. Special Telegram." ''Belfast News-Letter'' Saturday 03 July 1897: 5 [of 8], Col. 9 [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000038/18970703/015/0005.</ref>{{rp|p. 5, Col. 9a}} <ref name=":9">"By One Who Was There." “The Duchess’s Costume Ball.” ''Westminster Gazette'' 03 July 1897 Saturday: 5 [of 8], Cols. 1a–3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002947/18970703/035/0005.</ref> and constructed by the House of Worth. Comelli seems to have designed [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess and Her Entourage|the costumes of her retinue as well]]. According to Russell Harris,<blockquote>For her costume, the Duchess commissioned Monsieur Comelli (1858-1925), a well-known designer of opera costumes for the London theatre and opera stage, and then had the design made up by Worth of Paris. ''Munsey’s Magazine'' noted “it is safe to say that the Queen of Palmyra never owned such a sumptuous costume in her lifetime.”<ref>Harris, Russell. {{Cite web|url=http://www.rvondeh.dircon.co.uk/incalmprose/devonshiredss.html|title=Louise, Duchess of Devonshire, née Countess von Alten of Hanover (1832-1911), as Zenobia, Queen of Palmyra|website=www.rvondeh.dircon.co.uk|access-date=2022-05-05}} ''Narrated in Calm Prose: Photographs from the V&A's Lafayette Archive of Guests in Costume at the Duchess of Devonshire's Diamond Jubilee Ball, July 1897''. http://www.rvondeh.dircon.co.uk/incalmprose/devonshiredss.html.</ref></blockquote>Lafayette's portrait of "Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester)" in costume is photogravure #5 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref>"Devonshire House Fancy Dress Ball (1897): photogravures by Walker & Boutall after various photographers." 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.</ref> The printing on the portrait says, "The Duchess of Devonshire as Zenobia Queen of Palmyra," with a Long S in ''Duchess''.<ref>"Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester) as Zenobia, Queen of Palmyra." Devonshire House Fancy Dress Ball Album. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158357/Louise-Frederica-Augusta-Cavendish-ne-von-Alten-Duchess-of-Devonshire-formerly-Duchess-of-Manchester-as-Zenobia-Queen-of-Palmyra.</ref> Often, the V&A Lafayette Archive contains more than one portrait of a sitter for this ball, but the uncropped portrait (above right), which shows the unfinished end of the balustrade in front of the Duchess and the edge of the painted flat behind it, seems to have been the only portrait taken by Lafayette of the Duchess in costume. The copy owned by the National Portrait Gallery in London and the copy included in the album are cropped so that those unfinished edges do not show, but they appear to be from the same photograph. === Newspaper Descriptions of the Duchess's Costume === Newspaper articles about the Duchess's presence at the ball focused on her hosting, her costume, [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Jewelry|her jewelry]], and [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Entourage|her entourage]], often in the same story. These almost exactly identical descriptions suggest [[Social Victorians/1897 Fancy Dress Ball/anthology#Scissors-and-Paste Journalism|scissors-and-paste journalism]] or a shared primary source: * The London ''Evening Standard'', ''Morning Post'' and ''Times'' published almost identical descriptions, varying only by the occasional punctuation mark or short phrase ("supplied by Worth, of Paris"). The fullest version is this one: "The Duchess of Devonshire, as Zenobia, Queen of Palmyra, wore a magnificent costume, supplied by Worth, of Paris. The skirt of gold tissue was embroidered all over in a star-like design in emeralds, sapphires, diamonds, and other jewels, outlined with gold, the corners where it opened in front being elaborately wrought in the same jewels and gold to represent peacocks' outspread tails. This opened to show an under-dress of cream crêpe de chine, delicately embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was attached to the shoulders by two slender points, and was fastened at the waist with a large diamond ornament. It was of green velvet of a lovely shade, and was superbly embroidered in Oriental designs, introducing the lotus flower in rubies, sapphires, amethysts, emeralds, and diamonds, in four borderings on contrasting grounds, separated with gold cord. The train was lined with turquoise satin. The bodice was composed of gold tissue to match the skirt, and the front was of crêpe de chine, hidden with a stomacher of real diamonds, rubies, and emeralds, and there was a jewelled belt."<ref name=":16">“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.</ref>{{rp|p. 3, Col. 2b}} <ref name=":7" />{{rp|p. 7, Col. 7a}} <ref name=":6">"Ball at Devonshire House." The ''Times'' Saturday 3 July 1897: 12, Cols. 1A–4C ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref>{{rp|p. 12, Col. 3b}} *"The Duchess of Devonshire was a dazzling vision, dressed as 'Zenobia,' in a glistening gold gauze gown, elaborately ornamented with suns and discs, wrought in purple and green gems outlined with gold, and having a large diamond as centre. The space between was fluted with fine silver spangles. This robe was open in front over an under dress of white crépe de chine, delicately worked in crystals, and at each side of the opening on the gold robe were large fan-shaped groups of peacock feathers, worked in multicoloured jewels. The [[Social Victorians/Terminology#Corsage|corsage]] was to correspond, and had a magnificent [[Social Victorians/Terminology#Girdle|girdle]] of jewels, the train of bright green velvet, hung like a fan, without folds, being fastened at each side of the shoulders by diamond brooches, and caught at the waist with a similar ornament. It was a mass of gorgeous embroidery, carried out in heliotrope velvet, lotus flowers studded with tinted gems, and other devices in terra-cotta and electric blue velvet — all enriched with gold, diamond, and jewelled embroidery — and lined with pale blue satin. ... Attending the hostess were four children, four fan-bearers, and four trumpeters, all magnificently arrayed in artistically embroidered Assyrian robes, helmets, and other accessories, correct in every detail."<ref name=":15">"Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses." London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.</ref>{{rp|p. 5, Col. 6a}} * According to the article in ''The Graphic'' written by [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]], though this caption to the Lafayette photograph may not be hers, the Duchess of Devonshire wore a "Skirt of gold tissue, embroidered all over with emeralds, sapphires, diamonds, and other jewels outlined with gold. This opened to show an underdress of crème crêpe de chine, embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was green velvet, superbly embroidered in Oriental designs. The bodice was composed of gold tissue, and the front was of crêpe de chine hidden with a stomacher of diamonds, rubies, and emeralds."<ref name=":10">Greville, Violet, Lady. "Devonshire House Ball." The ''Graphic'' Saturday 10 July 1897: 15 [of 24]: Col. 1a–16, Col. 1c. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000057/18970710/019/0015.</ref>{{rp|p. 15, Col. 3b}} The photograph of the Duchess taken by Lafayette in her costume could not do justice to the colors of the costume, of course, because it is in black and white. Fortunately, this costume has survived and is held in Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth). Published photographs of it '''show X'''.<ref>Burlington, Julia, and Hamish Bowles. ''House Style: Five Centuries of Fashion at Chatsworth''. Skira Rizzoli, 2017. Based on an exhibition 25 March to 22 October 2017.</ref> It has been worn by '''X, the xth duchess'''. == The Duchess's Jewelry == The Duchess was known for her jewelry, and as she aged and after her marriage to the 8th Duke in 1892, the jewelry she wore only increased in scale and notability. It is not surprising that her jewelry would make up copy for the newspapers. Gossipy newspaper reports before the ball reported on the jewelry associated with the costumes being developed. For example, according to the Edinburgh ''Evening News'' on 21 June 1897, less than two weeks before the party, "The ball being a fancy dress one, men as well as women will be able in certain characters to wear jewels. The Duchess of Devonshire, who is to appear as Zenobia, is getting her jewels reset after the antique style."<ref>“The Duchess of Devonshire’s Ball.” Edinburgh ''Evening News'' 21 June 1897, Monday: 4 [of 6], Col. 5c [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18970621/079/0004.</ref> Some accounts of her costume suggest that the reporter thought that all the jewels sewn onto the dress were actual precious or semiprecious stones, but the London ''Daily News'' says some of them were crystals.<ref name=":15" />(p. 5, Col. 6a) While almost all descriptions of her mention her jewels because they were so much a topic of conversation and because they were sewn onto the costume itself, these emphasize her jewelry and most are very similar to each other: * "The Duchess was attired with great Oriental magnificence as Zenobia. Her dress was a tissue of silver, embroidered with gold and jewels, an overmantle of cloth of gold embroidered in the same manner hung from the shoulders, and she wore a bandeau of gold studded with gems, and surrounded by hanging chains of pearls over her elaborate headdress; strings and ropes of jewels and pearls were worn round the neck, and hung down almost to the knees."<ref>“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032. </ref>{{rp|p. 32, Cols. 1c–2a}} <ref name=":17">"Duchess of Devonshire's Fancy-Dress Ball. Brilliant Spectacle." The [Guernsey] ''Star'' 6 July 1897, Tuesday: 1 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000184/18970706/003/0001.</ref>{{rp|p. 1, Col. 2a}} * "A wonderfully beautiful dress was that which was worn by the Duchess of Devonshire as Zenobia, Queen of Palmyra. It was of golden tissue, sewn with silver paillettes, and jewelled with diamonds and other precious stones. In front there were silk embroideries, in many vivid shades of colour, and here the golden draperies opened to show a petticoat of white crêpe de chine, embroidered with pearls and gold. The short train was of brilliant green velvet, exquisitely embroidered. One of the Duchess of Devonshire’s beautiful diamond and emerald tiaras had been taken to pieces to form a stomacher, the effect of which was dazzling in its brilliancy. Long chains of pearls and other wonderful jewels were worn with this beautiful dress."<ref>“The Devonshire House Ball. A Brilliant Gathering.” The ''Pall Mall Gazette'' 3 July 1897, Saturday: 7 [of 10], Col. 2a–3a [of 3]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970703/019/0007.</ref>{{rp|p. 7, Col. 2b}} * In the article about the ball in the ''Graphic'', [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]] says, "The Ducal hostess herself elected to appear as Zenobia, Queen of Palmyra, with lavish magnificence, and wearing a corruscation of jewels which must have eclipsed the state of even the all-subduing majesty the Duchess impersonated."<ref name=":10" />{{rp|p. 16, Col. 1a}} *The Duchess was dressed "as Zenobia, Queen of Palmyra, her dress a marvel of soft tissues and exquisite ornament, and her tiara a still greater marvel of the jeweller's art."<ref name=":6" />{{rp|p. 12, Col. 2a}} <ref>"The Duchess of Devonshire’s Historic Ball. Some of the Fancy Costumes." Supplement. The ''Leicester Chronicle and Leicestershire Mercury'' 10 July 1897, Saturday: 11 [of 12], Cols. 4a–b [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000173/18970710/141/0011.</ref>{{rp|p. 11, 4a}} As with the colors, Lafayette's photograph of the Duchess in costume does not show the jewels very clearly either. We cannot see the stomacher or the "long chains of pearls and other wonderful jewels" or the pearls that "hung down almost to the knees" (although, of course, any newspaper account can be wrong because reporters were not present at the ball). === Zenobia's Crown<ref name=":10" /> === The crown that the Duchess wore as Zenobia is difficult to see clearly in the Lafayette photograph (above right). It was lavish, "encrusted" with jewels and featuring pearls: * London ''Daily News'' says, "The crown worn with this was high, and of filigree gold, surmounted with two horns, each tipped with a large diamond. It was encrusted with large diamonds, rubies, and emeralds, and long chains of pearls fell under the chin and about the head — one magnificent pear-shaped pearl resting on the forehead."<ref name=":15" />(p. 5, Col. 6a) * These descriptions are all identical except for the addition or subtraction of an occasional comma: The London ''Evening Standard'' says, "A gold crown encrusted with emeralds, diamonds, and rubies, with a diamond drop at each curved end, and two upstanding white ostrich feathers in the centre, and round the front were festoons of pearls, with a large pear-shaped pearl in the centre falling on the forehead."<ref name=":16" />{{rp|p. 3, Col. 2b}} <ref name=":7" />{{rp|p. 7, Col. 7a}} <ref name=":6" />{{rp|p. 12, Col. 3b}} <ref name=":10" />{{rp|p. 15, Col. 3b}} * The Guernsey ''Star'' says, "Her Grace wore a bandeau of gold round her head, studded with diamonds, turquoise, and emeralds, and surrounded by hanging chains of superb pearls."<ref name=":17" />{{rp|p. 1, Col. 2a}} === Goldsmith, Pearl & Diamond Merchant, & Silversmith === The Duchess's pearls, which were an important feature of her costume, occasioned a great deal of direct commentary in the newspaper accounts. The Duchess's jewelry occasioned a great deal of reportage in the articles about the ball. '''It was reported that she had her jewels restrung to be used in the costume. stomacher and review of jewelry in more general articles''' An invoice and receipt in the Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) is from a concern whose preprinted stationery has a crown in the upper-left corner, suggesting that they had a royal warrant, and no name other than Goldsmith, Pearl & Diamond Merchant, & Silversmith. This document offers a unique view into the evolution of one necklace, at least, over the years. It lists what are apparently three restringing of some pearls of Louise, Duchess of Devonshire. The three restringings appear to be dated: The first necklace is a "Pearl Necklet in original 4 rows," dated 20 October 1892 (but the stationery was printed to assume the invoice would be used in the 1880s, so the 9 is written over the second 8, and the 2 has been added).<ref name=":14">Invoice and receipt. Goldsmith, Pearl & Diamond Merchant & Silversmith. Date of itemized invoices for restringing pearls: 20 October 1892, 1 March 1897, 1909. The Devonshire Collections, Chatsworth, Reference number FIS/4/1/2.</ref>(p. 1) The necklet contained a "Total [of] Total 224 large pearls": # 1st [row] 51 large pearls # 2nd 53 large pearls # 3rd 57 large pearls # 4th 63 large pearls The second necklace is a "Necklet as re-strung on October 15th 1892, with addition of small pearls supplied, now consists of 5 rows, containing" a total of "224 large pearls & 227 small" <ref name=":14" />(p. 1) # 1st 41 large pearls & 40 small # 2nd 42 large pearls & 42 small # 3rd 44 large pearls & 45 small # 4th 47 large pearls & 48 small # 5th 50 large pearls & 51 small The third necklace is a "Pearl Necklet as again re-strung with additional pearls supplied 1 March 1897, now consisting of 5 Rows containing" a total of "262 Large Pearls & 267 Small"<ref name=":14" />(p. 2): # 1st Row 45 Large Pearls & 44 Small # 2nd Row 48 large Pearls & 49 Small # 3rd Row 51 Large Pearls & 52 Small # 4th Row 56 Large Pearls & 65 small Possibly these pearls may have been restrung in 1909 into a cornet?<ref name=":14" />(p. 2) If the Duchess wore one of these stringings of her pearls for the ball, then it must have been the second necklet, strung in 1892, a 5-strand necklace. None of the newspaper accounts refer to a 5-strand pearl necklace, although her pearls are often mentioned. == The Duchess's Entourage == Besides the Duke of Devonshire, the retinue of Louise, Duchess of Devonshire as Zenobia, Queen of Palmyra, included her grandson, [[Social Victorians/People/William Angus Drogo Montagu|William Angus Drago Montagu, 9th Duke of Manchester]], dressed as a Georgian courtier. Four newspapers say that the Duchess's entourage included three groups, all in costume: children, trumpeters and fan-bearers. According to two sources, the London ''Daily News''<ref name=":15" />(p. 5, Col. 6a) and the Belfast ''News-Letter,''<ref name=":8" />{{rp|p. 5, Col. 9a}} these groups each had four members. The London ''Daily News'' is likely the source (though not the only one) for the Belfast ''News-Letter'', which took part in [[Social Victorians/1897 Fancy Dress Ball/anthology#Scissors-and-Paste Journalism|scissors-and-paste journalism]], like so many other newspapers of the 19th century. ['''check this: which one was published earlier in the day, and on which day?'''] The ''Man of Ross'' and the ''Westminster Gazette'' do not address the number of members of the groups. These four sources describe the Duchess's retinue and how the people in it were dressed: *"Attending the hostess were four children, four fan-bearers, and four trumpeters, all magnificently arrayed in artistically embroidered Assyrian robes, helmets, and other accessories, correct in every detail."<ref name=":15" />{{rp|p. 5, Col. 6a}} *"The Duchess of Devonshire was dazzingly [sic] magnificent as 'Zenobia,' arrayed in the glistening fabrics and massive jewels in which artists have delighted to depict the Warrior Queen, the costume in this case being specially designed by the clever French artist, M. Comelli, who was also responsible for the splendid attire of the Queen's suite. This was composed of four children in white Assyrian robes, draped with pink shawls; four trumpeters in white cloth robes, embroidered in subdued tones of silks, with a purple shawl draped over, beautifully ornamented with embroidery, and wearing fringed steel helmets and leather cuirasses embossed in steel; and four fan-bearers attired in pale blue robes, with crimson shawls, enriched with gold and jewelled embroidery, adorned with jewelled diadems, and holding long-handled fans of white feathers, mounted in blue and gold — a gloriously magnificent pageant."<ref name=":8" />{{rp|p. 5, Col. 9a}} *"The duchess was dressed as Zenobia, in gold cloth, gorgeously embroidered in gold, brilliants, and coloured stones, and opening over an under dress of white crêpe de Chine, worked finely in brilliants. The train of light green velvet was lined with blue, and sumptuously embroidered in jewels and gold, the colouring being particularly artistic. With this dress were worn splendid jewels, and a large horn crown, encrusted with diamonds, emeralds, and rubies. The duchess was attended by a suite of children, trumpeters, and fan-bearers, all picturesquely attired in Assyian [sic] costumes — the whole group being specially designed by M. Comelli."<ref name=":5" /> *"The host was dressed as Charles V. of Germany, in black velvet, satin, and fur; and the Duchess made the most gorgeous of Zenobias, in a gown of gold gauze, and a green velvet train — both a mass of exquisite oriental embroidery. The crown and hanging ropes of pearls, the jewelled girdle, and the train of children, fan-bearers, and trumpeters — all in Babylonish garb — as designed by M. Comelli, made a gloriously imposing and picturesque group."<ref name=":9" /> === Details of the Costumes in the Entourage === The Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) has "receipts" or invoices that functioned as receipts for several commercial concerns that were involved in making costumes or accessories for costumes for this ball. They are the following: * [[Social Victorians/People/Louisa Montagu Cavendish#M. (Attillo Giuseppe) Comelli|M. (Attillo Giuseppe) Comelli]] * [[Social Victorians/People/Louisa Montagu Cavendish#B. Burnet & Co.|B. Burnet & Co.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Arthur Millward, Theatrical Jeweller|Arthur Millward, Theatrical Jeweller]] * [[Social Victorians/People/Louisa Montagu Cavendish#Liberty & Co., Ltd.|Liberty & Co., Ltd.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Lafayette, Ltd.|Lafayette, Ltd.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Goldsmith, Pearl & Diamond Merchant, & Silversmith|Goldsmith, Pearl & Diamond Merchant, & Silversmith]] This list of commercial concerns almost certainly cannot be the complete list of all concerns that contributed to the costumes. These are the only receipts or invoices about expenses for the ball, however, that the Chatsworth Archive contains; similar documents were likely not even kept or were destroyed with other papers not retained at some point in time. The business concerns listed above were specialized and likely used for different elements of the costumes. As a theatrical designer, Comelli would have depended on the suppliers he knew and arranged with them for the construction of these costumes. The Chatsworth Archive calls these documents ''receipts'', which indeed they are because they were returned to Devonshire House as receipts for payment. From our perspective, though, they are invoices that contain specifics about what was used to make the costumes. The analysis of these invoices has led to an understanding of what the people who attended the Duchess in her entourage wore and a clearer sense, perhaps, of how many people walked in that entourage. This analysis is based on the items listed on the invoices and their pricing, most of which is included in the section for each invoice, below. While the Belfast ''News-Letter'' says that each group contained four members,<ref name=":8" />{{rp|p. 5, Col. 9a}} the invoices and receipts suggest that the newspaper, the single source for this information, was wrong about the number of people in each group. It is theoretically possible, of course, that suppliers other than the ones in the Chatsworth Archive made some of these costumes and that other invoices and receipts must have existed at that time. But the [[Social Victorians/People/Louisa Montagu Cavendish#M. (Attillo Giuseppe) Comelli|Comelli memo, below]], seems definitive: he designed and seems to have overseen the construction of the costumes, which numbered six rather than twelve. '''[Collier discussion?]''' Besides providing welcome detail about the costumes of the people in the Duchess's entourage, which is available nowhere else, these invoices also raise at least as many questions as they answer. ==== M. (Attillo Giuseppe) Comelli ==== Attillo Giuseppe Comelli was a designer for opera, ballet and theatre in Europe, the UK and North America.<ref name=":13" /> The receipt in the Chatsworth Archive was sent from Covent Garden. The invoice lists £4 for "Making six costumes," 3''s'' for "Extras" and 12s for "Cab fares for men paid by the request of M. Comelli."<ref>M. Comelli, Covent Garden, to Duchess of Devonshire. Date of invoice 13 July 1897; postmarked 25 October 1897, for receipt of payment(?). The Devonshire Collections, Chatsworth, Reference number L/109/4(3).</ref> Three other names are on this invoice and receipt: * L. L[?] Collier [?], written under and perhaps as part of the direction to the Duchess of Devonshire * Mr. Strong ("Forwarded to Mr Strong. [sic] by the instructions of M. Comelli," written in the same hand as wrote the majority of the memo) * Floyd [?] Collier [??] ("Received with Thanks," presumably thanking for the payment, in a different hand) ==== B. Burnet & Co. ==== An invoice and receipt from B. Burnet & Co., held in the Archives of the Duke of Devonshire, has specific information about some of the fabrics, trims and accessories purchased for the costumes of the Duchess's retinue.<ref name=":11">B. Burnet & Co. to Louise, Duchess of Devonshire. Date of invoice 2 and 6 July 1897; postmarked 25 October 1897, for receipt of payment(?). The Devonshire Collections, Chatsworth, Reference number L/109/4(3).</ref> Besides itemizing some costume or accessory elements that seem clearly to be for the groups, the invoice also lists items not easy to associate with particular costumes, like the following: * 12 yards of White silk fringe 8in deep<ref name=":11" />{{rp|back left}} * 12 1/2 yards of "wht cloth"<ref name=":11" />{{rp|back left}} * 9 yards of "[[Social Victorians/Terminology#Selesia|Selesia]]"<ref name=":11" />{{rp|back left}} * 2 yards of Canvas<ref name=":11" />{{rp|back right}} * 4 Tan Wool Tights<ref name=":11" />{{rp|back right}} * 2 Tan Boys Tights<ref name=":11" />{{rp|back right}} At this time, we are not sure which costumes these elements were used for. Possibly the white silk fringe and the white cloth would have been used to construct the robes for the children and trumpeters in the entourage. The number of tights suggests that the six costumes on this invoice all included tights. With other elements of the trumpeters' costumes, the Burnet invoice also lists "6 prs Assyrian Buskins." Probably, to a late Victorian, buskins would have been "defensive leggings"<ref>Demmin, Auguste. An illustrated History of Arms and Armour: From the Earliest Period to the Present Time. George Bell, 1894. Google Books https://books.google.com/books?id=ArRCAAAAYAAJ: 106.</ref> laced together and covering the lower leg and often feet of a soldier. To a clothing and military historian, buskins (or greaves) were worn by people in a number of cultures over millennia and varied widely in style and construction. Buskins appear in Assyrian art held at the time by the British Museum. Listing six pairs of buskins suggests that every costume in the Duchess's entrourage included buskins, possibly worn over the tan tights. The Burnet invoice lists "4 Broad Belts," which may have held "4 Skins Fleshers."<ref name=":11" />{{rp|p. 1, front of invoice}} (A skin flesher is a kind of knife used to separate the skin from the meat in animals.) If each group included only two members, then perhaps the belts and fleshers were worn not only by the trumpeters but also by the fan-bearers. The Millward invoice (specifics in the section on the Millward invoice below) lists "8 Doz 'Plaques' for Belts'" with a drawing of an upright rectangle with a circle in the middle, which might have been a jewel. Double lines around the rectangle suggest that the plaques were not flat or the metal was not thin. The drawing does not give any ideas about how the plaques were attached to the belts, if they were. It is impossible to tell if the plaques were attached to the "4 Broad Belts" (likely for the trumpeters and fan-bearers), but unless they were quite tiny, "8 Doz 'Plaques'" would be far too many for the belts of only the two children. A different hand, probably "[L.??] L. Collier," wrote the following sentence at the end of the invoice and receipt, above the postmark:<blockquote>All the above named articles were used for the six [?] dresses made for the Devonshire Ball.<ref name=":11" />(back right)</blockquote>This same hand, signing what is possibly "Floyd Collier," also signed the postmark of the Comelli invoice and receipt. On the Burnet document, this writer, possibly an assistant or employee of the Duchess of Devonshire, says that "six dresses" were made (if in fact, that word is ''six''). (No "Collier" is listed among the staff or servants of the Duke of Devonshire at the end of the 19th century.<ref>"Servants A-H." ''Historic Servants and Staff. Servants and Staff Database''. Retrieved 18 July 2022 https://www.chatsworth.org/media/11528/servants-a-h.pdf.</ref> The invoice appears to itemize materials used for six costumes: two children, two trumpeters and two fan-bearers. ==== Arthur Millward, Theatrical Jeweller ==== An invoice and request for payment from Arthur Millward, Theatrical Jeweller, held in the Archives of Chatsworth House, has more specifics about some of the fabrics, trims and accessories for the costumes of the Duchess's retinue.<ref name=":12">Memorandum. Arthur Millward, Theatrical Jeweller, to Louise, Duchess of Devonshire. Date of itemized invoice 1 July 1897; date of request for payment(?) 27 August 1897. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> This invoice lists the following, which could have been used in any of the costumes for the entourage: * 8 Doz 'Plaques' for Belts [discussed with the belts in the section on the Burnet invoice, above] * 4 Large Armlets * 4 Bracelets * 8 Armlets<ref name=":12" />{{rp|p. 2, back}} Because Millward was a Theatrical Jeweller, it seems likely that most (if not all) of the items listed on the invoice were made of metal and the jewels mentioned were artificial, made of glass or paste. Other items on the invoice seem to belong to the costumes of the trumpeters, which the Belfast ''News-Letter'' says included helmets: * 2 Helmets * 2 Centre pieces The Millward invoice shows tiny line drawings next to the words ''2 Helmets'' and ''2 Centre pieces''. These drawings suggest that the Centre pieces were attached to the helmets rather than being anything that would have been put on a table as decoration. Other items seem to belong to the costumes of the fan-bearers: * 2 Pearl & Gold Headdresses * 2 Fan properties with Feathers The "Pearl & Gold Headdresses" were likely the "jewelled diadems" mentioned in the Belfast ''News-Letter''. The "Fan properties with Feathers" are likely to have been the "long-handled fans of white feathers, mounted in blue and gold" mentioned in the newspaper report.<ref name=":8" />{{rp|p. 5, Col. 9a}} At the end of the Millward invoice, a "reduction as agreed with M [Mr?] Commelli [sic]" of £1 10''s'' is subtracted from a total of £22 3''s''. No reason for this reduction is given.<ref name=":12" />{{rp|p. 2, back}} ==== Liberty & Co., Ltd. ==== One invoice and receipt from the Chatsworth Archive, dated 12 July, to the Duchess of Devonshire, lists "13 yds S&W Satin[?]," 7 yards of blue and 6 of purple.<ref>Invoice and receipt. Liberty & Co. Ltd. To Her Grace, The Duchess of Devonshire. Date of itemized invoice 12 July [1897]. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> Because the fabric is satin and from Liberty, it is possible that it was not used in the costumes of the people in the entourage but perhaps for the costume of the Duchess herself??? ==== Lafayette, Ltd. ==== The invoice and receipt from Lafayette, Ltd., the photographer that set up a temporary studio in the garden to take portraits of people at the ball in their costumes, may not be related to the ball at all.<ref>Invoice and receipt. Lafayette, Ltd. To His Grace The Duke of Devonshire. Date of itemized invoice 12 April 1897; addressed to the Duke, 18 February 1898; receipt and thanks for payment, 7 April 1898. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> Three dates are written on the preprinted stationery: # 18/2/98 (18 February 1898), under the direction to "His Grace The Duke of Devonshire" # 4/12/97 (4 December 1897), next to the single item on the invoice for which a charge is listed: "6 [??] £1.10.0" # 7/4/98 (7 April 1898), in a different hand, with "Recd by cheque 7/4/98 Lafayette Ltd pp[?] [??] thanks" At the bottom of the page, in the hand that wrote all of the invoice except the receipt and thanks, is "With Lafayette Ltds Compliments." ==== Details for the Children in the Entourage ==== According to the ''Belfast News-Letter'', four children were "in white Assyrian robes, draped with pink shawls."<ref name=":8" />{{rp|p. 5, Col. 9a}} According to the B. Burnet invoice, the following was purchased for "White Cloth Dresses":<ref name=":11" />{{rp|p. 2, back left of invoice}} * "2 Terra Gown draperies with Stars 200 in all" * "2 Cloth fronts embroidered with Square Medallions down centre" * "2 do do [ditto ditto, that is, cloth fronts] embroidered double border down front each side and collar" * "4 Sleeves embroidered Small Medallions" The Burnet & Co. invoice lists 6 yards of "Terra" Silk Fringe, which perhaps was used to trim the "terra draperies," or shawls, made from 3 1/4 yards of "Light Terra Satinette" worn by the children? ==== Details for the Trumpeters in the Entourage ==== According to the ''Belfast News-Letter'', four trumpeters were "in white cloth robes, embroidered in subdued tones of silks, with a purple shawl draped over, beautifully ornamented with embroidery, and wearing fringed steel helmets and leather cuirasses embossed in steel."<ref name=":8" />{{rp|p. 5, Col. 9a}} The trumpeters appear to have been dressed as soldiers or military men. According to the B. Burnet invoice, the following was purchased for the trumpeters' costumes:<ref name=":11" />{{rp|p. 1, front of invoice}} * 7 '''units (yards?)''' of purple silk [probably used for shawls?] * "2 skirt fronts with border alround [sic]" * "2 sets of Leather Cuarasses [sic] Embroidered front & back" * "4 Sleeves embroidered loop stitch" The Millward invoice lists * 2 Helmets * 2 Centre Pieces [probably for helmets rather than table decorations] ==== Details for the Fan-bearers in the Entourage ==== According to the ''Belfast News-Letter'', four fan-bearers were "attired in pale blue robes, with crimson shawls, enriched with gold and jewelled embroidery, adorned with jewelled diadems, and holding long-handled fans of white feathers, mounted in blue and gold."<ref name=":8" />{{rp|p. 5, Col. 9a}} According to the B. Burnet invoice, the following was purchased for the fan bearers's costumes:<ref name=":11" />{{rp|pp. 1–2, front and left-back of invoice}} * "Embroidering 2 Crimson draperies with Stars 334 in all" * "2 Top [?] fronts embroidered & round necks" * "4 Sleeves embroidered long stitch"The Millward invoice lists * 2 Pearl & Gold Headdresses * 2 Fan properties with Feathers<ref name=":12" />{{rp|p. 2, back}} The Burnet & Co. invoice lists 12 yards of "Red Silk Fringe," which perhaps was used to trim the "crimson shawls" or "Crimson draperies," which may have been made from the 5 yards of "Red Satinette." Again, this list suggests two rather than four costumes. === The Historical Zenobia === Zenobia (240 – c. 274) was queen of the Syrian Palmyrene Empire, ruling as regent for her son after her husband's assassination.<ref>{{Cite journal|date=2022-05-03|title=Zenobia|url=https://en.wikipedia.org/w/index.php?title=Zenobia&oldid=1086005949|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Zenobia.</ref> She was the subject of much art in the 19th century, including literature, opera, sculpture, and paintings. Middle-eastern traveller Lady Hester Stanhope (1776–1839) discussed Zenobia in her memoirs, published in 1847.<ref>{{Cite journal|date=2022-03-07|title=Lady Hester Stanhope|url=https://en.wikipedia.org/w/index.php?title=Lady_Hester_Stanhope&oldid=1075838273|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lady_Hester_Stanhope.</ref> == Demographics == *Nationality: born in Hanover, in what is now Germany<ref name=":0">{{Cite journal|date=2020-07-27|title=Louisa Cavendish, Duchess of Devonshire|url=https://en.wikipedia.org/w/index.php?title=Louisa_Cavendish,_Duchess_of_Devonshire&oldid=969824214|journal=Wikipedia|language=en}}</ref> === Residences === ==== As Duchess of Manchester ==== *Kimbolton Castle, Huntingdonshire *Manchester House, London ==== As Duchess of Devonshire ==== *Devonshire House, London (mid-April until mid-July, for the Season) *Compton Place, Eastbourne (mid-July until 12 August<ref name=":1" />{{rp|p. 32}}) *Bolton Abbey, Yorkshire (12 August until the middle of September<ref name=":1" />{{rp|p. 32}}) *Chatsworth, Derbyshire (middle of September until early Spring<ref name=":1" />{{rp|p. 32}}) *Lismore Castle, County Waterford (early Spring until the middle of April<ref name=":1" />{{rp|p. 32}}) == Family == *Louisa (or Luise) Friederike Auguste Gräfin von Alten Montagu Cavendish (15 January 1832 – 15 November 1911)<ref name=":2" /><ref name=":0" /> *William Drogo Montagu, 7th Duke of Manchester (15 October 1823 – 22 March 1890)<ref name=":3" /><ref>{{Cite journal|date=2020-09-07|title=William Montagu, 7th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=William_Montagu,_7th_Duke_of_Manchester&oldid=977197445|journal=Wikipedia|language=en}}</ref> #George Victor Drogo Montagu, 8th Duke of Manchester (17 June 1853 – 18 August 1892) #Mary Louise [Louisa?] Elizabeth Montagu Douglas-Hamilton Forster (27 December 1854 – 10 February 1934) #Louisa Augusta Beatrice Montagu Acheson (c. 1856 – 3 March 1944) #Charles William Augustus Montagu (23 November 1860 – 10 November 1939) #Alice Maude Olivia Montagu Stanley (15 August 1862 – 23 July 1957) *[[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish]], 8th Duke of Devonshire (23 July 1833 – 24 March 1908) == Notes and Questions == #As Duchess of Manchester Luise was not invited to the wedding between Bertie and Alix, Victoria's punishment for Luise's having gotten the Duke of Derby to promise her the position of Mistress of the Robes (and then exacting that promise).<ref>Leslie, Anita. ''The Marlborough House Set''. New York: Doubleday, 1973.</ref>{{rp|pp. 47–48}} #"As a young woman she was extremely beautiful; Princess Catherine Radziwill saw her at a reception given by the Empress of Germany and recalls on being introduced to her 'how she struck me as the loveliest creature I had ever set eyes upon. Indeed I have only met three women in my whole existence who could be compared to her.'"<ref name=":1" />{{rp|p. 21}} == Footnotes == {{reflist}} s432x0r5av9vk53t6kjhod8jhcxh08c 0 269289 2410807 2410780 2022-08-01T18:31:21Z Maneesh 2723004 /* 11-Oxygenated Androgen Pathways */ 11KT and T wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> and the potential clinical relevance in conditions involving androgen biosynthesis was proposed. Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref><ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid11514561">{{cite journal|last1=Nahoum|first1=Virginie|last2=Gangloff|first2=Anne|last3=Legrand|first3=Pierre|last4=Zhu|first4=Dao-Wei|last5=Cantin|first5=Line|last6=Zhorov|first6=Boris S.|last7=Luu-The|first7=Van|last8=Labrie|first8=Fernand|last9=Breton|first9=Rock|year=2001|title=Structure of the human 3alpha-hydroxysteroid dehydrogenase type 3 in complex with testosterone and NADP at 1.25-A resolution|journal=J Biol Chem|volume=276|issue=45|pages=42091–8|doi=10.1074/jbc.M105610200|pmid=11514561|doi-access=free|last10=Lin|first10=Sheng-Xiang}}</ref><ref name="pmid18923939">{{cite journal|last1=Dozmorov|first1=Mikhail G.|last2=Yang|first2=Qing|last3=Matwalli|first3=Adam|last4=Hurst|first4=Robert E.|last5=Culkin|first5=Daniel J.|last6=Kropp|first6=Bradley P.|last7=Lin|first7=Hsueh-Kung|year=2007|title=5alpha-androstane-3alpha,17beta-diol selectively activates the canonical PI3K/AKT pathway: a bioinformatics-based evidence for androgen-activated cytoplasmic signaling|journal=Genomic Med|volume=1|issue=3–4|pages=139–46|doi=10.1007/s11568-008-9018-9|pmc=2269037|pmid=18923939}}</ref><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref><ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}}The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}}The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}}The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608" />{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}}While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780" /> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. 11-Oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid30959151">{{cite journal |title=Circulating 11-oxygenated androgens across species |journal=J Steroid Biochem Mol Biol |volume=190 |pages=242–249 |year=2019 |pmid=30959151 |pmc=6733521 |doi=10.1016/j.jsbmb.2019.04.005|last1=Rege |first1=Juilee |last2=Garber |first2=Scott |last3=Conley |first3=Alan J. |last4=Elsey |first4=Ruth M. |last5=Turcu |first5=Adina F. |last6=Auchus |first6=Richard J. |last7=Rainey |first7=William E. }}</ref><ref name="pmid27519632" /><ref name="pmid34171490" /><ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646" /> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="NBK557634">{{cite book|url=https://www.ncbi.nlm.nih.gov/books/NBK557634/|title=Biochemistry, Dihydrotestosterone|publisher=StatPearls|year=2022}}</ref><ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |pages=431 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|pages=868|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|pages=66|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=441 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref><ref name="pmid23386646" /> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid29277706" /><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405" /> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-), therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the conversion of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1.<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRDA51 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid18923939"/><ref name="Nishiyama2011"/><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and continue to a lattice of transformations between 19-carbon steroid products. All the steroid products of this pathway have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on circulating levels, e.g., it may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels do not decline with age as T does. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== From Androstenedione or Testosterone Towards 11-Oxygenated Androgens ==== Of the pathways from A4 and T to 11-oxygenated androgens, A4 is the main initial substrate. Therefore, 11OHA4 is the first major metabolite, biosythesized as a result of 11β-hydroxylation of A4. Although some 11OHT originates from the 11β-hydroxylation of T, this probably makes a very small contribution. <ref name="pmid29936123"/><ref name="pmid23386646"/> 11OHA4 is not a substrate for AKR1C3 and thus requires the conversion to 11KA4 by HSD11B2.<ref name="pmid29936123"/> Given that AKR1C3 catalyses the conversion of 11KA4 to 11KT even more efficiently than the conversion of A4 to T,<ref name="pmid29936123">{{cite journal |title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer |journal=J Steroid Biochem Mol Biol |volume=183 |issue= |pages=192–201 |year=2018 |pmid=29936123 |pmc=6283102 |doi=10.1016/j.jsbmb.2018.06.013|last1=Barnard |first1=Monique |last2=Quanson |first2=Jonathan L. |last3=Mostaghel |first3=Elahe |last4=Pretorius |first4=Elzette |last5=Snoep |first5=Jacky L. |last6=Storbeck |first6=Karl-Heinz }}</ref><ref name="pmid33444228" /><ref name="pmid35560164">{{cite journal |title=Conversion of Classical and 11-Oxygenated Androgens by Insulin-Induced AKR1C3 in a Model of Human PCOS Adipocytes |journal=Endocrinology |volume=163 |issue=7 |year=2022 |pmid=35560164 |doi=10.1210/endocr/bqac068 |last1=Paulukinas |first1=Ryan D. |last2=Mesaros |first2=Clementina A. |last3=Penning |first3=Trevor M. |s2cid=248776966 }}</ref> it is therefore believed that more 11KT is produced from 11KA4 than from 11OHT.<ref name="pmid23386646" /><ref name="pmid29936123" /> As such, while all the routes from A4 and T can be seen on Figure 4, given that SRD5A1 does not efficiently catalyze the 5α-reduction of 11KT or 11KA4,<ref name="pmid32629108"/> the predominant route in normal conditions in humans to 11-oxygenated androgens can be described as follows:<ref name="pmid23386646"/><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref><ref name="pmid33444228">{{cite journal |title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens |journal=Eur J Endocrinol |volume=184 |issue=3 |pages=353–363 |pmid=33444228 |pmc=7923147 |doi=10.1530/EJE-20-1077| last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021 |issn=1479-683X}}</ref><ref name="pmid35560164">{{cite journal|last1=Paulukinas|first1=Ryan D.|last2=Mesaros|first2=Clementina A.|last3=Penning|first3=Trevor M.|year=2022|title=Conversion of Classical and 11-Oxygenated Androgens by Insulin-Induced AKR1C3 in a Model of Human PCOS Adipocytes|journal=Endocrinology|volume=163|issue=7|doi=10.1210/endocr/bqac068|pmid=35560164|s2cid=248776966}}</ref> {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} ==== From Progesterone and 17α-Hydroxyprogesterone Towards 11-Oxygenated Androgens ==== The pathways from P4 and 17-OHP to 11-oxygenated androgens can be described as follows. The 17,20-lyase activity of CYP17A1 is needed to cleave a side-chain (C17-C20 bond) from the steroid nucleus to convert an initial pregnane to a final androgen. Human CYP17A1 cannot efficiently catalyze this reaction for steroids that have the oxo- functional group at carbon position 3.<ref name="pmid32007561"/> Examples of such steroids are 11OHPdione or 11KPdione. Therefore, such C₂₁ steroid should be 3α-reduced by AKR1C4 before it can be converted to a C₁₉ steroid by CYP17A1. After the side-chain cleavage by CYP17A1, the oxo- group at position 3 is restored back in a 3α-oxidation reaction (by an enzyme such as AKR1C4 or HSD17B6) to convert an inactive androgen such as 11K-3αdiol to the active one such as 11KDHT.<ref name="pmid31626910"/> As you see, the order of steps in metabolic routes from P4 and 17-OHP towards 11-oxygenated androgens (11KDHT and 11OHDHT) is similar to the conversion of P4 and 17-OHP to DHT in a backdoor pathway.<ref name="pmid28774496" /> ==Clinical Significance == === Biological Role of 11-Oxygenated Androgens === 11-oxygenated androgens are produced in physiological quantities in healthy primate organisms (including humans).<ref name="pmid30959151" /><ref name="pmid30753518" /><ref name="pmid32629108" /> Humans have two isozymes with 11β-hydroxylase activity, encoded by the genes ''CYP11B1'' (regulated by the adrenocorticotropic hormone (ACTH)) and ''CYP11B2'' (regulated by angiotensin II).<ref name="pmid22217826">{{cite journal|name-list-style=vanc|title=Molecular biology of 11β-hydroxylase and 11β-hydroxysteroid dehydrogenase enzymes |journal=J Steroid Biochem Mol Biol |volume=43 |issue=8 |pages=827–35 |pmid=22217826 |doi=10.1016/0960-0760(92)90309-7 |s2cid=19379671 |last1=White |first1=Perrin C. |last2=Pascoe |first2=Leigh |last3=Curnow |first3=Kathleen M. |last4=Tannin |first4=Grace |last5=Rösler |first5=Ariel |year=1992 }}</ref> Since the first step in the biosynthesis of 11-oxygenated androgens involves 11β-hydroxylation of a steroid substrate by CYP11B1/2 isozymes that are generally associated with their expression in the adrenal gland, 11-oxygenated androgens are considered androgens of adrenal origin. They follow the circadian rhythm of cortisol but correlate very weakly with T.<ref name="pmid34867794">{{cite journal |title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=751191 |pmid=34867794 |pmc=8636728 |doi=10.3389/fendo.2021.751191 |doi-access=free |last1=Turcu |first1=Adina F. |last2=Mallappa |first2=Ashwini |last3=Nella |first3=Aikaterini A. |last4=Chen |first4=Xuan |last5=Zhao |first5=Lili |last6=Nanba |first6=Aya T. |last7=Byrd |first7=James Brian |last8=Auchus |first8=Richard J. |last9=Merke |first9=Deborah P. |year=2021 }}</ref><ref name="pmid34324429">{{cite journal|title=Circadian rhythms of 11-oxygenated C19 steroids and ∆5-steroid sulfates in healthy men |journal=Eur J Endocrinol |volume=185 |issue=4 |pages=K1–K6 |pmid=34324429 |doi=10.1530/EJE-21-0348 |pmc=8826489 |pmc-embargo-date=August 27, 2022 |last1=Turcu |first1=Adina F. |last2=Zhao |first2=Lili |last3=Chen |first3=Xuan |last4=Yang |first4=Rebecca |last5=Rege |first5=Juilee |last6=Rainey |first6=William E. |last7=Veldhuis |first7=Johannes D. |last8=Auchus |first8=Richard J. |year=2021 }}</ref> The levels of 11-oxygenated androgens raise after ACTH stimulation<ref name="pmid23386646"/><ref name="pmid13211802">{{cite journal |vauthors=DOBRINER K, KAPPAS A, GALLAGHER TF |title=Studies in steroid metabolism. XXVI. Steroid isolation studies in human leukemia |journal=J Clin Invest |volume=33 |issue=11 |pages=1481–6 |date=November 1954 |pmid=13211802 |pmc=1072573 |doi=10.1172/JCI103026 |url=}}</ref> that further supports their adrenal origin. However, in addition to the adrenal glands, CYP11B1 is also expressed in Leydig cells and ovarian theca cells, albeit at far lower levels, so the production of 11KT precursors may be one of the most important functions of 11β-hydroxylase activity in the gonads.<ref name="pmid27428878">{{cite journal|title=11-Ketotestosterone Is a Major Androgen Produced in Human Gonads |journal=J Clin Endocrinol Metab |volume=101 |issue=10 |pages=3582–3591 |pmid=27428878 |doi=10.1210/jc.2016-2311 |last1=Imamichi |first1=Yoshitaka |last2=Yuhki |first2=Koh-Ichi |last3=Orisaka |first3=Makoto |last4=Kitano |first4=Takeshi |last5=Mukai |first5=Kuniaki |last6=Ushikubi |first6=Fumitaka |last7=Taniguchi |first7=Takanobu |last8=Umezawa |first8=Akihiro |last9=Miyamoto |first9=Kaoru |last10=Yazawa |first10=Takashi |year=2016 }}</ref> In an in vitro study by Strushkevich et al. published in 2013, both isozymes have been shown to convert Δ⁴ steroids (P4, 17-OHP, A4 and T), but they are very specific to the configuration of the A-ring (carbon positions 1 to 5) of steroids, i.e., they cannot convert Δ⁵ steroids with a hydroxy group at the carbon position 3.<ref name="pmid23322723">{{cite journal |pmc=5417327|year=2013|last1=Strushkevich|first1=N.|last2=Gilep|first2=A. A.|last3=Shen|first3=L.|last4=Arrowsmith|first4=C. H.|last5=Edwards|first5=A. M.|last6=Usanov|first6=S. A.|last7=Park|first7=H. W.|title=Structural Insights into Aldosterone Synthase Substrate Specificity and Targeted Inhibition|journal=Molecular Endocrinology (Baltimore, Md.)|volume=27|issue=2|pages=315–324|doi=10.1210/me.2012-1287|pmid=23322723}}</ref> The relative biological significance of the 11-oxygenated androgens has been subject to some debate. There have been enough studies to establish that 11OHA4 itself does not have any androgenic activity but an important precursor to 11KT and 11KDHT.{{cn}} Androgen activity has been established for 11KT, 11KDHT<ref name="pmid27442248" /> as well as 11OHT and 11OHDHT but circulating levels of all of these androgens have not been firmly established. 11KT may serve as a primary androgen for healthy women,<ref name="pmid32629108">{{cite journal|last1=Barnard|first1=Lise|last2=Nikolaou|first2=Nikolaos|last3=Louw|first3=Carla|last4=Schiffer|first4=Lina|last5=Gibson|first5=Hylton|last6=Gilligan|first6=Lorna C.|last7=Gangitano|first7=Elena|last8=Snoep|first8=Jacky|last9=Arlt|first9=Wiebke|year=2020|title=The A-ring reduction of 11-ketotestosterone is efficiently catalysed by AKR1D1 and SRD5A2 but not SRD5A1|url=|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=202|pages=105724|doi=10.1016/j.jsbmb.2020.105724|pmid=32629108|s2cid=220323715|last10=Tomlinson|first10=Jeremy W.|last11=Storbeck|first11=Karl-Heinz}}</ref><ref name="pmid30753518" /> as it circulates at similar levels to T, but unlike T, the levels of 11KT are stable across the menstrual cycle.<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref> There are conflicting reports on whether 11-oxygenated androgens decline in women with age, and whether the relative contribution of 11KT as compared with T is higher in postmenopausal women than in younger ones — Nanba et al. (2019)<ref name="pmid30753518" /> and Davio et al. (2020)<ref name="pmid32498089">{{cite journal|last1=Davio|first1=Angela|last2=Woolcock|first2=Helen|last3=Nanba|first3=Aya T.|last4=Rege|first4=Juilee|last5=o'Day|first5=Patrick|last6=Ren|first6=Jianwei|last7=Zhao|first7=Lili|last8=Ebina|first8=Hiroki|last9=Auchus|first9=Richard|year=2020|title=Sex Differences in 11-Oxygenated Androgen Patterns Across Adulthood|journal=J Clin Endocrinol Metab|volume=105|issue=8|pages=e2921–e2929|doi=10.1210/clinem/dgaa343|pmc=7340191|pmid=32498089|last10=Rainey|first10=William E.|last11=Turcu|first11=Adina F.}}</ref> found that 11KT do not decline with age in women, however, Skiba et al. (2019)<ref name="pmid31390028" /> reported that the levels do decline. The decline of circulating 11-androgens with age may be associated with declining levels of DHEA and A4, which serve as precursors. It should also be noted that about half of circulating A4 quantities and almost all DHEA quantities are of adrenal origin.<ref name="pmid25428847">{{cite journal |vauthors=Turcu A, Smith JM, Auchus R, Rainey WE |title=Adrenal androgens and androgen precursors-definition, synthesis, regulation and physiologic actions |journal=Compr Physiol |volume=4 |issue=4 |pages=1369–81 |date=October 2014 |pmid=25428847 |pmc=4437668 |doi=10.1002/cphy.c140006 |url=}}</ref> Therefore the decline of 11-oxygenated androgens - steroids of adrenal origin - may be in line with a decline with their adrenal androgen precursors - DHEA and A4. Studies to date also strongly suggest that 11KT is the primary active 11-oxygenated androgen. Notably, 11KDHT has been found to circulate at substantially lower levels than DHT at least in prostate cancer<ref name="pmid30472582"/> suggesting that the 5α-reduction of 11KT is not as physiologically relevant in that condition. While early studies did show that 11KT and other 11-oxygenated androgens are substrates for SRD5A1 and SRD5A2, these did not perform full kinetic analyses.<ref name="pmid23856005"/>. Barnard et al., in a study published in 2020 showed that SRD5A1 does not efficiently catalyze the 5α-reduction of 11KT or 11KA4,<ref name="pmid32629108"/> confirming that 11KT may be the more relevant active 11-oxygenated androgen given the abundant peripheral expression of SRD5A2. While this does not rule out the potential for 11KDHT to be produced by SRD5A2 (or to a lesser degree by SRD5A1) in specific tissues, current evidence does suggest that the emphasis should be taken off the production of 11KDHT and rather placed on 11KT. As for the measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal |title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens |journal=Eur J Endocrinol |volume=184 |issue=3 |pages=353–363 |pmid=33444228 |pmc=7923147 |doi=10.1530/EJE-20-1077| last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021 |issn=1479-683X}}</ref> === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal |title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report |journal=Int J Mol Sci |year=2020 |volume=21 |issue=13 |pmid=32610579 |pmc=7369945 |doi=10.3390/ijms21134622 |doi-access=free |last1=Sumińska |first1=Marta |last2=Bogusz-Górna |first2=Klaudia |last3=Wegner |first3=Dominika |last4=Fichna |first4=Marta |page=4622 }}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal | last1=Yildiz | first1=Bulent O. | title=Diagnosis of hyperandrogenism: clinical criteria | journal=Best Practice & Research. Clinical Endocrinology & Metabolism | publisher=Elsevier BV | volume=20 | issue=2 | year=2006 | issn=1521-690X | pmid=16772149 | doi=10.1016/j.beem.2006.02.004 | pages=167–176}}</ref><ref name="pmid24184282">{{cite journal | last1=Peigné | first1=Maëliss | last2=Villers-Capelle | first2=Anne | last3=Robin | first3=Geoffroy | last4=Dewailly | first4=Didier | title=Hyperandrogénie féminine | journal=Presse Medicale (Paris, France) | publisher=Elsevier BV | volume=42 | issue=11 | year=2013 | issn=0755-4982 | pmid=24184282 | doi=10.1016/j.lpm.2013.07.016 | pages=1487–1499 | s2cid=28921380 | language=fr}}</ref> Relying on T levels alone in conditions associated with hyperandrogenism may read to diagnostic pitfalls and confusion.<ref name="pmid32610579"/> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|title=Clinical significance of 11-oxygenated androgens |journal=Curr Opin Endocrinol Diabetes Obes |volume=24 |issue=3 |pages=252–259 |pmid=28234803 |pmc=5819755 |doi=10.1097/MED.0000000000000334 |last1=Turcu |first1=Adina F. |last2=Auchus |first2=Richard J. |year=2017 }}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome |journal=Reprod Biomed Online |volume=41 |issue=4 |pages=734–742 |pmid=32912651 |doi=10.1016/j.rbmo.2020.07.013 |s2cid=221625488 |last1=Yang |first1=Yabo |last2=Ouyang |first2=Nengyong |last3=Ye |first3=Yang |last4=Hu |first4=Qin |last5=Du |first5=Tao |last6=Di |first6=Na |last7=Xu |first7=Wenming |last8=Azziz |first8=Ricardo |last9=Yang |first9=Dongzi |last10=Zhao |first10=Xiaomiao |year=2020 }}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803"/> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by very potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal | last1=Balsamo | first1=Antonio | last2=Baronio | first2=Federico | last3=Ortolano | first3=Rita | last4=Menabo | first4=Soara | last5=Baldazzi | first5=Lilia | last6=Di Natale | first6=Valeria | last7=Vissani | first7=Sofia | last8=Cassio | first8=Alessandra | title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant | journal=Frontiers in Pediatrics | year=2020 | publisher=Frontiers Media SA | volume=8 | page=593315 | issn=2296-2360 | pmid=33415088 | pmc=7783414 | doi=10.3389/fped.2020.593315| doi-access=free }}</ref><ref name="pmid29277706">{{cite journal | last1=Kamrath | first1=Clemens | last2=Wettstaedt | first2=Lisa | last3=Boettcher | first3=Claudia | last4=Hartmann | first4=Michaela F. | last5=Wudy | first5=Stefan A. | title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia | journal=The Journal of Steroid Biochemistry and Molecular Biology | publisher=Elsevier BV | volume=178 | year=2018 | issn=0960-0760 | pmid=29277706 | doi=10.1016/j.jsbmb.2017.12.016 | pages=221–228| s2cid=3709499 }}</ref> Another issue with the use of T as a biomarker of androgen excess is the low circulating levels in women and the specificity and sensitivity of the assays used.<ref name="pmid29306916">{{cite journal |title=Falsely elevated plasma testosterone concentrations in neonates: importance of LC-MS/MS measurements |journal=Clin Chem Lab Med |volume=56 |issue=6 |pages=e141–e143 |pmid=29306916 |doi=10.1515/cclm-2017-1028 |last1=Hamer |first1=Henrike M. |last2=Finken |first2=Martijn J.J. |last3=Van Herwaarden |first3=Antonius E. |last4=Du Toit |first4=Therina |last5=Swart |first5=Amanda C. |last6=Heijboer |first6=Annemieke C. |year=2018 |hdl=10019.1/106715 |s2cid=13917408 }}</ref><ref name="pmid32912651" /><ref name="pmid30753518">{{cite journal|last1=Nanba|first1=Aya T.|last2=Rege|first2=Juilee|last3=Ren|first3=Jianwei|last4=Auchus|first4=Richard J.|last5=Rainey|first5=William E.|last6=Turcu|first6=Adina F.|year=2019|title=11-Oxygenated C19 Steroids Do Not Decline With Age in Women|journal=J Clin Endocrinol Metab|volume=104|issue=7|pages=2615–2622|doi=10.1210/jc.2018-02527|pmc=6525564|pmid=30753518}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterised, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women |journal=Fertil Steril |volume=58 |issue=1 |pages=148–52 |pmid=1623996 |doi=10.1016/s0015-0282(16)55152-8 |last1=Carmina |first1=E. |last2=Stanczyk |first2=F. Z. |last3=Chang |first3=L. |last4=Miles |first4=R. A. |last5=Lobo |first5=R. A. |year=1992 }}</ref><ref name="pmid14417423">{{cite journal |title=Urinary ketosteroids and pregnanetriol in hirsutism |journal=J Clin Endocrinol Metab |volume=20 |issue= 2|pages=180–6 |pmid=14417423 |doi=10.1210/jcem-20-2-180|last1=Lipsett |first1=Mortimer B. |last2=Riter |first2=Barbara |year=1960 }}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries |journal=J Clin Endocrinol Metab |volume=66 |issue=5 |pages=946–50 |pmid=3129451 |doi=10.1210/jcem-66-5-946 |last1=Polson |first1=D. W. |last2=Reed |first2=M. J. |last3=Franks |first3=S. |last4=Scanlon |first4=M. J. |last5=James |first5=V. H. T. |year=1988 }}</ref> However, due to to conflicting reports ratios did not find a firm footing in the clinical as a diagnostic tool. === On The Aromatization of Androgens === Unlike T and A4, 11-oxygenated androgens are unlikely to be converted by aromatase into estrogens ''in vivo'',<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> that was first predicted in 2016 by Imamichi at al. in an ''in vitro'' study.<ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women when not taking into account DHEA. However, DHEA has a very low affinity for the androgen receptor and thus should not be an important contributor, if at all, for receptor activation under normal conditions.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, Barnard et al., incubating ''in vitro'' three different aromatase-expressing cell cultures and ''ex vivo'' human placenta explant cultures with normal and radiolabeled steroids, detected conversion of 11-oxygenated and conventional androgens into 11-oxygenated estrogens; however, 11-oxygenated estrogens were not detected ''in vivo'': neither in pregnant women who have high aromatase expression nor in patients who have high 11-androgens levels due to CAH or adrenocortical carcinoma, probably due to relatively low aromatase activity towards 11-oxygenated androgens compared to conventional androgens.<ref name="pmid33340399"/> However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Therefore, the role of DHT and 11-oxygenated androgens should be seriously considered in women patients. === Disorders of Sex Development === Since both the canonical and backdoor pathways of androgen biosynthesis towards DHT lead to early male sexual differentiation<ref name="pmid30763313" /><ref name="pmid30943210">{{cite journal|title = The "backdoor pathway" of androgen synthesis in human male sexual development | journal = PLOS Biology | volume = 17 | issue = 4 | pages = e3000198 | pmid = 30943210 | pmc = 6464227 | doi = 10.1371/journal.pbio.3000198 | last1 = Miller | first1 = Walter L. | last2 = Auchus | first2 = Richard J. |year = 2019 }}</ref><ref name="pmid11035809" /><ref name="pmid15249131" /> and are required for normal human male genital development,<ref name="pmid30943210" /><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of male fetuses,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 acts as a substrate during the biosynthesis of DHT in the backdoor pathway.<ref name="pmid30763313"/> Flück et al. described in 2011 a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. These findings confirmed that DHT produced in a backdoor pathway, while not necessary for the sexual development of females, is important for that of males. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were enough to disrupt it.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). As such, this syndrome leads to DSD in both sexes, while affected girls go usually unrecognized until puberty, when they show amenorrhea. This syndrome is also rare with only a few cases reported.<ref name="pmid34711511"/> Besides that, 11-oxygenated androgens may play previously overlooked role in DSD.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> === Congenital Adrenal Hyperplasia === Another cause of androgen excess is congenital adrenal hyperplasia (CAH), a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal |vauthors=El-Maouche D, Arlt W, Merke DP |title=Congenital adrenal hyperplasia |journal=Lancet |volume=390 |issue=10108 |pages=2194–2210 |date=November 2017 |pmid=28576284 |doi=10.1016/S0140-6736(17)31431-9 |url=}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal |vauthors=Speiser PW, White PC |title=Congenital adrenal hyperplasia |journal=N Engl J Med |volume=349 |issue=8 |pages=776–88 |date=August 2003 |pmid=12930931 |doi=10.1056/NEJMra021561 |url=}}</ref><ref name="pmid30272171">{{cite journal | title = Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline | journal = The Journal of Clinical Endocrinology and Metabolism | volume = 103 | issue = 11 | pages = 4043–4088 | year = 2018 | pmid = 30272171 | pmc = 6456929 | doi = 10.1210/jc.2018-01865 }}</ref> Such deficiency leads to an excessive accumulation of a respective cortisol precursor, that becomes to serve as a substrate to androgens. In CYP21A2 deficiency<ref name="pmid22170725" /> including the mild forms (which are not always diagnosed)<ref name="pmid32966723">{{cite journal |vauthors=Merke DP, Auchus RJ |title=Congenital Adrenal Hyperplasia Due to 21-Hydroxylase Deficiency |journal=N Engl J Med |volume=383 |issue=13 |pages=1248–1261 |date=September 2020 |pmid=32966723 |doi=10.1056/NEJMra1909786 |url=}}</ref><ref name="pmid31499506">{{cite book|title=Hyperandrogenism in Women|last1=Pignatelli|first1=Duarte|last2=Pereira|first2=Sofia S.|last3=Pasquali|first3=Renato|year=2019|isbn=978-3-318-06470-4|series=Frontiers of Hormone Research|volume=53|pages=65–76|chapter=Androgens in Congenital Adrenal Hyperplasia|doi=10.1159/000494903|pmid=31499506|s2cid=202412336}}</ref> or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> elevated 17-OHP levels starts the backdoor pathway to DHT. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> The reason why 17-OHP serves as a prerequisite substrate for DHT within the backdoor pathway roundabout of T rather then an immediate substrate within the Δ⁴ pathway for A4, and then T, is because the catalytic activity 17,20-lyase reaction (which cleaves a side-chain from the steroid nucleus converting a pregnane to an androstane (androgen), i.e., from 17OPH5 to DHEA; from 17-OHP to A4) performed by CYP17A1 in humans is approximately 100 times more efficient in the Δ⁵ pathway than in the Δ⁴ pathway. Therefore, the catalytic efficiency of CYP17A1 for 17-OHP is about 100 times lower than for 17OHP5, resulting in negligible A4 being produced from 17-OHP in the Δ⁴ reaction pathway in humans.<ref name="pmid8325965">{{cite journal|last1=Swart|first1=P.|last2=Swart|first2=A. C.|last3=Waterman|first3=M. R.|last4=Estabrook|first4=R. W.|last5=Mason|first5=J. I.|year=1993|title=Progesterone 16 alpha-hydroxylase activity is catalyzed by human cytochrome P450 17 alpha-hydroxylase|journal=J Clin Endocrinol Metab|volume=77|issue=1|pages=98–102|doi=10.1210/jcem.77.1.8325965|pmid=8325965}}</ref><ref name="pmid12915666">{{cite journal|last1=Flück|first1=Christa E.|last2=Miller|first2=Walter L.|last3=Auchus|first3=Richard J.|year=2003|title=The 17, 20-lyase activity of cytochrome CYP17A1 from human fetal testis favors the delta5 steroidogenic pathway|url=https://pubmed.ncbi.nlm.nih.gov/12915666|journal=The Journal of Clinical Endocrinology and Metabolism|volume=88|issue=8|pages=3762–3766|doi=10.1210/jc.2003-030143|issn=0021-972X|pmid=12915666}}</ref><ref name="pmid15774560">{{cite journal|last1=Miller|first1=Walter L.|year=2005|title=Minireview: regulation of steroidogenesis by electron transfer|url=https://pubmed.ncbi.nlm.nih.gov/15774560|journal=Endocrinology|volume=146|issue=6|pages=2544–2550|doi=10.1210/en.2005-0096|issn=0013-7227|pmid=15774560}}</ref><ref name="pmid32007561"/> The accumulation of 17-OHP in CYP21A2 deficiency in CAH can be attributed to the fact that the primary enzyme for 17-OHP in normal conditions is CYP21A2, that is expressed in the adrenal and not the gonads.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|year=2019 |volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086 }}</ref><ref name="pmid26038201"/> In a 1998 study, Auchus et al. demonstrated that human CYP17A1 efficiently catalyzed the conversion of P4 to 17-OHP, but the conversion of 17-OHP to A4 was much less efficient than the corresponding conversion of 17OHP5 to DHEA.<ref name="pmid9452426"/> In rodents, quite contrary, the conversion of 17-OHP to A4 is very efficient.<ref name="pmid9452426">{{cite journal | last1=Auchus | first1=Richard J. | last2=Lee | first2=Tim C. | last3=Miller | first3=Walter L. | title=Cytochrome b 5 Augments the 17,20-Lyase Activity of Human P450c17 without Direct Electron Transfer | journal=The Journal of Biological Chemistry | year=1998 | publisher=Elsevier BV | volume=273 | issue=6 | issn=0021-9258 | pmid=9452426 | doi=10.1074/jbc.273.6.3158 | pages=3158–3165| doi-access=free }}</ref> This explains significant accumulation of 17-OHP in CYP21A2 deficiency or POR deficiency in humans, so that 17-OHP, while not 21-hydroxylated in sufficient quantities, and being better a substrate for 5α-reductase than for CYP17A1, is 5α-reduced serving as the prerequisite for this backdoor pathway. Hence, fetal excess of 17-OHP in CAH may provoke activation of this pathway to DHT and lead to external genital virilization in newborn girls, thus explaining DSD in girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025"/><ref name="pmid31505456">{{cite journal |vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE |title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia |journal=Eur J Endocrinol |volume=181 |issue=5 |pages=481–488 |date=November 2019 |pmid=31505456 |doi=10.1530/EJE-19-0379 |url=}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal |vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M |title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia |journal=J Obstet Gynaecol Res |volume=43 |issue=5 |pages=946–950 |date=May 2017 |pmid=28188961 |doi=10.1111/jog.13288 |url=}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal | last1=Turcu | first1=Adina F | last2=Mallappa | first2=Ashwini | last3=Elman | first3=Meredith S | last4=Avila | first4=Nilo A | last5=Marko | first5=Jamie | last6=Rao | first6=Hamsini | last7=Tsodikov | first7=Alexander | last8=Auchus | first8=Richard J | last9=Merke | first9=Deborah P | title = 11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency | journal = The Journal of Clinical Endocrinology and Metabolism | year=2017 | volume = 102 | issue = 8 | pages = 2701–2710 | pmid = 28472487 | pmc = 5546849 | doi = 10.1210/jc.2016-3989}}</ref><ref name="pmid26865584">{{cite journal|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency |journal=Eur J Endocrinol |volume=174 |issue=5 |pages=601–9 |pmid=26865584 |pmc=4874183 |doi=10.1530/EJE-15-1181 |last1=Turcu |first1=Adina F. |last2=Nanba |first2=Aya T. |last3=Chomic |first3=Robert |last4=Upadhyay |first4=Sunil K. |last5=Giordano |first5=Thomas J. |last6=Shields |first6=James J. |last7=Merke |first7=Deborah P. |last8=Rainey |first8=William E. |last9=Auchus |first9=Richard J. |year=2016 }}</ref><ref name="pmid29718004">{{cite journal|title = Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency | journal = Current Opinion in Endocrinology, Diabetes, and Obesity | volume = 25 | issue = 3 | pages = 178–184 | pmid = 29718004 | doi = 10.1097/MED.0000000000000402 | s2cid = 26072848 |last1 = White |first1 = Perrin C. |year = 2018 }}</ref><ref name="pmid34867794"/> In males with CAH, 11-oxygenated androgens may lead to devlopment of testicular adrenal rest tumors.<ref name="pmid25850025">{{cite journal|pmc=4454804|year=2015|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|last10=Auchus|first10=R. J.|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmid=25850025}}</ref><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors |journal=J Clin Endocrinol Metab |volume=107 |issue=1 |pages=e272–e280 |pmid=34390337 |pmc=8684463 |doi=10.1210/clinem/dgab598 |last1=Schröder |first1=Mariska A M. |last2=Turcu |first2=Adina F. |last3=o'Day |first3=Patrick |last4=Van Herwaarden |first4=Antonius E. |last5=Span |first5=Paul N. |last6=Auchus |first6=Richard J. |last7=Sweep |first7=Fred C G J. |last8=Claahsen-Van Der Grinten |first8=Hedi L. |year=2022 }}</ref> The biosynthesis of 11OHP4 from P4 and 21dF from 17-OHP by CYP11B1/2 in CAH may be attributed to CYP21A2 deficiency resulting in increased P4 and 17-OHP concentrations and, together with the unavailability of CYP11B1/2's main substrates, 11-deoxycortisol (11dF) and 11-deoxycorticosterone (DOC), drive the production of 11-oxygenated pregnanes.<ref name="pmid3546944" /> We have reasons to believe that this may be aggravated by elevated ACTH due to a feedback loop in hypothalamic-pituitary-adrenal axis caused by impaired cortisol synthesis associated with CYP21A2 deficiency; higher ACTH causes higher CYP11B1 expression. Multiple studies demonstrated that in CAH due to CYP21A2 deficiency, both 21dF levels<ref name="pmid4372245">{{cite journal |title=Plasma 17-hydroxyprogesterone, 21-deoxycortisol and cortisol in congenital adrenal hyperplasia |journal=J Clin Endocrinol Metab |volume=39 |issue=6 |pages=1099–102 |year=1974 |pmid=4372245 |doi=10.1210/jcem-39-6-1099 |last1=Franks |first1=Robert C. }}</ref><ref name="pmid476971">{{cite journal |title=Rapid assay of plasma 21-deoxycortisol and 11-deoxycortisol in congenital adrenal hyperplasia |journal=Clin Endocrinol (Oxf) |volume=10 |issue=4 |pages=367–75 |year=1979 |pmid=476971 |doi=10.1111/j.1365-2265.1979.tb02091.x |url=|last1=Fukushima |first1=D. K. |last2=Nishina |first2=T. |last3=Wu |first3=R. H. K. |last4=Hellman |first4=L. |last5=Finkelstein |first5=J. W. |s2cid=2979354 }}</ref><ref name="pmid6090811">{{cite journal |title=Development of plasma 21-deoxycortisol radioimmunoassay and application to the diagnosis of patients with 21-hydroxylase deficiency |journal=J Steroid Biochem |volume=21 |issue=2 |pages=185–91 |year=1984 |pmid=6090811 |doi=10.1016/0022-4731(84)90382-0 |last1=Milewicz |first1=A. |last2=Vecsei |first2=P. |last3=Korth-Schütz |first3=S. |last4=Haack |first4=D. |last5=Rösler |first5=A. |last6=Lichtwald |first6=K. |last7=Lewicka |first7=S. |last8=Mittelstaedt |first8=G.v. }}</ref><ref name="pmid2986404">{{cite journal |title=Radioimmunoassay for 21-deoxycortisol: clinical applications |journal=Acta Endocrinol (Copenh) |volume=108 |issue=4 |pages=537–44 |year=1985 |pmid=2986404 |doi=10.1530/acta.0.1080537 |last1=Gueux |first1=B. |last2=Fiet |first2=J. |last3=Pham-Huu-Trung |first3=M. T. |last4=Villette |first4=J. M. |last5=Gourmelen |first5=M. |last6=Galons |first6=H. |last7=Brerault |first7=J. L. |last8=Vexiau |first8=P. |last9=Julien |first9=R. }}</ref><ref name="pmid25850025" /> and 11OPH4 levels<ref name="pmid3546944">{{cite journal |last1=Gueux |first1=Bernard |last2=Fiet |first2=Jean |last3=Galons |first3=Hervé |last4=Boneté |first4=Rémi |last5=Villette |first5=Jean-Marie |last6=Vexiau |first6=Patrick |last7=Pham-Huu-Trung |first7=Marie-Thérèse |last8=Raux-Eurin |first8=Marie-Charles |last9=Gourmelen |first9=Micheline |last10=Brérault |first10=Jean-Louis |last11=Julien |first11=René |last12=Dreux |first12=Claude |title=The measurement of 11β-hydroxy-4-pregnene-3,20-dione (21-Deoxycorticosterone) by radioimmunoassay in human plasma |journal=Journal of Steroid Biochemistry |year=1987 |volume=26 |issue=1 |pages=145–150 |doi=10.1016/0022-4731(87)90043-4 |pmid=3546944 }}</ref><ref name="pmid2537337">{{cite journal |last1=Fiet |first1=Jean |last2=Gueux |first2=Bernard |last3=Rauxdemay |first3=Marie-Charles |last4=Kuttenn |first4=Frederique |last5=Vexiau |first5=Patrick |last6=Brerault |first6=Jeanlouis |last7=Couillin |first7=Philippe |last8=Galons |first8=Herve |last9=Villette |first9=Jeanmarie |last10=Julien |first10=Rene |last11=Dreux |first11=Claude |title=Increased Plasma 21-Deoxycorticosterone (21-DB) Levels in Late-Onset Adrenal 21-Hydroxylase Deficiency Suggest a Mild Defect of the Mineralocorticoid Pathway |journal=The Journal of Clinical Endocrinology & Metabolism |year=1989 |volume=68 |issue=3 |pages=542–547 |doi=10.1210/jcem-68-3-542 |pmid=2537337 }}</ref><ref name="pmid29264476">{{cite journal |last1=Fiet |first1=Jean |last2=Le Bouc |first2=Yves |last3=Guéchot |first3=Jérôme |last4=Hélin |first4=Nicolas |last5=Maubert |first5=Marie-Anne |last6=Farabos |first6=Dominique |last7=Lamazière |first7=Antonin |title=A Liquid Chromatography/Tandem Mass Spectometry Profile of 16 Serum Steroids, Including 21-Deoxycortisol and 21-Deoxycorticosterone, for Management of Congenital Adrenal Hyperplasia |journal=Journal of the Endocrine Society |year=2017 |volume=1 |issue=3 |pages=186–201 |doi=10.1210/js.2016-1048 |pmid=29264476 |pmc=5686660 }}</ref><ref name="pmid31821037">{{cite journal |title=Interaction between accumulated 21-deoxysteroids and mineralocorticoid signaling in 21-hydroxylase deficiency |journal=Am J Physiol Endocrinol Metab |volume=318 |issue=2 |pages=E102–E110 |year=2020 |pmid=31821037 |doi=10.1152/ajpendo.00368.2019 |last1=Travers |first1=Simon |last2=Bouvattier |first2=Claire |last3=Fagart |first3=Jérôme |last4=Martinerie |first4=Laetitia |last5=Viengchareun |first5=Say |last6=Pussard |first6=Eric |last7=Lombès |first7=Marc |s2cid=209314028 }}</ref> are increased. It was Robert Franks in who first published a study, in 1974, that compared 21dF levels of CAH patients with those of healthy controls. He measured 21dF plasma levels in twelve CAH patients before treatment, three after treatment, and four healthy controls following ACTH administration. Mean values of 21dF in CAH patients was 88 ng/ml while in healthy controls it was not detected. In untreated patients, values decreased after therapy. Even that, there were earlier reports about unique cases where 21dF was detected in CAH patients, but without direct comparison to healthy controls.<ref name="pmid5845501">{{cite journal |title=Detection of 21-deoxycortisol in blood from a patient with congenital adrenal hyperplasia |journal=Metabolism |year=1965 |volume=14 |issue=12 |pages=1276–81 |pmid=5845501 |doi=10.1016/s0026-0495(65)80008-7|last1=Wieland |first1=Ralph G. |last2=Maynard |first2=Donald E. |last3=Riley |first3=Thomas R. |last4=Hamwi |first4=George J. }}</ref><ref name="pmid13271547">{{cite journal|title=17alpha-hydroxyprogesterone and 21-desoxyhydrocortisone; their metabolism and possible role in congenital adrenal virilism |journal=J Clin Invest |volume=34 |issue=11 |pages=1639–46 |year=1955 |pmid=13271547 |pmc=438744 |doi=10.1172/JCI103217|last1=Jailer |first1=Joseph W. |last2=Gold |first2=Jay J. |last3=Vande Wiele |first3=Raymond |last4=Lieberman |first4=Seymour }}</ref> As for 11OHP4, it were Gueux et al. who first demonstrated, in 1987, elevated plasma levels of 11OHP4 in CAH. In that study, in treated classical CAH patients, some of which had salt-wasting form, mean levels of 11OHP4 (5908.7 pmol/l) were 332 times higher than in healthy controls (17.8 pmol/l). There was no difference in 11OHP4 in healthy controls depending on sex or phase of a menstrual cycle; ACTH stimulation in those control increased 11OHP4 four- to six-fold, while dexamethasone 1 mg at midnight decreased 11OHP4 to almost undetectable levels 12 hours later. Therefore, the authors hypothesized that at least in healthy people 11OHP4 is biosythesized exclusively in the adrenal, while gonads are not involved.<ref name="pmid3546944" /> Nevertheless, in studies focusing on CAH caused by CYP21A2 deficiency, 11OHP4 received less attention than 21dF.<ref name="pmid29277707"/> However, it was not until 2017 when 11OHP4 or 21dF were viewed as potential substrates in pathways towards potent 11-ogygenated androgens in ''in vitro'' studies.<ref name="pmid32007561"/><ref name="pmid29277707"/> In a 2016 study, Turcu et al. showed that in classic CAH due to CYP21A2 deficiency, in male and female patients who received glucocorticoid therapy, both conventional and 11-oxygenated androgens were elevated 3-4 fold compared to healthy controls. The exceptions were dehydroepiandrosterone (DHEA), dehydroepiandrosterone sulfate (DHEA-S), and androstenediol sulfate (A5-S), whose levels were 6.0, 7.5, and 9.4 times lower, respectively, in the patients with the condition compared to healthy controls, due to suppression by glucocorticoid treatment. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in a backdoor pathway from 17-OHP or P4 as consequence of abnormally upregulated SRD5A1.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Besides that, the levels of 11OHA4 and 11KA4 correlated with the levels of markers of insulin resistance; therefore, the study suggests that androgen excess precedes androgen-driven insulin resistance in PCOS.<ref name="pmid27901631" /> While earlier studies had commonly only measured 11OHA4 or 11OHAST and 11β-hydroxyetiocholanolone (11OHEt), urinary metabolites of 11OHA,<ref name="pmid33539964" /> while 11OHEt is also a metabolite of cortisol,<ref name="pmid31362062">{{cite journal |title=Human steroid biosynthesis, metabolism and excretion are differentially reflected by serum and urine steroid metabolomes: A comprehensive review |journal=J Steroid Biochem Mol Biol |volume=194 |issue= |pages=105439 |year=2019 |pmid=31362062 |pmc=6857441 |doi=10.1016/j.jsbmb.2019.105439 |url=|last1=Schiffer |first1=Lina |last2=Barnard |first2=Lise |last3=Baranowski |first3=Elizabeth S. |last4=Gilligan |first4=Lorna C. |last5=Taylor |first5=Angela E. |last6=Arlt |first6=Wiebke |last7=Shackleton |first7=Cedric H.L. |last8=Storbeck |first8=Karl-Heinz }}</ref><ref name="pmid27845856">{{cite journal |title=Modified-Release and Conventional Glucocorticoids and Diurnal Androgen Excretion in Congenital Adrenal Hyperplasia |journal=J Clin Endocrinol Metab |volume=102 |issue=6 |pages=1797–1806 |year=2017 |pmid=27845856 |pmc=5470768 |doi=10.1210/jc.2016-2855|last1=Jones |first1=Christopher M. |last2=Mallappa |first2=Ashwini |last3=Reisch |first3=Nicole |last4=Nikolaou |first4=Nikolaos |last5=Krone |first5=Nils |last6=Hughes |first6=Beverly A. |last7=o'Neil |first7=Donna M. |last8=Whitaker |first8=Martin J. |last9=Tomlinson |first9=Jeremy W. |last10=Storbeck |first10=Karl-Heinz |last11=Merke |first11=Deborah P. |last12=Ross |first12=Richard J. |last13=Arlt |first13=Wiebke }}</ref> more recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Another study reported that 11OHT was the only significantly elevated 11-oxygeated androgen in PCOS and together with 11KT, correlated with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227"/> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Premature Adrenarche === In a 2018 study, Rege et al. demonstrated that levels of 11KT in girls aged between 4 and 7 years during normal adrenarche (healthy controls) exceeded those of T by 2.43 times, and in those with premature adrenarche by 3.48 times. However, the levels of T in girls with premature adrenarche were higher by just 13% compared to age-matched healthy controls.<ref name="pmid30137510">{{cite journal | last1=Rege | first1=Juilee | last2=Turcu | first2=Adina | last3=Kasa-Vubu | first3=Josephine Z | last4=Lerario | first4=Antonio M | last5=Auchus | first5=Gabriela C | last6=Auchus | first6=Richard J | last7=Smith | first7=Joshua M | last8=White | first8=Perrin C | last9=Rainey | first9=William E | title=11KT is the dominant circulating bioactive androgen during normal and premature adrenarche | journal=The Journal of Clinical Endocrinology and Metabolism | year=2018 | publisher=The Endocrine Society | volume=103 | issue=12 | pages=4589–4598 | issn=0021-972X | pmid=30137510 | pmc=6226603 | doi=10.1210/jc.2018-00736 }}</ref> === Prostate Cancer === In some cases of advanced prostate cancer, androgen deprivation therapy related to gonadal T depletion does not produce long-term effects, and metastatic tumors may develop into castration-resistant prostate cancer (CRPC). The development of CRPC depends on adrenal precursor steroids to produce DHT in the tumor in a metabolic pathway called the "5α-dione" pathway - the pathway in which T is not involved. SRD5A1, the expression of which increases in CRPC, reduces A4 to 5α-dione, which is then converted to DHT.<ref name="pmid21795608"/><ref name="pmid31900912"/> Therefore, the DHT produced within the "5α-dione" pathway hampers the androgen deprivation therapy. Although blood levels of T are decreased by 90-95% in men whose testicles have been removed, DHT in the prostate is only decreased by 50%, thus indicating the presence of a metabolic pathway in the prostate that does not require testicular T to produce DHT.<ref name="pmid18471780">{{cite journal | last1=Luu-The | first1=Van | last2=Bélanger | first2=Alain | last3=Labrie | first3=Fernand | title=Androgen biosynthetic pathways in the human prostate | journal=Best Practice & Research. Clinical Endocrinology & Metabolism | publisher=Elsevier BV | volume=22 | issue=2 | year=2008 | issn=1521-690X | pmid=18471780 | doi=10.1016/j.beem.2008.01.008 | pages=207–221}}</ref> Chang et al., incubating six established human prostate cancer cell lines from patients with CRPC in the presence of radiolabeled A4, showed in their experiment published in 2011<ref name="pmid21795608">{{cite journal|last1=Chang | first1=K.-H. | last2=Li | first2=R. | last3=Papari-Zareei | first3=M. | last4=Watumull | first4=L. | last5=Zhao | first5=Y. D. | last6=Auchus | first6=R. J. | last7=Sharifi | first7=N. | title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer | journal=Proceedings of the National Academy of Sciences of the United States of America |year=2011 | publisher=Proceedings of the National Academy of Sciences | volume=108 | issue=33 | issn=0027-8424 | pmid=21795608 | pmc=3158152 | doi=10.1073/pnas.1107898108 | pages=13728–13733|bibcode=2011PNAS..10813728C |doi-access=free }}</ref> the presence of this pathway to DHT which bypasses T and they called this the "alternative" pathway, that became later commonly called as the "5α-dione" pathway.<ref name="pmid23856005"/> The authors demonstrated that this was the dominant pathway in prostate cancer (over the direct conversion of A4 to T) with SRD5A1 (which is upregulated in prostate cancer) first converting A4 to androstanedione (5α-dione), also known as 5α-androstane-3,17-dione, and then HSD17B3 / AKR1C3 converting 5α-dione to DHT (not necessarily via AST and 3α-diol). The study also found that the SRD5A2 is not involved in this "alternative" pathway.<ref name="pmid31900912"/> Therefore, the study showed the importance of taking into consideration this "alternative" pathway in selecting drugs that inhibit 5α-reductase activity.<ref name="pmid21901017">{{cite journal |title=Prostate cancer: DHT bypasses testosterone to drive progression to castration resistance |journal=Nat Rev Urol |volume=8 |issue=9 |pages=470 |year=September 2011 |pmid=21901017 |doi=10.1038/nrurol.2011.122 }}</ref><ref name="pmid22064602" /><ref name="pmid22336886">{{cite journal |title=Dihydrotestosterone synthesis from adrenal precursors does not involve testosterone in castration-resistant prostate cancer |journal=Cancer Biol Ther |volume=13 |issue=5 |pages=237–8 |year=2012 |pmid=22336886 |doi=10.4161/cbt.19608}}</ref> Another pathway that may be activated in CRPC, which may also hamper the androgen deprivation therapy, is the backdoor pathway from P4 to DHT. Chen et al. in a study published in 2014<ref name="pmid25320358">{{cite journal |vauthors=Chen EJ, Sowalsky AG, Gao S, Cai C, Voznesensky O, Schaefer R, Loda M, True LD, Ye H, Troncoso P, Lis RL, Kantoff PW, Montgomery RB, Nelson PS, Bubley GJ, Balk SP, Taplin ME |title=Abiraterone treatment in castration-resistant prostate cancer selects for progesterone responsive mutant androgen receptors |journal=Clin Cancer Res |volume=21 |issue=6 |pages=1273–80 |date=March 2015 |pmid=25320358 |pmc=4359958 |doi=10.1158/1078-0432.CCR-14-1220 |url=}}</ref> predicted that abiraterone, a CYP17A1 inhibitor, with about 6-fold more selective for inhibition of 17α-hydroxylase over 17,20-lyase,<ref name="pmid28890368">{{cite journal |vauthors=de Mello Martins AGG, Allegretta G, Unteregger G, Haupenthal J, Eberhard J, Hoffmann M, van der Zee JA, Junker K, Stöckle M, Müller R, Hartmann RW, Ohlmann CH |title=CYP17A1-independent production of the neurosteroid-derived 5α-pregnan-3β,6α-diol-20-one in androgen-responsive prostate cancer cell lines under serum starvation and inhibition by Abiraterone |journal=J Steroid Biochem Mol Biol |volume=174 |issue= |pages=183–191 |date=November 2017 |pmid=28890368 |doi=10.1016/j.jsbmb.2017.09.006 |url=}}</ref><ref name="pmid28373265">{{cite journal |vauthors=Petrunak EM, Rogers SA, Aubé J, Scott EE |title=Structural and Functional Evaluation of Clinically Relevant Inhibitors of Steroidogenic Cytochrome P450 17A1 |journal=Drug Metab Dispos |volume=45 |issue=6 |pages=635–645 |date=June 2017 |pmid=28373265 |pmc=5438109 |doi=10.1124/dmd.117.075317 |url=}}</ref><ref name="pmid29710837">{{cite journal |vauthors=Fernández-Cancio M, Camats N, Flück CE, Zalewski A, Dick B, Frey BM, Monné R, Torán N, Audí L, Pandey AV |title=Mechanism of the Dual Activities of Human CYP17A1 and Binding to Anti-Prostate Cancer Drug Abiraterone Revealed by a Novel V366M Mutation Causing 17,20 Lyase Deficiency |journal=Pharmaceuticals (Basel) |volume=11 |issue=2 |pages= |date=April 2018 |pmid=29710837 |pmc=6027421 |doi=10.3390/ph11020037 |url=}}</ref> although disrupting canonical androgen biosynthesis, while lowering levels of T, causes elevation of P4, that can be 5α-reduced hence start a backdoor pathway from P4 to DHT with roundabout of T.<ref name="pmid25320358"/> Besides that, in CRPC, 11-oxygenated androgens contribute significantly to the androgen pool.<ref name="pmid23856005"/><ref name="pmid31900912"/> 11-oxygenated androgens play a previously overlooked role in the reactivation of androgen signaling in CRPC,<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> because after eliminating testicular T biosynthesis by chemical or physical castration, CRPC has been shown to develop the ability to convert inactive circulating adrenal androgen precursors, DHEA and A4, to potent 11-oxygenated androgens in the 11-oxygenated pathway in addition to the 5α-dione pathway.<ref name="pmid31672619">{{cite journal |title=The role of adrenal derived androgens in castration resistant prostate cancer |journal=J Steroid Biochem Mol Biol |volume=197 |issue= |pages=105506 |year=2020 |pmid=31672619 |doi=10.1016/j.jsbmb.2019.105506|last1=Barnard |first1=Monique |last2=Mostaghel |first2=Elahe A. |last3=Auchus |first3=Richard J. |last4=Storbeck |first4=Karl-Heinz |pmc=7883395 }}</ref><ref name="pmid33974560" /> In a 2021 study, Snaterse et al. demonstrated that 11KT is the most circulating active androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool. They also demonstrated that 11KT levels are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> In a 2018 study by du Toit et al., the full range of androgen pathway metabolites have been shown in normal prostate and various prostate cancer cell models. 11OHA4 and 11OHT were both converted to potent androgens, 11KT and 11KDHT. Compared to T and DHT, 11-oxygenated androgens were the most predominant androgens. High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant. Furthermore, glucuronidation of the 11-oxygenated androgens is hampered by the presence of an oxo- or a hydroxy- group at position 11 of androgens in prostate cancer cell models while in prostate cancer patients' plasma 11KDHT was present only in the unconjugated form, with 11KT also predominantly unconjugated.<ref name="pmid28939401"/> Of all 11-oxygenated androgens in prostate cancer, it may be that 11KT is the primary active 11-oxygenated androgen, rather than 11KDHT, as observed in experiments on prostate tissue homogenates by Häkkinen et al. published in 2018<ref name="pmid30472582">{{cite journal|last1=Häkkinen|first1=Merja R.|last2=Murtola|first2=Teemu|last3=Voutilainen|first3=Raimo|last4=Poutanen|first4=Matti|last5=Linnanen|first5=Tero|last6=Koskivuori|first6=Johanna|last7=Lakka|first7=Timo|last8=Jääskeläinen|first8=Jarmo|last9=Auriola|first9=Seppo|year=2019|title=Simultaneous analysis by LC-MS/MS of 22 ketosteroids with hydroxylamine derivatization and underivatized estradiol from human plasma, serum and prostate tissue|journal=J Pharm Biomed Anal|volume=164|issue=|pages=642–652|doi=10.1016/j.jpba.2018.11.035|pmid=30472582|s2cid=53729550}}</ref> Ventura-Bahena et al., in a 2021 study, describing results of epidemiological studies related to androgens and prostate cancer that focused on specific androgen concentrations (such as T, A4, and DHEA) as inconsistent, hypothesized that the differences in androgen biosynthetic pathways rather than differences in specific androgen levels are associated with prostate cancer development.<ref name="pmid34520388"/> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but also in clinical conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> The contribution of the 11-oxygenated androgens, as well as the biosynthesis of 11-oxygenated pregnanes to active androgens via a backdoor pathway, have also been demonstrated in BPH cell models showing the conversion of 11OHP4 and 11KP4 in the backdoor pathway resulting in the production of 11KDHT. Backdoor pathway intermediates were also detected in BPH tissue as well as in circulation in BPH patients.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> In a paper published in 2008, Dimitrakov et al. hypothesized that CP/CPPS may be associated with a mild CYP21A2 deficiency, a cause of non-classic CAH that leads to androgen excesses.<ref name="pmid18308097"/> Non-classic CAH was generally thought to be asymptomatic in men.<ref name="pmid28582566">{{cite journal |title=Non-classic congenital adrenal hyperplasia due to 21-hydroxylase deficiency revisited: an update with a special focus on adolescent and adult women |journal=Hum Reprod Update |volume=23 |issue=5 |pages=580–599 |year=2017 |pmid=28582566 |doi=10.1093/humupd/dmx014 |last1=Carmina |first1=Enrico |last2=Dewailly |first2=Didier |last3=Escobar-Morreale |first3=Héctor F. |last4=Kelestimur |first4=Fahrettin |last5=Moran |first5=Carlos |last6=Oberfield |first6=Sharon |last7=Witchel |first7=Selma F. |last8=Azziz |first8=Ricardo }}</ref><ref name="pmid20671993">{{cite journal |title=Nonclassic congenital adrenal hyperplasia |journal=Int J Pediatr Endocrinol |volume=2010 |pages=625105 |year=2010 |pmid=20671993 |pmc=2910408 |doi=10.1155/2010/625105|doi-access=free |last1=Witchel |first1=Selma Feldman |last2=Azziz |first2=Ricardo }}</ref> The authors of that 2008 paper, therefore, concluded that CP/CPPS may be a consequence of a systemic condition of androgen excess but not a disease that originates in the prostate such as a localized prostate infection, inflammation, or dysfunction. We hypothesize that CYP21A2 deficiency in CP/CPPS may be associated with elevated androgens produced by pathways activated by such deficiency, i.e. backdoor pathway from P4 or 17-OHP to DHT and the pathways towards 11-oxygenated androgens. ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} llmxsuy636lf7zjl5ch0qx2fh2vz1mm 2410905 2410807 2022-08-02T05:24:18Z Maneesh 2723004 /* From Androstenedione or Testosterone Towards 11-Oxygenated Androgens */ saving progress wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> and the potential clinical relevance in conditions involving androgen biosynthesis was proposed. Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref><ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid11514561">{{cite journal|last1=Nahoum|first1=Virginie|last2=Gangloff|first2=Anne|last3=Legrand|first3=Pierre|last4=Zhu|first4=Dao-Wei|last5=Cantin|first5=Line|last6=Zhorov|first6=Boris S.|last7=Luu-The|first7=Van|last8=Labrie|first8=Fernand|last9=Breton|first9=Rock|year=2001|title=Structure of the human 3alpha-hydroxysteroid dehydrogenase type 3 in complex with testosterone and NADP at 1.25-A resolution|journal=J Biol Chem|volume=276|issue=45|pages=42091–8|doi=10.1074/jbc.M105610200|pmid=11514561|doi-access=free|last10=Lin|first10=Sheng-Xiang}}</ref><ref name="pmid18923939">{{cite journal|last1=Dozmorov|first1=Mikhail G.|last2=Yang|first2=Qing|last3=Matwalli|first3=Adam|last4=Hurst|first4=Robert E.|last5=Culkin|first5=Daniel J.|last6=Kropp|first6=Bradley P.|last7=Lin|first7=Hsueh-Kung|year=2007|title=5alpha-androstane-3alpha,17beta-diol selectively activates the canonical PI3K/AKT pathway: a bioinformatics-based evidence for androgen-activated cytoplasmic signaling|journal=Genomic Med|volume=1|issue=3–4|pages=139–46|doi=10.1007/s11568-008-9018-9|pmc=2269037|pmid=18923939}}</ref><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref><ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}}The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}}The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}}The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608" />{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}}While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780" /> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. 11-Oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid30959151">{{cite journal |title=Circulating 11-oxygenated androgens across species |journal=J Steroid Biochem Mol Biol |volume=190 |pages=242–249 |year=2019 |pmid=30959151 |pmc=6733521 |doi=10.1016/j.jsbmb.2019.04.005|last1=Rege |first1=Juilee |last2=Garber |first2=Scott |last3=Conley |first3=Alan J. |last4=Elsey |first4=Ruth M. |last5=Turcu |first5=Adina F. |last6=Auchus |first6=Richard J. |last7=Rainey |first7=William E. }}</ref><ref name="pmid27519632" /><ref name="pmid34171490" /><ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646" /> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="NBK557634">{{cite book|url=https://www.ncbi.nlm.nih.gov/books/NBK557634/|title=Biochemistry, Dihydrotestosterone|publisher=StatPearls|year=2022}}</ref><ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |pages=431 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|pages=868|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|pages=66|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=441 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref><ref name="pmid23386646" /> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid29277706" /><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405" /> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-), therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the conversion of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1.<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRDA51 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid18923939"/><ref name="Nishiyama2011"/><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels do not decline with age as T does. While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== From Androstenedione or Testosterone Towards 11-Oxygenated Androgens ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== From Progesterone and 17α-Hydroxyprogesterone Towards 11-Oxygenated Androgens ==== The pathways from P4 and 17-OHP to 11-oxygenated androgens can be described as follows. The 17,20-lyase activity of CYP17A1 is needed to cleave a side-chain (C17-C20 bond) from the steroid nucleus to convert an initial pregnane to a final androgen. Human CYP17A1 cannot efficiently catalyze this reaction for steroids that have the oxo- functional group at carbon position 3.<ref name="pmid32007561"/> Examples of such steroids are 11OHPdione or 11KPdione. Therefore, such C₂₁ steroid should be 3α-reduced by AKR1C4 before it can be converted to a C₁₉ steroid by CYP17A1. After the side-chain cleavage by CYP17A1, the oxo- group at position 3 is restored back in a 3α-oxidation reaction (by an enzyme such as AKR1C4 or HSD17B6) to convert an inactive androgen such as 11K-3αdiol to the active one such as 11KDHT.<ref name="pmid31626910"/> As you see, the order of steps in metabolic routes from P4 and 17-OHP towards 11-oxygenated androgens (11KDHT and 11OHDHT) is similar to the conversion of P4 and 17-OHP to DHT in a backdoor pathway.<ref name="pmid28774496" /> ==Clinical Significance == === Biological Role of 11-Oxygenated Androgens === 11-oxygenated androgens are produced in physiological quantities in healthy primate organisms (including humans).<ref name="pmid30959151" /><ref name="pmid30753518" /><ref name="pmid32629108" /> Humans have two isozymes with 11β-hydroxylase activity, encoded by the genes ''CYP11B1'' (regulated by the adrenocorticotropic hormone (ACTH)) and ''CYP11B2'' (regulated by angiotensin II).<ref name="pmid22217826">{{cite journal|name-list-style=vanc|title=Molecular biology of 11β-hydroxylase and 11β-hydroxysteroid dehydrogenase enzymes |journal=J Steroid Biochem Mol Biol |volume=43 |issue=8 |pages=827–35 |pmid=22217826 |doi=10.1016/0960-0760(92)90309-7 |s2cid=19379671 |last1=White |first1=Perrin C. |last2=Pascoe |first2=Leigh |last3=Curnow |first3=Kathleen M. |last4=Tannin |first4=Grace |last5=Rösler |first5=Ariel |year=1992 }}</ref> Since the first step in the biosynthesis of 11-oxygenated androgens involves 11β-hydroxylation of a steroid substrate by CYP11B1/2 isozymes that are generally associated with their expression in the adrenal gland, 11-oxygenated androgens are considered androgens of adrenal origin. They follow the circadian rhythm of cortisol but correlate very weakly with T.<ref name="pmid34867794">{{cite journal |title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=751191 |pmid=34867794 |pmc=8636728 |doi=10.3389/fendo.2021.751191 |doi-access=free |last1=Turcu |first1=Adina F. |last2=Mallappa |first2=Ashwini |last3=Nella |first3=Aikaterini A. |last4=Chen |first4=Xuan |last5=Zhao |first5=Lili |last6=Nanba |first6=Aya T. |last7=Byrd |first7=James Brian |last8=Auchus |first8=Richard J. |last9=Merke |first9=Deborah P. |year=2021 }}</ref><ref name="pmid34324429">{{cite journal|title=Circadian rhythms of 11-oxygenated C19 steroids and ∆5-steroid sulfates in healthy men |journal=Eur J Endocrinol |volume=185 |issue=4 |pages=K1–K6 |pmid=34324429 |doi=10.1530/EJE-21-0348 |pmc=8826489 |pmc-embargo-date=August 27, 2022 |last1=Turcu |first1=Adina F. |last2=Zhao |first2=Lili |last3=Chen |first3=Xuan |last4=Yang |first4=Rebecca |last5=Rege |first5=Juilee |last6=Rainey |first6=William E. |last7=Veldhuis |first7=Johannes D. |last8=Auchus |first8=Richard J. |year=2021 }}</ref> The levels of 11-oxygenated androgens raise after ACTH stimulation<ref name="pmid23386646"/><ref name="pmid13211802">{{cite journal |vauthors=DOBRINER K, KAPPAS A, GALLAGHER TF |title=Studies in steroid metabolism. XXVI. Steroid isolation studies in human leukemia |journal=J Clin Invest |volume=33 |issue=11 |pages=1481–6 |date=November 1954 |pmid=13211802 |pmc=1072573 |doi=10.1172/JCI103026 |url=}}</ref> that further supports their adrenal origin. However, in addition to the adrenal glands, CYP11B1 is also expressed in Leydig cells and ovarian theca cells, albeit at far lower levels, so the production of 11KT precursors may be one of the most important functions of 11β-hydroxylase activity in the gonads.<ref name="pmid27428878">{{cite journal|title=11-Ketotestosterone Is a Major Androgen Produced in Human Gonads |journal=J Clin Endocrinol Metab |volume=101 |issue=10 |pages=3582–3591 |pmid=27428878 |doi=10.1210/jc.2016-2311 |last1=Imamichi |first1=Yoshitaka |last2=Yuhki |first2=Koh-Ichi |last3=Orisaka |first3=Makoto |last4=Kitano |first4=Takeshi |last5=Mukai |first5=Kuniaki |last6=Ushikubi |first6=Fumitaka |last7=Taniguchi |first7=Takanobu |last8=Umezawa |first8=Akihiro |last9=Miyamoto |first9=Kaoru |last10=Yazawa |first10=Takashi |year=2016 }}</ref> In an in vitro study by Strushkevich et al. published in 2013, both isozymes have been shown to convert Δ⁴ steroids (P4, 17-OHP, A4 and T), but they are very specific to the configuration of the A-ring (carbon positions 1 to 5) of steroids, i.e., they cannot convert Δ⁵ steroids with a hydroxy group at the carbon position 3.<ref name="pmid23322723">{{cite journal |pmc=5417327|year=2013|last1=Strushkevich|first1=N.|last2=Gilep|first2=A. A.|last3=Shen|first3=L.|last4=Arrowsmith|first4=C. H.|last5=Edwards|first5=A. M.|last6=Usanov|first6=S. A.|last7=Park|first7=H. W.|title=Structural Insights into Aldosterone Synthase Substrate Specificity and Targeted Inhibition|journal=Molecular Endocrinology (Baltimore, Md.)|volume=27|issue=2|pages=315–324|doi=10.1210/me.2012-1287|pmid=23322723}}</ref> The relative biological significance of the 11-oxygenated androgens has been subject to some debate. There have been enough studies to establish that 11OHA4 itself does not have any androgenic activity but an important precursor to 11KT and 11KDHT.{{cn}} Androgen activity has been established for 11KT, 11KDHT<ref name="pmid27442248" /> as well as 11OHT and 11OHDHT but circulating levels of all of these androgens have not been firmly established. 11KT may serve as a primary androgen for healthy women,<ref name="pmid32629108">{{cite journal|last1=Barnard|first1=Lise|last2=Nikolaou|first2=Nikolaos|last3=Louw|first3=Carla|last4=Schiffer|first4=Lina|last5=Gibson|first5=Hylton|last6=Gilligan|first6=Lorna C.|last7=Gangitano|first7=Elena|last8=Snoep|first8=Jacky|last9=Arlt|first9=Wiebke|year=2020|title=The A-ring reduction of 11-ketotestosterone is efficiently catalysed by AKR1D1 and SRD5A2 but not SRD5A1|url=|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=202|pages=105724|doi=10.1016/j.jsbmb.2020.105724|pmid=32629108|s2cid=220323715|last10=Tomlinson|first10=Jeremy W.|last11=Storbeck|first11=Karl-Heinz}}</ref><ref name="pmid30753518" /> as it circulates at similar levels to T, but unlike T, the levels of 11KT are stable across the menstrual cycle.<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref> There are conflicting reports on whether 11-oxygenated androgens decline in women with age, and whether the relative contribution of 11KT as compared with T is higher in postmenopausal women than in younger ones — Nanba et al. (2019)<ref name="pmid30753518" /> and Davio et al. (2020)<ref name="pmid32498089">{{cite journal|last1=Davio|first1=Angela|last2=Woolcock|first2=Helen|last3=Nanba|first3=Aya T.|last4=Rege|first4=Juilee|last5=o'Day|first5=Patrick|last6=Ren|first6=Jianwei|last7=Zhao|first7=Lili|last8=Ebina|first8=Hiroki|last9=Auchus|first9=Richard|year=2020|title=Sex Differences in 11-Oxygenated Androgen Patterns Across Adulthood|journal=J Clin Endocrinol Metab|volume=105|issue=8|pages=e2921–e2929|doi=10.1210/clinem/dgaa343|pmc=7340191|pmid=32498089|last10=Rainey|first10=William E.|last11=Turcu|first11=Adina F.}}</ref> found that 11KT do not decline with age in women, however, Skiba et al. (2019)<ref name="pmid31390028" /> reported that the levels do decline. The decline of circulating 11-androgens with age may be associated with declining levels of DHEA and A4, which serve as precursors. It should also be noted that about half of circulating A4 quantities and almost all DHEA quantities are of adrenal origin.<ref name="pmid25428847">{{cite journal |vauthors=Turcu A, Smith JM, Auchus R, Rainey WE |title=Adrenal androgens and androgen precursors-definition, synthesis, regulation and physiologic actions |journal=Compr Physiol |volume=4 |issue=4 |pages=1369–81 |date=October 2014 |pmid=25428847 |pmc=4437668 |doi=10.1002/cphy.c140006 |url=}}</ref> Therefore the decline of 11-oxygenated androgens - steroids of adrenal origin - may be in line with a decline with their adrenal androgen precursors - DHEA and A4. Studies to date also strongly suggest that 11KT is the primary active 11-oxygenated androgen. Notably, 11KDHT has been found to circulate at substantially lower levels than DHT at least in prostate cancer<ref name="pmid30472582"/> suggesting that the 5α-reduction of 11KT is not as physiologically relevant in that condition. While early studies did show that 11KT and other 11-oxygenated androgens are substrates for SRD5A1 and SRD5A2, these did not perform full kinetic analyses.<ref name="pmid23856005"/>. Barnard et al., in a study published in 2020 showed that SRD5A1 does not efficiently catalyze the 5α-reduction of 11KT or 11KA4,<ref name="pmid32629108"/> confirming that 11KT may be the more relevant active 11-oxygenated androgen given the abundant peripheral expression of SRD5A2. While this does not rule out the potential for 11KDHT to be produced by SRD5A2 (or to a lesser degree by SRD5A1) in specific tissues, current evidence does suggest that the emphasis should be taken off the production of 11KDHT and rather placed on 11KT. As for the measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal |title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report |journal=Int J Mol Sci |year=2020 |volume=21 |issue=13 |pmid=32610579 |pmc=7369945 |doi=10.3390/ijms21134622 |doi-access=free |last1=Sumińska |first1=Marta |last2=Bogusz-Górna |first2=Klaudia |last3=Wegner |first3=Dominika |last4=Fichna |first4=Marta |page=4622 }}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal | last1=Yildiz | first1=Bulent O. | title=Diagnosis of hyperandrogenism: clinical criteria | journal=Best Practice & Research. Clinical Endocrinology & Metabolism | publisher=Elsevier BV | volume=20 | issue=2 | year=2006 | issn=1521-690X | pmid=16772149 | doi=10.1016/j.beem.2006.02.004 | pages=167–176}}</ref><ref name="pmid24184282">{{cite journal | last1=Peigné | first1=Maëliss | last2=Villers-Capelle | first2=Anne | last3=Robin | first3=Geoffroy | last4=Dewailly | first4=Didier | title=Hyperandrogénie féminine | journal=Presse Medicale (Paris, France) | publisher=Elsevier BV | volume=42 | issue=11 | year=2013 | issn=0755-4982 | pmid=24184282 | doi=10.1016/j.lpm.2013.07.016 | pages=1487–1499 | s2cid=28921380 | language=fr}}</ref> Relying on T levels alone in conditions associated with hyperandrogenism may read to diagnostic pitfalls and confusion.<ref name="pmid32610579"/> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|title=Clinical significance of 11-oxygenated androgens |journal=Curr Opin Endocrinol Diabetes Obes |volume=24 |issue=3 |pages=252–259 |pmid=28234803 |pmc=5819755 |doi=10.1097/MED.0000000000000334 |last1=Turcu |first1=Adina F. |last2=Auchus |first2=Richard J. |year=2017 }}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome |journal=Reprod Biomed Online |volume=41 |issue=4 |pages=734–742 |pmid=32912651 |doi=10.1016/j.rbmo.2020.07.013 |s2cid=221625488 |last1=Yang |first1=Yabo |last2=Ouyang |first2=Nengyong |last3=Ye |first3=Yang |last4=Hu |first4=Qin |last5=Du |first5=Tao |last6=Di |first6=Na |last7=Xu |first7=Wenming |last8=Azziz |first8=Ricardo |last9=Yang |first9=Dongzi |last10=Zhao |first10=Xiaomiao |year=2020 }}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803"/> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by very potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal | last1=Balsamo | first1=Antonio | last2=Baronio | first2=Federico | last3=Ortolano | first3=Rita | last4=Menabo | first4=Soara | last5=Baldazzi | first5=Lilia | last6=Di Natale | first6=Valeria | last7=Vissani | first7=Sofia | last8=Cassio | first8=Alessandra | title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant | journal=Frontiers in Pediatrics | year=2020 | publisher=Frontiers Media SA | volume=8 | page=593315 | issn=2296-2360 | pmid=33415088 | pmc=7783414 | doi=10.3389/fped.2020.593315| doi-access=free }}</ref><ref name="pmid29277706">{{cite journal | last1=Kamrath | first1=Clemens | last2=Wettstaedt | first2=Lisa | last3=Boettcher | first3=Claudia | last4=Hartmann | first4=Michaela F. | last5=Wudy | first5=Stefan A. | title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia | journal=The Journal of Steroid Biochemistry and Molecular Biology | publisher=Elsevier BV | volume=178 | year=2018 | issn=0960-0760 | pmid=29277706 | doi=10.1016/j.jsbmb.2017.12.016 | pages=221–228| s2cid=3709499 }}</ref> Another issue with the use of T as a biomarker of androgen excess is the low circulating levels in women and the specificity and sensitivity of the assays used.<ref name="pmid29306916">{{cite journal |title=Falsely elevated plasma testosterone concentrations in neonates: importance of LC-MS/MS measurements |journal=Clin Chem Lab Med |volume=56 |issue=6 |pages=e141–e143 |pmid=29306916 |doi=10.1515/cclm-2017-1028 |last1=Hamer |first1=Henrike M. |last2=Finken |first2=Martijn J.J. |last3=Van Herwaarden |first3=Antonius E. |last4=Du Toit |first4=Therina |last5=Swart |first5=Amanda C. |last6=Heijboer |first6=Annemieke C. |year=2018 |hdl=10019.1/106715 |s2cid=13917408 }}</ref><ref name="pmid32912651" /><ref name="pmid30753518">{{cite journal|last1=Nanba|first1=Aya T.|last2=Rege|first2=Juilee|last3=Ren|first3=Jianwei|last4=Auchus|first4=Richard J.|last5=Rainey|first5=William E.|last6=Turcu|first6=Adina F.|year=2019|title=11-Oxygenated C19 Steroids Do Not Decline With Age in Women|journal=J Clin Endocrinol Metab|volume=104|issue=7|pages=2615–2622|doi=10.1210/jc.2018-02527|pmc=6525564|pmid=30753518}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterised, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women |journal=Fertil Steril |volume=58 |issue=1 |pages=148–52 |pmid=1623996 |doi=10.1016/s0015-0282(16)55152-8 |last1=Carmina |first1=E. |last2=Stanczyk |first2=F. Z. |last3=Chang |first3=L. |last4=Miles |first4=R. A. |last5=Lobo |first5=R. A. |year=1992 }}</ref><ref name="pmid14417423">{{cite journal |title=Urinary ketosteroids and pregnanetriol in hirsutism |journal=J Clin Endocrinol Metab |volume=20 |issue= 2|pages=180–6 |pmid=14417423 |doi=10.1210/jcem-20-2-180|last1=Lipsett |first1=Mortimer B. |last2=Riter |first2=Barbara |year=1960 }}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries |journal=J Clin Endocrinol Metab |volume=66 |issue=5 |pages=946–50 |pmid=3129451 |doi=10.1210/jcem-66-5-946 |last1=Polson |first1=D. W. |last2=Reed |first2=M. J. |last3=Franks |first3=S. |last4=Scanlon |first4=M. J. |last5=James |first5=V. H. T. |year=1988 }}</ref> However, due to to conflicting reports ratios did not find a firm footing in the clinical as a diagnostic tool. === On The Aromatization of Androgens === Unlike T and A4, 11-oxygenated androgens are unlikely to be converted by aromatase into estrogens ''in vivo'',<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> that was first predicted in 2016 by Imamichi at al. in an ''in vitro'' study.<ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women when not taking into account DHEA. However, DHEA has a very low affinity for the androgen receptor and thus should not be an important contributor, if at all, for receptor activation under normal conditions.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, Barnard et al., incubating ''in vitro'' three different aromatase-expressing cell cultures and ''ex vivo'' human placenta explant cultures with normal and radiolabeled steroids, detected conversion of 11-oxygenated and conventional androgens into 11-oxygenated estrogens; however, 11-oxygenated estrogens were not detected ''in vivo'': neither in pregnant women who have high aromatase expression nor in patients who have high 11-androgens levels due to CAH or adrenocortical carcinoma, probably due to relatively low aromatase activity towards 11-oxygenated androgens compared to conventional androgens.<ref name="pmid33340399"/> However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Therefore, the role of DHT and 11-oxygenated androgens should be seriously considered in women patients. === Disorders of Sex Development === Since both the canonical and backdoor pathways of androgen biosynthesis towards DHT lead to early male sexual differentiation<ref name="pmid30763313" /><ref name="pmid30943210">{{cite journal|title = The "backdoor pathway" of androgen synthesis in human male sexual development | journal = PLOS Biology | volume = 17 | issue = 4 | pages = e3000198 | pmid = 30943210 | pmc = 6464227 | doi = 10.1371/journal.pbio.3000198 | last1 = Miller | first1 = Walter L. | last2 = Auchus | first2 = Richard J. |year = 2019 }}</ref><ref name="pmid11035809" /><ref name="pmid15249131" /> and are required for normal human male genital development,<ref name="pmid30943210" /><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of male fetuses,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 acts as a substrate during the biosynthesis of DHT in the backdoor pathway.<ref name="pmid30763313"/> Flück et al. described in 2011 a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. These findings confirmed that DHT produced in a backdoor pathway, while not necessary for the sexual development of females, is important for that of males. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were enough to disrupt it.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). As such, this syndrome leads to DSD in both sexes, while affected girls go usually unrecognized until puberty, when they show amenorrhea. This syndrome is also rare with only a few cases reported.<ref name="pmid34711511"/> Besides that, 11-oxygenated androgens may play previously overlooked role in DSD.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> === Congenital Adrenal Hyperplasia === Another cause of androgen excess is congenital adrenal hyperplasia (CAH), a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal |vauthors=El-Maouche D, Arlt W, Merke DP |title=Congenital adrenal hyperplasia |journal=Lancet |volume=390 |issue=10108 |pages=2194–2210 |date=November 2017 |pmid=28576284 |doi=10.1016/S0140-6736(17)31431-9 |url=}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal |vauthors=Speiser PW, White PC |title=Congenital adrenal hyperplasia |journal=N Engl J Med |volume=349 |issue=8 |pages=776–88 |date=August 2003 |pmid=12930931 |doi=10.1056/NEJMra021561 |url=}}</ref><ref name="pmid30272171">{{cite journal | title = Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline | journal = The Journal of Clinical Endocrinology and Metabolism | volume = 103 | issue = 11 | pages = 4043–4088 | year = 2018 | pmid = 30272171 | pmc = 6456929 | doi = 10.1210/jc.2018-01865 }}</ref> Such deficiency leads to an excessive accumulation of a respective cortisol precursor, that becomes to serve as a substrate to androgens. In CYP21A2 deficiency<ref name="pmid22170725" /> including the mild forms (which are not always diagnosed)<ref name="pmid32966723">{{cite journal |vauthors=Merke DP, Auchus RJ |title=Congenital Adrenal Hyperplasia Due to 21-Hydroxylase Deficiency |journal=N Engl J Med |volume=383 |issue=13 |pages=1248–1261 |date=September 2020 |pmid=32966723 |doi=10.1056/NEJMra1909786 |url=}}</ref><ref name="pmid31499506">{{cite book|title=Hyperandrogenism in Women|last1=Pignatelli|first1=Duarte|last2=Pereira|first2=Sofia S.|last3=Pasquali|first3=Renato|year=2019|isbn=978-3-318-06470-4|series=Frontiers of Hormone Research|volume=53|pages=65–76|chapter=Androgens in Congenital Adrenal Hyperplasia|doi=10.1159/000494903|pmid=31499506|s2cid=202412336}}</ref> or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> elevated 17-OHP levels starts the backdoor pathway to DHT. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> The reason why 17-OHP serves as a prerequisite substrate for DHT within the backdoor pathway roundabout of T rather then an immediate substrate within the Δ⁴ pathway for A4, and then T, is because the catalytic activity 17,20-lyase reaction (which cleaves a side-chain from the steroid nucleus converting a pregnane to an androstane (androgen), i.e., from 17OPH5 to DHEA; from 17-OHP to A4) performed by CYP17A1 in humans is approximately 100 times more efficient in the Δ⁵ pathway than in the Δ⁴ pathway. Therefore, the catalytic efficiency of CYP17A1 for 17-OHP is about 100 times lower than for 17OHP5, resulting in negligible A4 being produced from 17-OHP in the Δ⁴ reaction pathway in humans.<ref name="pmid8325965">{{cite journal|last1=Swart|first1=P.|last2=Swart|first2=A. C.|last3=Waterman|first3=M. R.|last4=Estabrook|first4=R. W.|last5=Mason|first5=J. I.|year=1993|title=Progesterone 16 alpha-hydroxylase activity is catalyzed by human cytochrome P450 17 alpha-hydroxylase|journal=J Clin Endocrinol Metab|volume=77|issue=1|pages=98–102|doi=10.1210/jcem.77.1.8325965|pmid=8325965}}</ref><ref name="pmid12915666">{{cite journal|last1=Flück|first1=Christa E.|last2=Miller|first2=Walter L.|last3=Auchus|first3=Richard J.|year=2003|title=The 17, 20-lyase activity of cytochrome CYP17A1 from human fetal testis favors the delta5 steroidogenic pathway|url=https://pubmed.ncbi.nlm.nih.gov/12915666|journal=The Journal of Clinical Endocrinology and Metabolism|volume=88|issue=8|pages=3762–3766|doi=10.1210/jc.2003-030143|issn=0021-972X|pmid=12915666}}</ref><ref name="pmid15774560">{{cite journal|last1=Miller|first1=Walter L.|year=2005|title=Minireview: regulation of steroidogenesis by electron transfer|url=https://pubmed.ncbi.nlm.nih.gov/15774560|journal=Endocrinology|volume=146|issue=6|pages=2544–2550|doi=10.1210/en.2005-0096|issn=0013-7227|pmid=15774560}}</ref><ref name="pmid32007561"/> The accumulation of 17-OHP in CYP21A2 deficiency in CAH can be attributed to the fact that the primary enzyme for 17-OHP in normal conditions is CYP21A2, that is expressed in the adrenal and not the gonads.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|year=2019 |volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086 }}</ref><ref name="pmid26038201"/> In a 1998 study, Auchus et al. demonstrated that human CYP17A1 efficiently catalyzed the conversion of P4 to 17-OHP, but the conversion of 17-OHP to A4 was much less efficient than the corresponding conversion of 17OHP5 to DHEA.<ref name="pmid9452426"/> In rodents, quite contrary, the conversion of 17-OHP to A4 is very efficient.<ref name="pmid9452426">{{cite journal | last1=Auchus | first1=Richard J. | last2=Lee | first2=Tim C. | last3=Miller | first3=Walter L. | title=Cytochrome b 5 Augments the 17,20-Lyase Activity of Human P450c17 without Direct Electron Transfer | journal=The Journal of Biological Chemistry | year=1998 | publisher=Elsevier BV | volume=273 | issue=6 | issn=0021-9258 | pmid=9452426 | doi=10.1074/jbc.273.6.3158 | pages=3158–3165| doi-access=free }}</ref> This explains significant accumulation of 17-OHP in CYP21A2 deficiency or POR deficiency in humans, so that 17-OHP, while not 21-hydroxylated in sufficient quantities, and being better a substrate for 5α-reductase than for CYP17A1, is 5α-reduced serving as the prerequisite for this backdoor pathway. Hence, fetal excess of 17-OHP in CAH may provoke activation of this pathway to DHT and lead to external genital virilization in newborn girls, thus explaining DSD in girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025"/><ref name="pmid31505456">{{cite journal |vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE |title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia |journal=Eur J Endocrinol |volume=181 |issue=5 |pages=481–488 |date=November 2019 |pmid=31505456 |doi=10.1530/EJE-19-0379 |url=}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal |vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M |title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia |journal=J Obstet Gynaecol Res |volume=43 |issue=5 |pages=946–950 |date=May 2017 |pmid=28188961 |doi=10.1111/jog.13288 |url=}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal | last1=Turcu | first1=Adina F | last2=Mallappa | first2=Ashwini | last3=Elman | first3=Meredith S | last4=Avila | first4=Nilo A | last5=Marko | first5=Jamie | last6=Rao | first6=Hamsini | last7=Tsodikov | first7=Alexander | last8=Auchus | first8=Richard J | last9=Merke | first9=Deborah P | title = 11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency | journal = The Journal of Clinical Endocrinology and Metabolism | year=2017 | volume = 102 | issue = 8 | pages = 2701–2710 | pmid = 28472487 | pmc = 5546849 | doi = 10.1210/jc.2016-3989}}</ref><ref name="pmid26865584">{{cite journal|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency |journal=Eur J Endocrinol |volume=174 |issue=5 |pages=601–9 |pmid=26865584 |pmc=4874183 |doi=10.1530/EJE-15-1181 |last1=Turcu |first1=Adina F. |last2=Nanba |first2=Aya T. |last3=Chomic |first3=Robert |last4=Upadhyay |first4=Sunil K. |last5=Giordano |first5=Thomas J. |last6=Shields |first6=James J. |last7=Merke |first7=Deborah P. |last8=Rainey |first8=William E. |last9=Auchus |first9=Richard J. |year=2016 }}</ref><ref name="pmid29718004">{{cite journal|title = Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency | journal = Current Opinion in Endocrinology, Diabetes, and Obesity | volume = 25 | issue = 3 | pages = 178–184 | pmid = 29718004 | doi = 10.1097/MED.0000000000000402 | s2cid = 26072848 |last1 = White |first1 = Perrin C. |year = 2018 }}</ref><ref name="pmid34867794"/> In males with CAH, 11-oxygenated androgens may lead to devlopment of testicular adrenal rest tumors.<ref name="pmid25850025">{{cite journal|pmc=4454804|year=2015|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|last10=Auchus|first10=R. J.|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmid=25850025}}</ref><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors |journal=J Clin Endocrinol Metab |volume=107 |issue=1 |pages=e272–e280 |pmid=34390337 |pmc=8684463 |doi=10.1210/clinem/dgab598 |last1=Schröder |first1=Mariska A M. |last2=Turcu |first2=Adina F. |last3=o'Day |first3=Patrick |last4=Van Herwaarden |first4=Antonius E. |last5=Span |first5=Paul N. |last6=Auchus |first6=Richard J. |last7=Sweep |first7=Fred C G J. |last8=Claahsen-Van Der Grinten |first8=Hedi L. |year=2022 }}</ref> The biosynthesis of 11OHP4 from P4 and 21dF from 17-OHP by CYP11B1/2 in CAH may be attributed to CYP21A2 deficiency resulting in increased P4 and 17-OHP concentrations and, together with the unavailability of CYP11B1/2's main substrates, 11-deoxycortisol (11dF) and 11-deoxycorticosterone (DOC), drive the production of 11-oxygenated pregnanes.<ref name="pmid3546944" /> We have reasons to believe that this may be aggravated by elevated ACTH due to a feedback loop in hypothalamic-pituitary-adrenal axis caused by impaired cortisol synthesis associated with CYP21A2 deficiency; higher ACTH causes higher CYP11B1 expression. Multiple studies demonstrated that in CAH due to CYP21A2 deficiency, both 21dF levels<ref name="pmid4372245">{{cite journal |title=Plasma 17-hydroxyprogesterone, 21-deoxycortisol and cortisol in congenital adrenal hyperplasia |journal=J Clin Endocrinol Metab |volume=39 |issue=6 |pages=1099–102 |year=1974 |pmid=4372245 |doi=10.1210/jcem-39-6-1099 |last1=Franks |first1=Robert C. }}</ref><ref name="pmid476971">{{cite journal |title=Rapid assay of plasma 21-deoxycortisol and 11-deoxycortisol in congenital adrenal hyperplasia |journal=Clin Endocrinol (Oxf) |volume=10 |issue=4 |pages=367–75 |year=1979 |pmid=476971 |doi=10.1111/j.1365-2265.1979.tb02091.x |url=|last1=Fukushima |first1=D. K. |last2=Nishina |first2=T. |last3=Wu |first3=R. H. K. |last4=Hellman |first4=L. |last5=Finkelstein |first5=J. W. |s2cid=2979354 }}</ref><ref name="pmid6090811">{{cite journal |title=Development of plasma 21-deoxycortisol radioimmunoassay and application to the diagnosis of patients with 21-hydroxylase deficiency |journal=J Steroid Biochem |volume=21 |issue=2 |pages=185–91 |year=1984 |pmid=6090811 |doi=10.1016/0022-4731(84)90382-0 |last1=Milewicz |first1=A. |last2=Vecsei |first2=P. |last3=Korth-Schütz |first3=S. |last4=Haack |first4=D. |last5=Rösler |first5=A. |last6=Lichtwald |first6=K. |last7=Lewicka |first7=S. |last8=Mittelstaedt |first8=G.v. }}</ref><ref name="pmid2986404">{{cite journal |title=Radioimmunoassay for 21-deoxycortisol: clinical applications |journal=Acta Endocrinol (Copenh) |volume=108 |issue=4 |pages=537–44 |year=1985 |pmid=2986404 |doi=10.1530/acta.0.1080537 |last1=Gueux |first1=B. |last2=Fiet |first2=J. |last3=Pham-Huu-Trung |first3=M. T. |last4=Villette |first4=J. M. |last5=Gourmelen |first5=M. |last6=Galons |first6=H. |last7=Brerault |first7=J. L. |last8=Vexiau |first8=P. |last9=Julien |first9=R. }}</ref><ref name="pmid25850025" /> and 11OPH4 levels<ref name="pmid3546944">{{cite journal |last1=Gueux |first1=Bernard |last2=Fiet |first2=Jean |last3=Galons |first3=Hervé |last4=Boneté |first4=Rémi |last5=Villette |first5=Jean-Marie |last6=Vexiau |first6=Patrick |last7=Pham-Huu-Trung |first7=Marie-Thérèse |last8=Raux-Eurin |first8=Marie-Charles |last9=Gourmelen |first9=Micheline |last10=Brérault |first10=Jean-Louis |last11=Julien |first11=René |last12=Dreux |first12=Claude |title=The measurement of 11β-hydroxy-4-pregnene-3,20-dione (21-Deoxycorticosterone) by radioimmunoassay in human plasma |journal=Journal of Steroid Biochemistry |year=1987 |volume=26 |issue=1 |pages=145–150 |doi=10.1016/0022-4731(87)90043-4 |pmid=3546944 }}</ref><ref name="pmid2537337">{{cite journal |last1=Fiet |first1=Jean |last2=Gueux |first2=Bernard |last3=Rauxdemay |first3=Marie-Charles |last4=Kuttenn |first4=Frederique |last5=Vexiau |first5=Patrick |last6=Brerault |first6=Jeanlouis |last7=Couillin |first7=Philippe |last8=Galons |first8=Herve |last9=Villette |first9=Jeanmarie |last10=Julien |first10=Rene |last11=Dreux |first11=Claude |title=Increased Plasma 21-Deoxycorticosterone (21-DB) Levels in Late-Onset Adrenal 21-Hydroxylase Deficiency Suggest a Mild Defect of the Mineralocorticoid Pathway |journal=The Journal of Clinical Endocrinology & Metabolism |year=1989 |volume=68 |issue=3 |pages=542–547 |doi=10.1210/jcem-68-3-542 |pmid=2537337 }}</ref><ref name="pmid29264476">{{cite journal |last1=Fiet |first1=Jean |last2=Le Bouc |first2=Yves |last3=Guéchot |first3=Jérôme |last4=Hélin |first4=Nicolas |last5=Maubert |first5=Marie-Anne |last6=Farabos |first6=Dominique |last7=Lamazière |first7=Antonin |title=A Liquid Chromatography/Tandem Mass Spectometry Profile of 16 Serum Steroids, Including 21-Deoxycortisol and 21-Deoxycorticosterone, for Management of Congenital Adrenal Hyperplasia |journal=Journal of the Endocrine Society |year=2017 |volume=1 |issue=3 |pages=186–201 |doi=10.1210/js.2016-1048 |pmid=29264476 |pmc=5686660 }}</ref><ref name="pmid31821037">{{cite journal |title=Interaction between accumulated 21-deoxysteroids and mineralocorticoid signaling in 21-hydroxylase deficiency |journal=Am J Physiol Endocrinol Metab |volume=318 |issue=2 |pages=E102–E110 |year=2020 |pmid=31821037 |doi=10.1152/ajpendo.00368.2019 |last1=Travers |first1=Simon |last2=Bouvattier |first2=Claire |last3=Fagart |first3=Jérôme |last4=Martinerie |first4=Laetitia |last5=Viengchareun |first5=Say |last6=Pussard |first6=Eric |last7=Lombès |first7=Marc |s2cid=209314028 }}</ref> are increased. It was Robert Franks in who first published a study, in 1974, that compared 21dF levels of CAH patients with those of healthy controls. He measured 21dF plasma levels in twelve CAH patients before treatment, three after treatment, and four healthy controls following ACTH administration. Mean values of 21dF in CAH patients was 88 ng/ml while in healthy controls it was not detected. In untreated patients, values decreased after therapy. Even that, there were earlier reports about unique cases where 21dF was detected in CAH patients, but without direct comparison to healthy controls.<ref name="pmid5845501">{{cite journal |title=Detection of 21-deoxycortisol in blood from a patient with congenital adrenal hyperplasia |journal=Metabolism |year=1965 |volume=14 |issue=12 |pages=1276–81 |pmid=5845501 |doi=10.1016/s0026-0495(65)80008-7|last1=Wieland |first1=Ralph G. |last2=Maynard |first2=Donald E. |last3=Riley |first3=Thomas R. |last4=Hamwi |first4=George J. }}</ref><ref name="pmid13271547">{{cite journal|title=17alpha-hydroxyprogesterone and 21-desoxyhydrocortisone; their metabolism and possible role in congenital adrenal virilism |journal=J Clin Invest |volume=34 |issue=11 |pages=1639–46 |year=1955 |pmid=13271547 |pmc=438744 |doi=10.1172/JCI103217|last1=Jailer |first1=Joseph W. |last2=Gold |first2=Jay J. |last3=Vande Wiele |first3=Raymond |last4=Lieberman |first4=Seymour }}</ref> As for 11OHP4, it were Gueux et al. who first demonstrated, in 1987, elevated plasma levels of 11OHP4 in CAH. In that study, in treated classical CAH patients, some of which had salt-wasting form, mean levels of 11OHP4 (5908.7 pmol/l) were 332 times higher than in healthy controls (17.8 pmol/l). There was no difference in 11OHP4 in healthy controls depending on sex or phase of a menstrual cycle; ACTH stimulation in those control increased 11OHP4 four- to six-fold, while dexamethasone 1 mg at midnight decreased 11OHP4 to almost undetectable levels 12 hours later. Therefore, the authors hypothesized that at least in healthy people 11OHP4 is biosythesized exclusively in the adrenal, while gonads are not involved.<ref name="pmid3546944" /> Nevertheless, in studies focusing on CAH caused by CYP21A2 deficiency, 11OHP4 received less attention than 21dF.<ref name="pmid29277707"/> However, it was not until 2017 when 11OHP4 or 21dF were viewed as potential substrates in pathways towards potent 11-ogygenated androgens in ''in vitro'' studies.<ref name="pmid32007561"/><ref name="pmid29277707"/> In a 2016 study, Turcu et al. showed that in classic CAH due to CYP21A2 deficiency, in male and female patients who received glucocorticoid therapy, both conventional and 11-oxygenated androgens were elevated 3-4 fold compared to healthy controls. The exceptions were dehydroepiandrosterone (DHEA), dehydroepiandrosterone sulfate (DHEA-S), and androstenediol sulfate (A5-S), whose levels were 6.0, 7.5, and 9.4 times lower, respectively, in the patients with the condition compared to healthy controls, due to suppression by glucocorticoid treatment. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in a backdoor pathway from 17-OHP or P4 as consequence of abnormally upregulated SRD5A1.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Besides that, the levels of 11OHA4 and 11KA4 correlated with the levels of markers of insulin resistance; therefore, the study suggests that androgen excess precedes androgen-driven insulin resistance in PCOS.<ref name="pmid27901631" /> While earlier studies had commonly only measured 11OHA4 or 11OHAST and 11β-hydroxyetiocholanolone (11OHEt), urinary metabolites of 11OHA,<ref name="pmid33539964" /> while 11OHEt is also a metabolite of cortisol,<ref name="pmid31362062">{{cite journal |title=Human steroid biosynthesis, metabolism and excretion are differentially reflected by serum and urine steroid metabolomes: A comprehensive review |journal=J Steroid Biochem Mol Biol |volume=194 |issue= |pages=105439 |year=2019 |pmid=31362062 |pmc=6857441 |doi=10.1016/j.jsbmb.2019.105439 |url=|last1=Schiffer |first1=Lina |last2=Barnard |first2=Lise |last3=Baranowski |first3=Elizabeth S. |last4=Gilligan |first4=Lorna C. |last5=Taylor |first5=Angela E. |last6=Arlt |first6=Wiebke |last7=Shackleton |first7=Cedric H.L. |last8=Storbeck |first8=Karl-Heinz }}</ref><ref name="pmid27845856">{{cite journal |title=Modified-Release and Conventional Glucocorticoids and Diurnal Androgen Excretion in Congenital Adrenal Hyperplasia |journal=J Clin Endocrinol Metab |volume=102 |issue=6 |pages=1797–1806 |year=2017 |pmid=27845856 |pmc=5470768 |doi=10.1210/jc.2016-2855|last1=Jones |first1=Christopher M. |last2=Mallappa |first2=Ashwini |last3=Reisch |first3=Nicole |last4=Nikolaou |first4=Nikolaos |last5=Krone |first5=Nils |last6=Hughes |first6=Beverly A. |last7=o'Neil |first7=Donna M. |last8=Whitaker |first8=Martin J. |last9=Tomlinson |first9=Jeremy W. |last10=Storbeck |first10=Karl-Heinz |last11=Merke |first11=Deborah P. |last12=Ross |first12=Richard J. |last13=Arlt |first13=Wiebke }}</ref> more recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Another study reported that 11OHT was the only significantly elevated 11-oxygeated androgen in PCOS and together with 11KT, correlated with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227"/> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Premature Adrenarche === In a 2018 study, Rege et al. demonstrated that levels of 11KT in girls aged between 4 and 7 years during normal adrenarche (healthy controls) exceeded those of T by 2.43 times, and in those with premature adrenarche by 3.48 times. However, the levels of T in girls with premature adrenarche were higher by just 13% compared to age-matched healthy controls.<ref name="pmid30137510">{{cite journal | last1=Rege | first1=Juilee | last2=Turcu | first2=Adina | last3=Kasa-Vubu | first3=Josephine Z | last4=Lerario | first4=Antonio M | last5=Auchus | first5=Gabriela C | last6=Auchus | first6=Richard J | last7=Smith | first7=Joshua M | last8=White | first8=Perrin C | last9=Rainey | first9=William E | title=11KT is the dominant circulating bioactive androgen during normal and premature adrenarche | journal=The Journal of Clinical Endocrinology and Metabolism | year=2018 | publisher=The Endocrine Society | volume=103 | issue=12 | pages=4589–4598 | issn=0021-972X | pmid=30137510 | pmc=6226603 | doi=10.1210/jc.2018-00736 }}</ref> === Prostate Cancer === In some cases of advanced prostate cancer, androgen deprivation therapy related to gonadal T depletion does not produce long-term effects, and metastatic tumors may develop into castration-resistant prostate cancer (CRPC). The development of CRPC depends on adrenal precursor steroids to produce DHT in the tumor in a metabolic pathway called the "5α-dione" pathway - the pathway in which T is not involved. SRD5A1, the expression of which increases in CRPC, reduces A4 to 5α-dione, which is then converted to DHT.<ref name="pmid21795608"/><ref name="pmid31900912"/> Therefore, the DHT produced within the "5α-dione" pathway hampers the androgen deprivation therapy. Although blood levels of T are decreased by 90-95% in men whose testicles have been removed, DHT in the prostate is only decreased by 50%, thus indicating the presence of a metabolic pathway in the prostate that does not require testicular T to produce DHT.<ref name="pmid18471780">{{cite journal | last1=Luu-The | first1=Van | last2=Bélanger | first2=Alain | last3=Labrie | first3=Fernand | title=Androgen biosynthetic pathways in the human prostate | journal=Best Practice & Research. Clinical Endocrinology & Metabolism | publisher=Elsevier BV | volume=22 | issue=2 | year=2008 | issn=1521-690X | pmid=18471780 | doi=10.1016/j.beem.2008.01.008 | pages=207–221}}</ref> Chang et al., incubating six established human prostate cancer cell lines from patients with CRPC in the presence of radiolabeled A4, showed in their experiment published in 2011<ref name="pmid21795608">{{cite journal|last1=Chang | first1=K.-H. | last2=Li | first2=R. | last3=Papari-Zareei | first3=M. | last4=Watumull | first4=L. | last5=Zhao | first5=Y. D. | last6=Auchus | first6=R. J. | last7=Sharifi | first7=N. | title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer | journal=Proceedings of the National Academy of Sciences of the United States of America |year=2011 | publisher=Proceedings of the National Academy of Sciences | volume=108 | issue=33 | issn=0027-8424 | pmid=21795608 | pmc=3158152 | doi=10.1073/pnas.1107898108 | pages=13728–13733|bibcode=2011PNAS..10813728C |doi-access=free }}</ref> the presence of this pathway to DHT which bypasses T and they called this the "alternative" pathway, that became later commonly called as the "5α-dione" pathway.<ref name="pmid23856005"/> The authors demonstrated that this was the dominant pathway in prostate cancer (over the direct conversion of A4 to T) with SRD5A1 (which is upregulated in prostate cancer) first converting A4 to androstanedione (5α-dione), also known as 5α-androstane-3,17-dione, and then HSD17B3 / AKR1C3 converting 5α-dione to DHT (not necessarily via AST and 3α-diol). The study also found that the SRD5A2 is not involved in this "alternative" pathway.<ref name="pmid31900912"/> Therefore, the study showed the importance of taking into consideration this "alternative" pathway in selecting drugs that inhibit 5α-reductase activity.<ref name="pmid21901017">{{cite journal |title=Prostate cancer: DHT bypasses testosterone to drive progression to castration resistance |journal=Nat Rev Urol |volume=8 |issue=9 |pages=470 |year=September 2011 |pmid=21901017 |doi=10.1038/nrurol.2011.122 }}</ref><ref name="pmid22064602" /><ref name="pmid22336886">{{cite journal |title=Dihydrotestosterone synthesis from adrenal precursors does not involve testosterone in castration-resistant prostate cancer |journal=Cancer Biol Ther |volume=13 |issue=5 |pages=237–8 |year=2012 |pmid=22336886 |doi=10.4161/cbt.19608}}</ref> Another pathway that may be activated in CRPC, which may also hamper the androgen deprivation therapy, is the backdoor pathway from P4 to DHT. Chen et al. in a study published in 2014<ref name="pmid25320358">{{cite journal |vauthors=Chen EJ, Sowalsky AG, Gao S, Cai C, Voznesensky O, Schaefer R, Loda M, True LD, Ye H, Troncoso P, Lis RL, Kantoff PW, Montgomery RB, Nelson PS, Bubley GJ, Balk SP, Taplin ME |title=Abiraterone treatment in castration-resistant prostate cancer selects for progesterone responsive mutant androgen receptors |journal=Clin Cancer Res |volume=21 |issue=6 |pages=1273–80 |date=March 2015 |pmid=25320358 |pmc=4359958 |doi=10.1158/1078-0432.CCR-14-1220 |url=}}</ref> predicted that abiraterone, a CYP17A1 inhibitor, with about 6-fold more selective for inhibition of 17α-hydroxylase over 17,20-lyase,<ref name="pmid28890368">{{cite journal |vauthors=de Mello Martins AGG, Allegretta G, Unteregger G, Haupenthal J, Eberhard J, Hoffmann M, van der Zee JA, Junker K, Stöckle M, Müller R, Hartmann RW, Ohlmann CH |title=CYP17A1-independent production of the neurosteroid-derived 5α-pregnan-3β,6α-diol-20-one in androgen-responsive prostate cancer cell lines under serum starvation and inhibition by Abiraterone |journal=J Steroid Biochem Mol Biol |volume=174 |issue= |pages=183–191 |date=November 2017 |pmid=28890368 |doi=10.1016/j.jsbmb.2017.09.006 |url=}}</ref><ref name="pmid28373265">{{cite journal |vauthors=Petrunak EM, Rogers SA, Aubé J, Scott EE |title=Structural and Functional Evaluation of Clinically Relevant Inhibitors of Steroidogenic Cytochrome P450 17A1 |journal=Drug Metab Dispos |volume=45 |issue=6 |pages=635–645 |date=June 2017 |pmid=28373265 |pmc=5438109 |doi=10.1124/dmd.117.075317 |url=}}</ref><ref name="pmid29710837">{{cite journal |vauthors=Fernández-Cancio M, Camats N, Flück CE, Zalewski A, Dick B, Frey BM, Monné R, Torán N, Audí L, Pandey AV |title=Mechanism of the Dual Activities of Human CYP17A1 and Binding to Anti-Prostate Cancer Drug Abiraterone Revealed by a Novel V366M Mutation Causing 17,20 Lyase Deficiency |journal=Pharmaceuticals (Basel) |volume=11 |issue=2 |pages= |date=April 2018 |pmid=29710837 |pmc=6027421 |doi=10.3390/ph11020037 |url=}}</ref> although disrupting canonical androgen biosynthesis, while lowering levels of T, causes elevation of P4, that can be 5α-reduced hence start a backdoor pathway from P4 to DHT with roundabout of T.<ref name="pmid25320358"/> Besides that, in CRPC, 11-oxygenated androgens contribute significantly to the androgen pool.<ref name="pmid23856005"/><ref name="pmid31900912"/> 11-oxygenated androgens play a previously overlooked role in the reactivation of androgen signaling in CRPC,<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> because after eliminating testicular T biosynthesis by chemical or physical castration, CRPC has been shown to develop the ability to convert inactive circulating adrenal androgen precursors, DHEA and A4, to potent 11-oxygenated androgens in the 11-oxygenated pathway in addition to the 5α-dione pathway.<ref name="pmid31672619">{{cite journal |title=The role of adrenal derived androgens in castration resistant prostate cancer |journal=J Steroid Biochem Mol Biol |volume=197 |issue= |pages=105506 |year=2020 |pmid=31672619 |doi=10.1016/j.jsbmb.2019.105506|last1=Barnard |first1=Monique |last2=Mostaghel |first2=Elahe A. |last3=Auchus |first3=Richard J. |last4=Storbeck |first4=Karl-Heinz |pmc=7883395 }}</ref><ref name="pmid33974560" /> In a 2021 study, Snaterse et al. demonstrated that 11KT is the most circulating active androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool. They also demonstrated that 11KT levels are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> In a 2018 study by du Toit et al., the full range of androgen pathway metabolites have been shown in normal prostate and various prostate cancer cell models. 11OHA4 and 11OHT were both converted to potent androgens, 11KT and 11KDHT. Compared to T and DHT, 11-oxygenated androgens were the most predominant androgens. High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant. Furthermore, glucuronidation of the 11-oxygenated androgens is hampered by the presence of an oxo- or a hydroxy- group at position 11 of androgens in prostate cancer cell models while in prostate cancer patients' plasma 11KDHT was present only in the unconjugated form, with 11KT also predominantly unconjugated.<ref name="pmid28939401"/> Of all 11-oxygenated androgens in prostate cancer, it may be that 11KT is the primary active 11-oxygenated androgen, rather than 11KDHT, as observed in experiments on prostate tissue homogenates by Häkkinen et al. published in 2018<ref name="pmid30472582">{{cite journal|last1=Häkkinen|first1=Merja R.|last2=Murtola|first2=Teemu|last3=Voutilainen|first3=Raimo|last4=Poutanen|first4=Matti|last5=Linnanen|first5=Tero|last6=Koskivuori|first6=Johanna|last7=Lakka|first7=Timo|last8=Jääskeläinen|first8=Jarmo|last9=Auriola|first9=Seppo|year=2019|title=Simultaneous analysis by LC-MS/MS of 22 ketosteroids with hydroxylamine derivatization and underivatized estradiol from human plasma, serum and prostate tissue|journal=J Pharm Biomed Anal|volume=164|issue=|pages=642–652|doi=10.1016/j.jpba.2018.11.035|pmid=30472582|s2cid=53729550}}</ref> Ventura-Bahena et al., in a 2021 study, describing results of epidemiological studies related to androgens and prostate cancer that focused on specific androgen concentrations (such as T, A4, and DHEA) as inconsistent, hypothesized that the differences in androgen biosynthetic pathways rather than differences in specific androgen levels are associated with prostate cancer development.<ref name="pmid34520388"/> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but also in clinical conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> The contribution of the 11-oxygenated androgens, as well as the biosynthesis of 11-oxygenated pregnanes to active androgens via a backdoor pathway, have also been demonstrated in BPH cell models showing the conversion of 11OHP4 and 11KP4 in the backdoor pathway resulting in the production of 11KDHT. Backdoor pathway intermediates were also detected in BPH tissue as well as in circulation in BPH patients.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> In a paper published in 2008, Dimitrakov et al. hypothesized that CP/CPPS may be associated with a mild CYP21A2 deficiency, a cause of non-classic CAH that leads to androgen excesses.<ref name="pmid18308097"/> Non-classic CAH was generally thought to be asymptomatic in men.<ref name="pmid28582566">{{cite journal |title=Non-classic congenital adrenal hyperplasia due to 21-hydroxylase deficiency revisited: an update with a special focus on adolescent and adult women |journal=Hum Reprod Update |volume=23 |issue=5 |pages=580–599 |year=2017 |pmid=28582566 |doi=10.1093/humupd/dmx014 |last1=Carmina |first1=Enrico |last2=Dewailly |first2=Didier |last3=Escobar-Morreale |first3=Héctor F. |last4=Kelestimur |first4=Fahrettin |last5=Moran |first5=Carlos |last6=Oberfield |first6=Sharon |last7=Witchel |first7=Selma F. |last8=Azziz |first8=Ricardo }}</ref><ref name="pmid20671993">{{cite journal |title=Nonclassic congenital adrenal hyperplasia |journal=Int J Pediatr Endocrinol |volume=2010 |pages=625105 |year=2010 |pmid=20671993 |pmc=2910408 |doi=10.1155/2010/625105|doi-access=free |last1=Witchel |first1=Selma Feldman |last2=Azziz |first2=Ricardo }}</ref> The authors of that 2008 paper, therefore, concluded that CP/CPPS may be a consequence of a systemic condition of androgen excess but not a disease that originates in the prostate such as a localized prostate infection, inflammation, or dysfunction. We hypothesize that CYP21A2 deficiency in CP/CPPS may be associated with elevated androgens produced by pathways activated by such deficiency, i.e. backdoor pathway from P4 or 17-OHP to DHT and the pathways towards 11-oxygenated androgens. ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} k12uyw90q9ldb2al4x7n9n1x4amhluy 2410906 2410905 2022-08-02T05:30:35Z Maneesh 2723004 /* 11-Oxygenated Androgen Pathways */ reader has to understand that only C19 is relevant for healthy humans, C21 can be discussed in next section where relevant wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> and the potential clinical relevance in conditions involving androgen biosynthesis was proposed. Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref><ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid11514561">{{cite journal|last1=Nahoum|first1=Virginie|last2=Gangloff|first2=Anne|last3=Legrand|first3=Pierre|last4=Zhu|first4=Dao-Wei|last5=Cantin|first5=Line|last6=Zhorov|first6=Boris S.|last7=Luu-The|first7=Van|last8=Labrie|first8=Fernand|last9=Breton|first9=Rock|year=2001|title=Structure of the human 3alpha-hydroxysteroid dehydrogenase type 3 in complex with testosterone and NADP at 1.25-A resolution|journal=J Biol Chem|volume=276|issue=45|pages=42091–8|doi=10.1074/jbc.M105610200|pmid=11514561|doi-access=free|last10=Lin|first10=Sheng-Xiang}}</ref><ref name="pmid18923939">{{cite journal|last1=Dozmorov|first1=Mikhail G.|last2=Yang|first2=Qing|last3=Matwalli|first3=Adam|last4=Hurst|first4=Robert E.|last5=Culkin|first5=Daniel J.|last6=Kropp|first6=Bradley P.|last7=Lin|first7=Hsueh-Kung|year=2007|title=5alpha-androstane-3alpha,17beta-diol selectively activates the canonical PI3K/AKT pathway: a bioinformatics-based evidence for androgen-activated cytoplasmic signaling|journal=Genomic Med|volume=1|issue=3–4|pages=139–46|doi=10.1007/s11568-008-9018-9|pmc=2269037|pmid=18923939}}</ref><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref><ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}}The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}}The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}}The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608" />{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}}While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780" /> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. 11-Oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid30959151">{{cite journal |title=Circulating 11-oxygenated androgens across species |journal=J Steroid Biochem Mol Biol |volume=190 |pages=242–249 |year=2019 |pmid=30959151 |pmc=6733521 |doi=10.1016/j.jsbmb.2019.04.005|last1=Rege |first1=Juilee |last2=Garber |first2=Scott |last3=Conley |first3=Alan J. |last4=Elsey |first4=Ruth M. |last5=Turcu |first5=Adina F. |last6=Auchus |first6=Richard J. |last7=Rainey |first7=William E. }}</ref><ref name="pmid27519632" /><ref name="pmid34171490" /><ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646" /> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="NBK557634">{{cite book|url=https://www.ncbi.nlm.nih.gov/books/NBK557634/|title=Biochemistry, Dihydrotestosterone|publisher=StatPearls|year=2022}}</ref><ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |pages=431 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|pages=868|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|pages=66|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=441 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref><ref name="pmid23386646" /> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid29277706" /><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405" /> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-), therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the conversion of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1.<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRDA51 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid18923939"/><ref name="Nishiyama2011"/><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels do not decline with age as T does. While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroids Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroids Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === Biological Role of 11-Oxygenated Androgens === 11-oxygenated androgens are produced in physiological quantities in healthy primate organisms (including humans).<ref name="pmid30959151" /><ref name="pmid30753518" /><ref name="pmid32629108" /> Humans have two isozymes with 11β-hydroxylase activity, encoded by the genes ''CYP11B1'' (regulated by the adrenocorticotropic hormone (ACTH)) and ''CYP11B2'' (regulated by angiotensin II).<ref name="pmid22217826">{{cite journal|name-list-style=vanc|title=Molecular biology of 11β-hydroxylase and 11β-hydroxysteroid dehydrogenase enzymes |journal=J Steroid Biochem Mol Biol |volume=43 |issue=8 |pages=827–35 |pmid=22217826 |doi=10.1016/0960-0760(92)90309-7 |s2cid=19379671 |last1=White |first1=Perrin C. |last2=Pascoe |first2=Leigh |last3=Curnow |first3=Kathleen M. |last4=Tannin |first4=Grace |last5=Rösler |first5=Ariel |year=1992 }}</ref> Since the first step in the biosynthesis of 11-oxygenated androgens involves 11β-hydroxylation of a steroid substrate by CYP11B1/2 isozymes that are generally associated with their expression in the adrenal gland, 11-oxygenated androgens are considered androgens of adrenal origin. They follow the circadian rhythm of cortisol but correlate very weakly with T.<ref name="pmid34867794">{{cite journal |title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=751191 |pmid=34867794 |pmc=8636728 |doi=10.3389/fendo.2021.751191 |doi-access=free |last1=Turcu |first1=Adina F. |last2=Mallappa |first2=Ashwini |last3=Nella |first3=Aikaterini A. |last4=Chen |first4=Xuan |last5=Zhao |first5=Lili |last6=Nanba |first6=Aya T. |last7=Byrd |first7=James Brian |last8=Auchus |first8=Richard J. |last9=Merke |first9=Deborah P. |year=2021 }}</ref><ref name="pmid34324429">{{cite journal|title=Circadian rhythms of 11-oxygenated C19 steroids and ∆5-steroid sulfates in healthy men |journal=Eur J Endocrinol |volume=185 |issue=4 |pages=K1–K6 |pmid=34324429 |doi=10.1530/EJE-21-0348 |pmc=8826489 |pmc-embargo-date=August 27, 2022 |last1=Turcu |first1=Adina F. |last2=Zhao |first2=Lili |last3=Chen |first3=Xuan |last4=Yang |first4=Rebecca |last5=Rege |first5=Juilee |last6=Rainey |first6=William E. |last7=Veldhuis |first7=Johannes D. |last8=Auchus |first8=Richard J. |year=2021 }}</ref> The levels of 11-oxygenated androgens raise after ACTH stimulation<ref name="pmid23386646"/><ref name="pmid13211802">{{cite journal |vauthors=DOBRINER K, KAPPAS A, GALLAGHER TF |title=Studies in steroid metabolism. XXVI. Steroid isolation studies in human leukemia |journal=J Clin Invest |volume=33 |issue=11 |pages=1481–6 |date=November 1954 |pmid=13211802 |pmc=1072573 |doi=10.1172/JCI103026 |url=}}</ref> that further supports their adrenal origin. However, in addition to the adrenal glands, CYP11B1 is also expressed in Leydig cells and ovarian theca cells, albeit at far lower levels, so the production of 11KT precursors may be one of the most important functions of 11β-hydroxylase activity in the gonads.<ref name="pmid27428878">{{cite journal|title=11-Ketotestosterone Is a Major Androgen Produced in Human Gonads |journal=J Clin Endocrinol Metab |volume=101 |issue=10 |pages=3582–3591 |pmid=27428878 |doi=10.1210/jc.2016-2311 |last1=Imamichi |first1=Yoshitaka |last2=Yuhki |first2=Koh-Ichi |last3=Orisaka |first3=Makoto |last4=Kitano |first4=Takeshi |last5=Mukai |first5=Kuniaki |last6=Ushikubi |first6=Fumitaka |last7=Taniguchi |first7=Takanobu |last8=Umezawa |first8=Akihiro |last9=Miyamoto |first9=Kaoru |last10=Yazawa |first10=Takashi |year=2016 }}</ref> In an in vitro study by Strushkevich et al. published in 2013, both isozymes have been shown to convert Δ⁴ steroids (P4, 17-OHP, A4 and T), but they are very specific to the configuration of the A-ring (carbon positions 1 to 5) of steroids, i.e., they cannot convert Δ⁵ steroids with a hydroxy group at the carbon position 3.<ref name="pmid23322723">{{cite journal |pmc=5417327|year=2013|last1=Strushkevich|first1=N.|last2=Gilep|first2=A. A.|last3=Shen|first3=L.|last4=Arrowsmith|first4=C. H.|last5=Edwards|first5=A. M.|last6=Usanov|first6=S. A.|last7=Park|first7=H. W.|title=Structural Insights into Aldosterone Synthase Substrate Specificity and Targeted Inhibition|journal=Molecular Endocrinology (Baltimore, Md.)|volume=27|issue=2|pages=315–324|doi=10.1210/me.2012-1287|pmid=23322723}}</ref> The relative biological significance of the 11-oxygenated androgens has been subject to some debate. There have been enough studies to establish that 11OHA4 itself does not have any androgenic activity but an important precursor to 11KT and 11KDHT.{{cn}} Androgen activity has been established for 11KT, 11KDHT<ref name="pmid27442248" /> as well as 11OHT and 11OHDHT but circulating levels of all of these androgens have not been firmly established. 11KT may serve as a primary androgen for healthy women,<ref name="pmid32629108">{{cite journal|last1=Barnard|first1=Lise|last2=Nikolaou|first2=Nikolaos|last3=Louw|first3=Carla|last4=Schiffer|first4=Lina|last5=Gibson|first5=Hylton|last6=Gilligan|first6=Lorna C.|last7=Gangitano|first7=Elena|last8=Snoep|first8=Jacky|last9=Arlt|first9=Wiebke|year=2020|title=The A-ring reduction of 11-ketotestosterone is efficiently catalysed by AKR1D1 and SRD5A2 but not SRD5A1|url=|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=202|pages=105724|doi=10.1016/j.jsbmb.2020.105724|pmid=32629108|s2cid=220323715|last10=Tomlinson|first10=Jeremy W.|last11=Storbeck|first11=Karl-Heinz}}</ref><ref name="pmid30753518" /> as it circulates at similar levels to T, but unlike T, the levels of 11KT are stable across the menstrual cycle.<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref> There are conflicting reports on whether 11-oxygenated androgens decline in women with age, and whether the relative contribution of 11KT as compared with T is higher in postmenopausal women than in younger ones — Nanba et al. (2019)<ref name="pmid30753518" /> and Davio et al. (2020)<ref name="pmid32498089">{{cite journal|last1=Davio|first1=Angela|last2=Woolcock|first2=Helen|last3=Nanba|first3=Aya T.|last4=Rege|first4=Juilee|last5=o'Day|first5=Patrick|last6=Ren|first6=Jianwei|last7=Zhao|first7=Lili|last8=Ebina|first8=Hiroki|last9=Auchus|first9=Richard|year=2020|title=Sex Differences in 11-Oxygenated Androgen Patterns Across Adulthood|journal=J Clin Endocrinol Metab|volume=105|issue=8|pages=e2921–e2929|doi=10.1210/clinem/dgaa343|pmc=7340191|pmid=32498089|last10=Rainey|first10=William E.|last11=Turcu|first11=Adina F.}}</ref> found that 11KT do not decline with age in women, however, Skiba et al. (2019)<ref name="pmid31390028" /> reported that the levels do decline. The decline of circulating 11-androgens with age may be associated with declining levels of DHEA and A4, which serve as precursors. It should also be noted that about half of circulating A4 quantities and almost all DHEA quantities are of adrenal origin.<ref name="pmid25428847">{{cite journal |vauthors=Turcu A, Smith JM, Auchus R, Rainey WE |title=Adrenal androgens and androgen precursors-definition, synthesis, regulation and physiologic actions |journal=Compr Physiol |volume=4 |issue=4 |pages=1369–81 |date=October 2014 |pmid=25428847 |pmc=4437668 |doi=10.1002/cphy.c140006 |url=}}</ref> Therefore the decline of 11-oxygenated androgens - steroids of adrenal origin - may be in line with a decline with their adrenal androgen precursors - DHEA and A4. Studies to date also strongly suggest that 11KT is the primary active 11-oxygenated androgen. Notably, 11KDHT has been found to circulate at substantially lower levels than DHT at least in prostate cancer<ref name="pmid30472582"/> suggesting that the 5α-reduction of 11KT is not as physiologically relevant in that condition. While early studies did show that 11KT and other 11-oxygenated androgens are substrates for SRD5A1 and SRD5A2, these did not perform full kinetic analyses.<ref name="pmid23856005"/>. Barnard et al., in a study published in 2020 showed that SRD5A1 does not efficiently catalyze the 5α-reduction of 11KT or 11KA4,<ref name="pmid32629108"/> confirming that 11KT may be the more relevant active 11-oxygenated androgen given the abundant peripheral expression of SRD5A2. While this does not rule out the potential for 11KDHT to be produced by SRD5A2 (or to a lesser degree by SRD5A1) in specific tissues, current evidence does suggest that the emphasis should be taken off the production of 11KDHT and rather placed on 11KT. As for the measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal |title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report |journal=Int J Mol Sci |year=2020 |volume=21 |issue=13 |pmid=32610579 |pmc=7369945 |doi=10.3390/ijms21134622 |doi-access=free |last1=Sumińska |first1=Marta |last2=Bogusz-Górna |first2=Klaudia |last3=Wegner |first3=Dominika |last4=Fichna |first4=Marta |page=4622 }}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal | last1=Yildiz | first1=Bulent O. | title=Diagnosis of hyperandrogenism: clinical criteria | journal=Best Practice & Research. Clinical Endocrinology & Metabolism | publisher=Elsevier BV | volume=20 | issue=2 | year=2006 | issn=1521-690X | pmid=16772149 | doi=10.1016/j.beem.2006.02.004 | pages=167–176}}</ref><ref name="pmid24184282">{{cite journal | last1=Peigné | first1=Maëliss | last2=Villers-Capelle | first2=Anne | last3=Robin | first3=Geoffroy | last4=Dewailly | first4=Didier | title=Hyperandrogénie féminine | journal=Presse Medicale (Paris, France) | publisher=Elsevier BV | volume=42 | issue=11 | year=2013 | issn=0755-4982 | pmid=24184282 | doi=10.1016/j.lpm.2013.07.016 | pages=1487–1499 | s2cid=28921380 | language=fr}}</ref> Relying on T levels alone in conditions associated with hyperandrogenism may read to diagnostic pitfalls and confusion.<ref name="pmid32610579"/> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|title=Clinical significance of 11-oxygenated androgens |journal=Curr Opin Endocrinol Diabetes Obes |volume=24 |issue=3 |pages=252–259 |pmid=28234803 |pmc=5819755 |doi=10.1097/MED.0000000000000334 |last1=Turcu |first1=Adina F. |last2=Auchus |first2=Richard J. |year=2017 }}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome |journal=Reprod Biomed Online |volume=41 |issue=4 |pages=734–742 |pmid=32912651 |doi=10.1016/j.rbmo.2020.07.013 |s2cid=221625488 |last1=Yang |first1=Yabo |last2=Ouyang |first2=Nengyong |last3=Ye |first3=Yang |last4=Hu |first4=Qin |last5=Du |first5=Tao |last6=Di |first6=Na |last7=Xu |first7=Wenming |last8=Azziz |first8=Ricardo |last9=Yang |first9=Dongzi |last10=Zhao |first10=Xiaomiao |year=2020 }}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803"/> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by very potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal | last1=Balsamo | first1=Antonio | last2=Baronio | first2=Federico | last3=Ortolano | first3=Rita | last4=Menabo | first4=Soara | last5=Baldazzi | first5=Lilia | last6=Di Natale | first6=Valeria | last7=Vissani | first7=Sofia | last8=Cassio | first8=Alessandra | title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant | journal=Frontiers in Pediatrics | year=2020 | publisher=Frontiers Media SA | volume=8 | page=593315 | issn=2296-2360 | pmid=33415088 | pmc=7783414 | doi=10.3389/fped.2020.593315| doi-access=free }}</ref><ref name="pmid29277706">{{cite journal | last1=Kamrath | first1=Clemens | last2=Wettstaedt | first2=Lisa | last3=Boettcher | first3=Claudia | last4=Hartmann | first4=Michaela F. | last5=Wudy | first5=Stefan A. | title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia | journal=The Journal of Steroid Biochemistry and Molecular Biology | publisher=Elsevier BV | volume=178 | year=2018 | issn=0960-0760 | pmid=29277706 | doi=10.1016/j.jsbmb.2017.12.016 | pages=221–228| s2cid=3709499 }}</ref> Another issue with the use of T as a biomarker of androgen excess is the low circulating levels in women and the specificity and sensitivity of the assays used.<ref name="pmid29306916">{{cite journal |title=Falsely elevated plasma testosterone concentrations in neonates: importance of LC-MS/MS measurements |journal=Clin Chem Lab Med |volume=56 |issue=6 |pages=e141–e143 |pmid=29306916 |doi=10.1515/cclm-2017-1028 |last1=Hamer |first1=Henrike M. |last2=Finken |first2=Martijn J.J. |last3=Van Herwaarden |first3=Antonius E. |last4=Du Toit |first4=Therina |last5=Swart |first5=Amanda C. |last6=Heijboer |first6=Annemieke C. |year=2018 |hdl=10019.1/106715 |s2cid=13917408 }}</ref><ref name="pmid32912651" /><ref name="pmid30753518">{{cite journal|last1=Nanba|first1=Aya T.|last2=Rege|first2=Juilee|last3=Ren|first3=Jianwei|last4=Auchus|first4=Richard J.|last5=Rainey|first5=William E.|last6=Turcu|first6=Adina F.|year=2019|title=11-Oxygenated C19 Steroids Do Not Decline With Age in Women|journal=J Clin Endocrinol Metab|volume=104|issue=7|pages=2615–2622|doi=10.1210/jc.2018-02527|pmc=6525564|pmid=30753518}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterised, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women |journal=Fertil Steril |volume=58 |issue=1 |pages=148–52 |pmid=1623996 |doi=10.1016/s0015-0282(16)55152-8 |last1=Carmina |first1=E. |last2=Stanczyk |first2=F. Z. |last3=Chang |first3=L. |last4=Miles |first4=R. A. |last5=Lobo |first5=R. A. |year=1992 }}</ref><ref name="pmid14417423">{{cite journal |title=Urinary ketosteroids and pregnanetriol in hirsutism |journal=J Clin Endocrinol Metab |volume=20 |issue= 2|pages=180–6 |pmid=14417423 |doi=10.1210/jcem-20-2-180|last1=Lipsett |first1=Mortimer B. |last2=Riter |first2=Barbara |year=1960 }}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries |journal=J Clin Endocrinol Metab |volume=66 |issue=5 |pages=946–50 |pmid=3129451 |doi=10.1210/jcem-66-5-946 |last1=Polson |first1=D. W. |last2=Reed |first2=M. J. |last3=Franks |first3=S. |last4=Scanlon |first4=M. J. |last5=James |first5=V. H. T. |year=1988 }}</ref> However, due to to conflicting reports ratios did not find a firm footing in the clinical as a diagnostic tool. === On The Aromatization of Androgens === Unlike T and A4, 11-oxygenated androgens are unlikely to be converted by aromatase into estrogens ''in vivo'',<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> that was first predicted in 2016 by Imamichi at al. in an ''in vitro'' study.<ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women when not taking into account DHEA. However, DHEA has a very low affinity for the androgen receptor and thus should not be an important contributor, if at all, for receptor activation under normal conditions.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, Barnard et al., incubating ''in vitro'' three different aromatase-expressing cell cultures and ''ex vivo'' human placenta explant cultures with normal and radiolabeled steroids, detected conversion of 11-oxygenated and conventional androgens into 11-oxygenated estrogens; however, 11-oxygenated estrogens were not detected ''in vivo'': neither in pregnant women who have high aromatase expression nor in patients who have high 11-androgens levels due to CAH or adrenocortical carcinoma, probably due to relatively low aromatase activity towards 11-oxygenated androgens compared to conventional androgens.<ref name="pmid33340399"/> However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Therefore, the role of DHT and 11-oxygenated androgens should be seriously considered in women patients. === Disorders of Sex Development === Since both the canonical and backdoor pathways of androgen biosynthesis towards DHT lead to early male sexual differentiation<ref name="pmid30763313" /><ref name="pmid30943210">{{cite journal|title = The "backdoor pathway" of androgen synthesis in human male sexual development | journal = PLOS Biology | volume = 17 | issue = 4 | pages = e3000198 | pmid = 30943210 | pmc = 6464227 | doi = 10.1371/journal.pbio.3000198 | last1 = Miller | first1 = Walter L. | last2 = Auchus | first2 = Richard J. |year = 2019 }}</ref><ref name="pmid11035809" /><ref name="pmid15249131" /> and are required for normal human male genital development,<ref name="pmid30943210" /><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of male fetuses,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 acts as a substrate during the biosynthesis of DHT in the backdoor pathway.<ref name="pmid30763313"/> Flück et al. described in 2011 a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. These findings confirmed that DHT produced in a backdoor pathway, while not necessary for the sexual development of females, is important for that of males. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were enough to disrupt it.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). As such, this syndrome leads to DSD in both sexes, while affected girls go usually unrecognized until puberty, when they show amenorrhea. This syndrome is also rare with only a few cases reported.<ref name="pmid34711511"/> Besides that, 11-oxygenated androgens may play previously overlooked role in DSD.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> === Congenital Adrenal Hyperplasia === Another cause of androgen excess is congenital adrenal hyperplasia (CAH), a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal |vauthors=El-Maouche D, Arlt W, Merke DP |title=Congenital adrenal hyperplasia |journal=Lancet |volume=390 |issue=10108 |pages=2194–2210 |date=November 2017 |pmid=28576284 |doi=10.1016/S0140-6736(17)31431-9 |url=}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal |vauthors=Speiser PW, White PC |title=Congenital adrenal hyperplasia |journal=N Engl J Med |volume=349 |issue=8 |pages=776–88 |date=August 2003 |pmid=12930931 |doi=10.1056/NEJMra021561 |url=}}</ref><ref name="pmid30272171">{{cite journal | title = Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline | journal = The Journal of Clinical Endocrinology and Metabolism | volume = 103 | issue = 11 | pages = 4043–4088 | year = 2018 | pmid = 30272171 | pmc = 6456929 | doi = 10.1210/jc.2018-01865 }}</ref> Such deficiency leads to an excessive accumulation of a respective cortisol precursor, that becomes to serve as a substrate to androgens. In CYP21A2 deficiency<ref name="pmid22170725" /> including the mild forms (which are not always diagnosed)<ref name="pmid32966723">{{cite journal |vauthors=Merke DP, Auchus RJ |title=Congenital Adrenal Hyperplasia Due to 21-Hydroxylase Deficiency |journal=N Engl J Med |volume=383 |issue=13 |pages=1248–1261 |date=September 2020 |pmid=32966723 |doi=10.1056/NEJMra1909786 |url=}}</ref><ref name="pmid31499506">{{cite book|title=Hyperandrogenism in Women|last1=Pignatelli|first1=Duarte|last2=Pereira|first2=Sofia S.|last3=Pasquali|first3=Renato|year=2019|isbn=978-3-318-06470-4|series=Frontiers of Hormone Research|volume=53|pages=65–76|chapter=Androgens in Congenital Adrenal Hyperplasia|doi=10.1159/000494903|pmid=31499506|s2cid=202412336}}</ref> or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> elevated 17-OHP levels starts the backdoor pathway to DHT. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> The reason why 17-OHP serves as a prerequisite substrate for DHT within the backdoor pathway roundabout of T rather then an immediate substrate within the Δ⁴ pathway for A4, and then T, is because the catalytic activity 17,20-lyase reaction (which cleaves a side-chain from the steroid nucleus converting a pregnane to an androstane (androgen), i.e., from 17OPH5 to DHEA; from 17-OHP to A4) performed by CYP17A1 in humans is approximately 100 times more efficient in the Δ⁵ pathway than in the Δ⁴ pathway. Therefore, the catalytic efficiency of CYP17A1 for 17-OHP is about 100 times lower than for 17OHP5, resulting in negligible A4 being produced from 17-OHP in the Δ⁴ reaction pathway in humans.<ref name="pmid8325965">{{cite journal|last1=Swart|first1=P.|last2=Swart|first2=A. C.|last3=Waterman|first3=M. R.|last4=Estabrook|first4=R. W.|last5=Mason|first5=J. I.|year=1993|title=Progesterone 16 alpha-hydroxylase activity is catalyzed by human cytochrome P450 17 alpha-hydroxylase|journal=J Clin Endocrinol Metab|volume=77|issue=1|pages=98–102|doi=10.1210/jcem.77.1.8325965|pmid=8325965}}</ref><ref name="pmid12915666">{{cite journal|last1=Flück|first1=Christa E.|last2=Miller|first2=Walter L.|last3=Auchus|first3=Richard J.|year=2003|title=The 17, 20-lyase activity of cytochrome CYP17A1 from human fetal testis favors the delta5 steroidogenic pathway|url=https://pubmed.ncbi.nlm.nih.gov/12915666|journal=The Journal of Clinical Endocrinology and Metabolism|volume=88|issue=8|pages=3762–3766|doi=10.1210/jc.2003-030143|issn=0021-972X|pmid=12915666}}</ref><ref name="pmid15774560">{{cite journal|last1=Miller|first1=Walter L.|year=2005|title=Minireview: regulation of steroidogenesis by electron transfer|url=https://pubmed.ncbi.nlm.nih.gov/15774560|journal=Endocrinology|volume=146|issue=6|pages=2544–2550|doi=10.1210/en.2005-0096|issn=0013-7227|pmid=15774560}}</ref><ref name="pmid32007561"/> The accumulation of 17-OHP in CYP21A2 deficiency in CAH can be attributed to the fact that the primary enzyme for 17-OHP in normal conditions is CYP21A2, that is expressed in the adrenal and not the gonads.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|year=2019 |volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086 }}</ref><ref name="pmid26038201"/> In a 1998 study, Auchus et al. demonstrated that human CYP17A1 efficiently catalyzed the conversion of P4 to 17-OHP, but the conversion of 17-OHP to A4 was much less efficient than the corresponding conversion of 17OHP5 to DHEA.<ref name="pmid9452426"/> In rodents, quite contrary, the conversion of 17-OHP to A4 is very efficient.<ref name="pmid9452426">{{cite journal | last1=Auchus | first1=Richard J. | last2=Lee | first2=Tim C. | last3=Miller | first3=Walter L. | title=Cytochrome b 5 Augments the 17,20-Lyase Activity of Human P450c17 without Direct Electron Transfer | journal=The Journal of Biological Chemistry | year=1998 | publisher=Elsevier BV | volume=273 | issue=6 | issn=0021-9258 | pmid=9452426 | doi=10.1074/jbc.273.6.3158 | pages=3158–3165| doi-access=free }}</ref> This explains significant accumulation of 17-OHP in CYP21A2 deficiency or POR deficiency in humans, so that 17-OHP, while not 21-hydroxylated in sufficient quantities, and being better a substrate for 5α-reductase than for CYP17A1, is 5α-reduced serving as the prerequisite for this backdoor pathway. Hence, fetal excess of 17-OHP in CAH may provoke activation of this pathway to DHT and lead to external genital virilization in newborn girls, thus explaining DSD in girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025"/><ref name="pmid31505456">{{cite journal |vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE |title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia |journal=Eur J Endocrinol |volume=181 |issue=5 |pages=481–488 |date=November 2019 |pmid=31505456 |doi=10.1530/EJE-19-0379 |url=}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal |vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M |title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia |journal=J Obstet Gynaecol Res |volume=43 |issue=5 |pages=946–950 |date=May 2017 |pmid=28188961 |doi=10.1111/jog.13288 |url=}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal | last1=Turcu | first1=Adina F | last2=Mallappa | first2=Ashwini | last3=Elman | first3=Meredith S | last4=Avila | first4=Nilo A | last5=Marko | first5=Jamie | last6=Rao | first6=Hamsini | last7=Tsodikov | first7=Alexander | last8=Auchus | first8=Richard J | last9=Merke | first9=Deborah P | title = 11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency | journal = The Journal of Clinical Endocrinology and Metabolism | year=2017 | volume = 102 | issue = 8 | pages = 2701–2710 | pmid = 28472487 | pmc = 5546849 | doi = 10.1210/jc.2016-3989}}</ref><ref name="pmid26865584">{{cite journal|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency |journal=Eur J Endocrinol |volume=174 |issue=5 |pages=601–9 |pmid=26865584 |pmc=4874183 |doi=10.1530/EJE-15-1181 |last1=Turcu |first1=Adina F. |last2=Nanba |first2=Aya T. |last3=Chomic |first3=Robert |last4=Upadhyay |first4=Sunil K. |last5=Giordano |first5=Thomas J. |last6=Shields |first6=James J. |last7=Merke |first7=Deborah P. |last8=Rainey |first8=William E. |last9=Auchus |first9=Richard J. |year=2016 }}</ref><ref name="pmid29718004">{{cite journal|title = Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency | journal = Current Opinion in Endocrinology, Diabetes, and Obesity | volume = 25 | issue = 3 | pages = 178–184 | pmid = 29718004 | doi = 10.1097/MED.0000000000000402 | s2cid = 26072848 |last1 = White |first1 = Perrin C. |year = 2018 }}</ref><ref name="pmid34867794"/> In males with CAH, 11-oxygenated androgens may lead to devlopment of testicular adrenal rest tumors.<ref name="pmid25850025">{{cite journal|pmc=4454804|year=2015|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|last10=Auchus|first10=R. J.|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmid=25850025}}</ref><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors |journal=J Clin Endocrinol Metab |volume=107 |issue=1 |pages=e272–e280 |pmid=34390337 |pmc=8684463 |doi=10.1210/clinem/dgab598 |last1=Schröder |first1=Mariska A M. |last2=Turcu |first2=Adina F. |last3=o'Day |first3=Patrick |last4=Van Herwaarden |first4=Antonius E. |last5=Span |first5=Paul N. |last6=Auchus |first6=Richard J. |last7=Sweep |first7=Fred C G J. |last8=Claahsen-Van Der Grinten |first8=Hedi L. |year=2022 }}</ref> The biosynthesis of 11OHP4 from P4 and 21dF from 17-OHP by CYP11B1/2 in CAH may be attributed to CYP21A2 deficiency resulting in increased P4 and 17-OHP concentrations and, together with the unavailability of CYP11B1/2's main substrates, 11-deoxycortisol (11dF) and 11-deoxycorticosterone (DOC), drive the production of 11-oxygenated pregnanes.<ref name="pmid3546944" /> We have reasons to believe that this may be aggravated by elevated ACTH due to a feedback loop in hypothalamic-pituitary-adrenal axis caused by impaired cortisol synthesis associated with CYP21A2 deficiency; higher ACTH causes higher CYP11B1 expression. Multiple studies demonstrated that in CAH due to CYP21A2 deficiency, both 21dF levels<ref name="pmid4372245">{{cite journal |title=Plasma 17-hydroxyprogesterone, 21-deoxycortisol and cortisol in congenital adrenal hyperplasia |journal=J Clin Endocrinol Metab |volume=39 |issue=6 |pages=1099–102 |year=1974 |pmid=4372245 |doi=10.1210/jcem-39-6-1099 |last1=Franks |first1=Robert C. }}</ref><ref name="pmid476971">{{cite journal |title=Rapid assay of plasma 21-deoxycortisol and 11-deoxycortisol in congenital adrenal hyperplasia |journal=Clin Endocrinol (Oxf) |volume=10 |issue=4 |pages=367–75 |year=1979 |pmid=476971 |doi=10.1111/j.1365-2265.1979.tb02091.x |url=|last1=Fukushima |first1=D. K. |last2=Nishina |first2=T. |last3=Wu |first3=R. H. K. |last4=Hellman |first4=L. |last5=Finkelstein |first5=J. W. |s2cid=2979354 }}</ref><ref name="pmid6090811">{{cite journal |title=Development of plasma 21-deoxycortisol radioimmunoassay and application to the diagnosis of patients with 21-hydroxylase deficiency |journal=J Steroid Biochem |volume=21 |issue=2 |pages=185–91 |year=1984 |pmid=6090811 |doi=10.1016/0022-4731(84)90382-0 |last1=Milewicz |first1=A. |last2=Vecsei |first2=P. |last3=Korth-Schütz |first3=S. |last4=Haack |first4=D. |last5=Rösler |first5=A. |last6=Lichtwald |first6=K. |last7=Lewicka |first7=S. |last8=Mittelstaedt |first8=G.v. }}</ref><ref name="pmid2986404">{{cite journal |title=Radioimmunoassay for 21-deoxycortisol: clinical applications |journal=Acta Endocrinol (Copenh) |volume=108 |issue=4 |pages=537–44 |year=1985 |pmid=2986404 |doi=10.1530/acta.0.1080537 |last1=Gueux |first1=B. |last2=Fiet |first2=J. |last3=Pham-Huu-Trung |first3=M. T. |last4=Villette |first4=J. M. |last5=Gourmelen |first5=M. |last6=Galons |first6=H. |last7=Brerault |first7=J. L. |last8=Vexiau |first8=P. |last9=Julien |first9=R. }}</ref><ref name="pmid25850025" /> and 11OPH4 levels<ref name="pmid3546944">{{cite journal |last1=Gueux |first1=Bernard |last2=Fiet |first2=Jean |last3=Galons |first3=Hervé |last4=Boneté |first4=Rémi |last5=Villette |first5=Jean-Marie |last6=Vexiau |first6=Patrick |last7=Pham-Huu-Trung |first7=Marie-Thérèse |last8=Raux-Eurin |first8=Marie-Charles |last9=Gourmelen |first9=Micheline |last10=Brérault |first10=Jean-Louis |last11=Julien |first11=René |last12=Dreux |first12=Claude |title=The measurement of 11β-hydroxy-4-pregnene-3,20-dione (21-Deoxycorticosterone) by radioimmunoassay in human plasma |journal=Journal of Steroid Biochemistry |year=1987 |volume=26 |issue=1 |pages=145–150 |doi=10.1016/0022-4731(87)90043-4 |pmid=3546944 }}</ref><ref name="pmid2537337">{{cite journal |last1=Fiet |first1=Jean |last2=Gueux |first2=Bernard |last3=Rauxdemay |first3=Marie-Charles |last4=Kuttenn |first4=Frederique |last5=Vexiau |first5=Patrick |last6=Brerault |first6=Jeanlouis |last7=Couillin |first7=Philippe |last8=Galons |first8=Herve |last9=Villette |first9=Jeanmarie |last10=Julien |first10=Rene |last11=Dreux |first11=Claude |title=Increased Plasma 21-Deoxycorticosterone (21-DB) Levels in Late-Onset Adrenal 21-Hydroxylase Deficiency Suggest a Mild Defect of the Mineralocorticoid Pathway |journal=The Journal of Clinical Endocrinology & Metabolism |year=1989 |volume=68 |issue=3 |pages=542–547 |doi=10.1210/jcem-68-3-542 |pmid=2537337 }}</ref><ref name="pmid29264476">{{cite journal |last1=Fiet |first1=Jean |last2=Le Bouc |first2=Yves |last3=Guéchot |first3=Jérôme |last4=Hélin |first4=Nicolas |last5=Maubert |first5=Marie-Anne |last6=Farabos |first6=Dominique |last7=Lamazière |first7=Antonin |title=A Liquid Chromatography/Tandem Mass Spectometry Profile of 16 Serum Steroids, Including 21-Deoxycortisol and 21-Deoxycorticosterone, for Management of Congenital Adrenal Hyperplasia |journal=Journal of the Endocrine Society |year=2017 |volume=1 |issue=3 |pages=186–201 |doi=10.1210/js.2016-1048 |pmid=29264476 |pmc=5686660 }}</ref><ref name="pmid31821037">{{cite journal |title=Interaction between accumulated 21-deoxysteroids and mineralocorticoid signaling in 21-hydroxylase deficiency |journal=Am J Physiol Endocrinol Metab |volume=318 |issue=2 |pages=E102–E110 |year=2020 |pmid=31821037 |doi=10.1152/ajpendo.00368.2019 |last1=Travers |first1=Simon |last2=Bouvattier |first2=Claire |last3=Fagart |first3=Jérôme |last4=Martinerie |first4=Laetitia |last5=Viengchareun |first5=Say |last6=Pussard |first6=Eric |last7=Lombès |first7=Marc |s2cid=209314028 }}</ref> are increased. It was Robert Franks in who first published a study, in 1974, that compared 21dF levels of CAH patients with those of healthy controls. He measured 21dF plasma levels in twelve CAH patients before treatment, three after treatment, and four healthy controls following ACTH administration. Mean values of 21dF in CAH patients was 88 ng/ml while in healthy controls it was not detected. In untreated patients, values decreased after therapy. Even that, there were earlier reports about unique cases where 21dF was detected in CAH patients, but without direct comparison to healthy controls.<ref name="pmid5845501">{{cite journal |title=Detection of 21-deoxycortisol in blood from a patient with congenital adrenal hyperplasia |journal=Metabolism |year=1965 |volume=14 |issue=12 |pages=1276–81 |pmid=5845501 |doi=10.1016/s0026-0495(65)80008-7|last1=Wieland |first1=Ralph G. |last2=Maynard |first2=Donald E. |last3=Riley |first3=Thomas R. |last4=Hamwi |first4=George J. }}</ref><ref name="pmid13271547">{{cite journal|title=17alpha-hydroxyprogesterone and 21-desoxyhydrocortisone; their metabolism and possible role in congenital adrenal virilism |journal=J Clin Invest |volume=34 |issue=11 |pages=1639–46 |year=1955 |pmid=13271547 |pmc=438744 |doi=10.1172/JCI103217|last1=Jailer |first1=Joseph W. |last2=Gold |first2=Jay J. |last3=Vande Wiele |first3=Raymond |last4=Lieberman |first4=Seymour }}</ref> As for 11OHP4, it were Gueux et al. who first demonstrated, in 1987, elevated plasma levels of 11OHP4 in CAH. In that study, in treated classical CAH patients, some of which had salt-wasting form, mean levels of 11OHP4 (5908.7 pmol/l) were 332 times higher than in healthy controls (17.8 pmol/l). There was no difference in 11OHP4 in healthy controls depending on sex or phase of a menstrual cycle; ACTH stimulation in those control increased 11OHP4 four- to six-fold, while dexamethasone 1 mg at midnight decreased 11OHP4 to almost undetectable levels 12 hours later. Therefore, the authors hypothesized that at least in healthy people 11OHP4 is biosythesized exclusively in the adrenal, while gonads are not involved.<ref name="pmid3546944" /> Nevertheless, in studies focusing on CAH caused by CYP21A2 deficiency, 11OHP4 received less attention than 21dF.<ref name="pmid29277707"/> However, it was not until 2017 when 11OHP4 or 21dF were viewed as potential substrates in pathways towards potent 11-ogygenated androgens in ''in vitro'' studies.<ref name="pmid32007561"/><ref name="pmid29277707"/> In a 2016 study, Turcu et al. showed that in classic CAH due to CYP21A2 deficiency, in male and female patients who received glucocorticoid therapy, both conventional and 11-oxygenated androgens were elevated 3-4 fold compared to healthy controls. The exceptions were dehydroepiandrosterone (DHEA), dehydroepiandrosterone sulfate (DHEA-S), and androstenediol sulfate (A5-S), whose levels were 6.0, 7.5, and 9.4 times lower, respectively, in the patients with the condition compared to healthy controls, due to suppression by glucocorticoid treatment. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in a backdoor pathway from 17-OHP or P4 as consequence of abnormally upregulated SRD5A1.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Besides that, the levels of 11OHA4 and 11KA4 correlated with the levels of markers of insulin resistance; therefore, the study suggests that androgen excess precedes androgen-driven insulin resistance in PCOS.<ref name="pmid27901631" /> While earlier studies had commonly only measured 11OHA4 or 11OHAST and 11β-hydroxyetiocholanolone (11OHEt), urinary metabolites of 11OHA,<ref name="pmid33539964" /> while 11OHEt is also a metabolite of cortisol,<ref name="pmid31362062">{{cite journal |title=Human steroid biosynthesis, metabolism and excretion are differentially reflected by serum and urine steroid metabolomes: A comprehensive review |journal=J Steroid Biochem Mol Biol |volume=194 |issue= |pages=105439 |year=2019 |pmid=31362062 |pmc=6857441 |doi=10.1016/j.jsbmb.2019.105439 |url=|last1=Schiffer |first1=Lina |last2=Barnard |first2=Lise |last3=Baranowski |first3=Elizabeth S. |last4=Gilligan |first4=Lorna C. |last5=Taylor |first5=Angela E. |last6=Arlt |first6=Wiebke |last7=Shackleton |first7=Cedric H.L. |last8=Storbeck |first8=Karl-Heinz }}</ref><ref name="pmid27845856">{{cite journal |title=Modified-Release and Conventional Glucocorticoids and Diurnal Androgen Excretion in Congenital Adrenal Hyperplasia |journal=J Clin Endocrinol Metab |volume=102 |issue=6 |pages=1797–1806 |year=2017 |pmid=27845856 |pmc=5470768 |doi=10.1210/jc.2016-2855|last1=Jones |first1=Christopher M. |last2=Mallappa |first2=Ashwini |last3=Reisch |first3=Nicole |last4=Nikolaou |first4=Nikolaos |last5=Krone |first5=Nils |last6=Hughes |first6=Beverly A. |last7=o'Neil |first7=Donna M. |last8=Whitaker |first8=Martin J. |last9=Tomlinson |first9=Jeremy W. |last10=Storbeck |first10=Karl-Heinz |last11=Merke |first11=Deborah P. |last12=Ross |first12=Richard J. |last13=Arlt |first13=Wiebke }}</ref> more recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Another study reported that 11OHT was the only significantly elevated 11-oxygeated androgen in PCOS and together with 11KT, correlated with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227"/> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Premature Adrenarche === In a 2018 study, Rege et al. demonstrated that levels of 11KT in girls aged between 4 and 7 years during normal adrenarche (healthy controls) exceeded those of T by 2.43 times, and in those with premature adrenarche by 3.48 times. However, the levels of T in girls with premature adrenarche were higher by just 13% compared to age-matched healthy controls.<ref name="pmid30137510">{{cite journal | last1=Rege | first1=Juilee | last2=Turcu | first2=Adina | last3=Kasa-Vubu | first3=Josephine Z | last4=Lerario | first4=Antonio M | last5=Auchus | first5=Gabriela C | last6=Auchus | first6=Richard J | last7=Smith | first7=Joshua M | last8=White | first8=Perrin C | last9=Rainey | first9=William E | title=11KT is the dominant circulating bioactive androgen during normal and premature adrenarche | journal=The Journal of Clinical Endocrinology and Metabolism | year=2018 | publisher=The Endocrine Society | volume=103 | issue=12 | pages=4589–4598 | issn=0021-972X | pmid=30137510 | pmc=6226603 | doi=10.1210/jc.2018-00736 }}</ref> === Prostate Cancer === In some cases of advanced prostate cancer, androgen deprivation therapy related to gonadal T depletion does not produce long-term effects, and metastatic tumors may develop into castration-resistant prostate cancer (CRPC). The development of CRPC depends on adrenal precursor steroids to produce DHT in the tumor in a metabolic pathway called the "5α-dione" pathway - the pathway in which T is not involved. SRD5A1, the expression of which increases in CRPC, reduces A4 to 5α-dione, which is then converted to DHT.<ref name="pmid21795608"/><ref name="pmid31900912"/> Therefore, the DHT produced within the "5α-dione" pathway hampers the androgen deprivation therapy. Although blood levels of T are decreased by 90-95% in men whose testicles have been removed, DHT in the prostate is only decreased by 50%, thus indicating the presence of a metabolic pathway in the prostate that does not require testicular T to produce DHT.<ref name="pmid18471780">{{cite journal | last1=Luu-The | first1=Van | last2=Bélanger | first2=Alain | last3=Labrie | first3=Fernand | title=Androgen biosynthetic pathways in the human prostate | journal=Best Practice & Research. Clinical Endocrinology & Metabolism | publisher=Elsevier BV | volume=22 | issue=2 | year=2008 | issn=1521-690X | pmid=18471780 | doi=10.1016/j.beem.2008.01.008 | pages=207–221}}</ref> Chang et al., incubating six established human prostate cancer cell lines from patients with CRPC in the presence of radiolabeled A4, showed in their experiment published in 2011<ref name="pmid21795608">{{cite journal|last1=Chang | first1=K.-H. | last2=Li | first2=R. | last3=Papari-Zareei | first3=M. | last4=Watumull | first4=L. | last5=Zhao | first5=Y. D. | last6=Auchus | first6=R. J. | last7=Sharifi | first7=N. | title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer | journal=Proceedings of the National Academy of Sciences of the United States of America |year=2011 | publisher=Proceedings of the National Academy of Sciences | volume=108 | issue=33 | issn=0027-8424 | pmid=21795608 | pmc=3158152 | doi=10.1073/pnas.1107898108 | pages=13728–13733|bibcode=2011PNAS..10813728C |doi-access=free }}</ref> the presence of this pathway to DHT which bypasses T and they called this the "alternative" pathway, that became later commonly called as the "5α-dione" pathway.<ref name="pmid23856005"/> The authors demonstrated that this was the dominant pathway in prostate cancer (over the direct conversion of A4 to T) with SRD5A1 (which is upregulated in prostate cancer) first converting A4 to androstanedione (5α-dione), also known as 5α-androstane-3,17-dione, and then HSD17B3 / AKR1C3 converting 5α-dione to DHT (not necessarily via AST and 3α-diol). The study also found that the SRD5A2 is not involved in this "alternative" pathway.<ref name="pmid31900912"/> Therefore, the study showed the importance of taking into consideration this "alternative" pathway in selecting drugs that inhibit 5α-reductase activity.<ref name="pmid21901017">{{cite journal |title=Prostate cancer: DHT bypasses testosterone to drive progression to castration resistance |journal=Nat Rev Urol |volume=8 |issue=9 |pages=470 |year=September 2011 |pmid=21901017 |doi=10.1038/nrurol.2011.122 }}</ref><ref name="pmid22064602" /><ref name="pmid22336886">{{cite journal |title=Dihydrotestosterone synthesis from adrenal precursors does not involve testosterone in castration-resistant prostate cancer |journal=Cancer Biol Ther |volume=13 |issue=5 |pages=237–8 |year=2012 |pmid=22336886 |doi=10.4161/cbt.19608}}</ref> Another pathway that may be activated in CRPC, which may also hamper the androgen deprivation therapy, is the backdoor pathway from P4 to DHT. Chen et al. in a study published in 2014<ref name="pmid25320358">{{cite journal |vauthors=Chen EJ, Sowalsky AG, Gao S, Cai C, Voznesensky O, Schaefer R, Loda M, True LD, Ye H, Troncoso P, Lis RL, Kantoff PW, Montgomery RB, Nelson PS, Bubley GJ, Balk SP, Taplin ME |title=Abiraterone treatment in castration-resistant prostate cancer selects for progesterone responsive mutant androgen receptors |journal=Clin Cancer Res |volume=21 |issue=6 |pages=1273–80 |date=March 2015 |pmid=25320358 |pmc=4359958 |doi=10.1158/1078-0432.CCR-14-1220 |url=}}</ref> predicted that abiraterone, a CYP17A1 inhibitor, with about 6-fold more selective for inhibition of 17α-hydroxylase over 17,20-lyase,<ref name="pmid28890368">{{cite journal |vauthors=de Mello Martins AGG, Allegretta G, Unteregger G, Haupenthal J, Eberhard J, Hoffmann M, van der Zee JA, Junker K, Stöckle M, Müller R, Hartmann RW, Ohlmann CH |title=CYP17A1-independent production of the neurosteroid-derived 5α-pregnan-3β,6α-diol-20-one in androgen-responsive prostate cancer cell lines under serum starvation and inhibition by Abiraterone |journal=J Steroid Biochem Mol Biol |volume=174 |issue= |pages=183–191 |date=November 2017 |pmid=28890368 |doi=10.1016/j.jsbmb.2017.09.006 |url=}}</ref><ref name="pmid28373265">{{cite journal |vauthors=Petrunak EM, Rogers SA, Aubé J, Scott EE |title=Structural and Functional Evaluation of Clinically Relevant Inhibitors of Steroidogenic Cytochrome P450 17A1 |journal=Drug Metab Dispos |volume=45 |issue=6 |pages=635–645 |date=June 2017 |pmid=28373265 |pmc=5438109 |doi=10.1124/dmd.117.075317 |url=}}</ref><ref name="pmid29710837">{{cite journal |vauthors=Fernández-Cancio M, Camats N, Flück CE, Zalewski A, Dick B, Frey BM, Monné R, Torán N, Audí L, Pandey AV |title=Mechanism of the Dual Activities of Human CYP17A1 and Binding to Anti-Prostate Cancer Drug Abiraterone Revealed by a Novel V366M Mutation Causing 17,20 Lyase Deficiency |journal=Pharmaceuticals (Basel) |volume=11 |issue=2 |pages= |date=April 2018 |pmid=29710837 |pmc=6027421 |doi=10.3390/ph11020037 |url=}}</ref> although disrupting canonical androgen biosynthesis, while lowering levels of T, causes elevation of P4, that can be 5α-reduced hence start a backdoor pathway from P4 to DHT with roundabout of T.<ref name="pmid25320358"/> Besides that, in CRPC, 11-oxygenated androgens contribute significantly to the androgen pool.<ref name="pmid23856005"/><ref name="pmid31900912"/> 11-oxygenated androgens play a previously overlooked role in the reactivation of androgen signaling in CRPC,<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> because after eliminating testicular T biosynthesis by chemical or physical castration, CRPC has been shown to develop the ability to convert inactive circulating adrenal androgen precursors, DHEA and A4, to potent 11-oxygenated androgens in the 11-oxygenated pathway in addition to the 5α-dione pathway.<ref name="pmid31672619">{{cite journal |title=The role of adrenal derived androgens in castration resistant prostate cancer |journal=J Steroid Biochem Mol Biol |volume=197 |issue= |pages=105506 |year=2020 |pmid=31672619 |doi=10.1016/j.jsbmb.2019.105506|last1=Barnard |first1=Monique |last2=Mostaghel |first2=Elahe A. |last3=Auchus |first3=Richard J. |last4=Storbeck |first4=Karl-Heinz |pmc=7883395 }}</ref><ref name="pmid33974560" /> In a 2021 study, Snaterse et al. demonstrated that 11KT is the most circulating active androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool. They also demonstrated that 11KT levels are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> In a 2018 study by du Toit et al., the full range of androgen pathway metabolites have been shown in normal prostate and various prostate cancer cell models. 11OHA4 and 11OHT were both converted to potent androgens, 11KT and 11KDHT. Compared to T and DHT, 11-oxygenated androgens were the most predominant androgens. High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant. Furthermore, glucuronidation of the 11-oxygenated androgens is hampered by the presence of an oxo- or a hydroxy- group at position 11 of androgens in prostate cancer cell models while in prostate cancer patients' plasma 11KDHT was present only in the unconjugated form, with 11KT also predominantly unconjugated.<ref name="pmid28939401"/> Of all 11-oxygenated androgens in prostate cancer, it may be that 11KT is the primary active 11-oxygenated androgen, rather than 11KDHT, as observed in experiments on prostate tissue homogenates by Häkkinen et al. published in 2018<ref name="pmid30472582">{{cite journal|last1=Häkkinen|first1=Merja R.|last2=Murtola|first2=Teemu|last3=Voutilainen|first3=Raimo|last4=Poutanen|first4=Matti|last5=Linnanen|first5=Tero|last6=Koskivuori|first6=Johanna|last7=Lakka|first7=Timo|last8=Jääskeläinen|first8=Jarmo|last9=Auriola|first9=Seppo|year=2019|title=Simultaneous analysis by LC-MS/MS of 22 ketosteroids with hydroxylamine derivatization and underivatized estradiol from human plasma, serum and prostate tissue|journal=J Pharm Biomed Anal|volume=164|issue=|pages=642–652|doi=10.1016/j.jpba.2018.11.035|pmid=30472582|s2cid=53729550}}</ref> Ventura-Bahena et al., in a 2021 study, describing results of epidemiological studies related to androgens and prostate cancer that focused on specific androgen concentrations (such as T, A4, and DHEA) as inconsistent, hypothesized that the differences in androgen biosynthetic pathways rather than differences in specific androgen levels are associated with prostate cancer development.<ref name="pmid34520388"/> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but also in clinical conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> The contribution of the 11-oxygenated androgens, as well as the biosynthesis of 11-oxygenated pregnanes to active androgens via a backdoor pathway, have also been demonstrated in BPH cell models showing the conversion of 11OHP4 and 11KP4 in the backdoor pathway resulting in the production of 11KDHT. Backdoor pathway intermediates were also detected in BPH tissue as well as in circulation in BPH patients.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> In a paper published in 2008, Dimitrakov et al. hypothesized that CP/CPPS may be associated with a mild CYP21A2 deficiency, a cause of non-classic CAH that leads to androgen excesses.<ref name="pmid18308097"/> Non-classic CAH was generally thought to be asymptomatic in men.<ref name="pmid28582566">{{cite journal |title=Non-classic congenital adrenal hyperplasia due to 21-hydroxylase deficiency revisited: an update with a special focus on adolescent and adult women |journal=Hum Reprod Update |volume=23 |issue=5 |pages=580–599 |year=2017 |pmid=28582566 |doi=10.1093/humupd/dmx014 |last1=Carmina |first1=Enrico |last2=Dewailly |first2=Didier |last3=Escobar-Morreale |first3=Héctor F. |last4=Kelestimur |first4=Fahrettin |last5=Moran |first5=Carlos |last6=Oberfield |first6=Sharon |last7=Witchel |first7=Selma F. |last8=Azziz |first8=Ricardo }}</ref><ref name="pmid20671993">{{cite journal |title=Nonclassic congenital adrenal hyperplasia |journal=Int J Pediatr Endocrinol |volume=2010 |pages=625105 |year=2010 |pmid=20671993 |pmc=2910408 |doi=10.1155/2010/625105|doi-access=free |last1=Witchel |first1=Selma Feldman |last2=Azziz |first2=Ricardo }}</ref> The authors of that 2008 paper, therefore, concluded that CP/CPPS may be a consequence of a systemic condition of androgen excess but not a disease that originates in the prostate such as a localized prostate infection, inflammation, or dysfunction. We hypothesize that CYP21A2 deficiency in CP/CPPS may be associated with elevated androgens produced by pathways activated by such deficiency, i.e. backdoor pathway from P4 or 17-OHP to DHT and the pathways towards 11-oxygenated androgens. ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} 79jzvzrjscy95fvxruhqn3hnsaeof2o 2410938 2410906 2022-08-02T07:23:40Z Maneesh 2723004 /* C19 Steroids Entry Points */ wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> and the potential clinical relevance in conditions involving androgen biosynthesis was proposed. Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref><ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid11514561">{{cite journal|last1=Nahoum|first1=Virginie|last2=Gangloff|first2=Anne|last3=Legrand|first3=Pierre|last4=Zhu|first4=Dao-Wei|last5=Cantin|first5=Line|last6=Zhorov|first6=Boris S.|last7=Luu-The|first7=Van|last8=Labrie|first8=Fernand|last9=Breton|first9=Rock|year=2001|title=Structure of the human 3alpha-hydroxysteroid dehydrogenase type 3 in complex with testosterone and NADP at 1.25-A resolution|journal=J Biol Chem|volume=276|issue=45|pages=42091–8|doi=10.1074/jbc.M105610200|pmid=11514561|doi-access=free|last10=Lin|first10=Sheng-Xiang}}</ref><ref name="pmid18923939">{{cite journal|last1=Dozmorov|first1=Mikhail G.|last2=Yang|first2=Qing|last3=Matwalli|first3=Adam|last4=Hurst|first4=Robert E.|last5=Culkin|first5=Daniel J.|last6=Kropp|first6=Bradley P.|last7=Lin|first7=Hsueh-Kung|year=2007|title=5alpha-androstane-3alpha,17beta-diol selectively activates the canonical PI3K/AKT pathway: a bioinformatics-based evidence for androgen-activated cytoplasmic signaling|journal=Genomic Med|volume=1|issue=3–4|pages=139–46|doi=10.1007/s11568-008-9018-9|pmc=2269037|pmid=18923939}}</ref><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref><ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}}The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}}The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}}The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608" />{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}}While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780" /> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. 11-Oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid30959151">{{cite journal |title=Circulating 11-oxygenated androgens across species |journal=J Steroid Biochem Mol Biol |volume=190 |pages=242–249 |year=2019 |pmid=30959151 |pmc=6733521 |doi=10.1016/j.jsbmb.2019.04.005|last1=Rege |first1=Juilee |last2=Garber |first2=Scott |last3=Conley |first3=Alan J. |last4=Elsey |first4=Ruth M. |last5=Turcu |first5=Adina F. |last6=Auchus |first6=Richard J. |last7=Rainey |first7=William E. }}</ref><ref name="pmid27519632" /><ref name="pmid34171490" /><ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646" /> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="NBK557634">{{cite book|url=https://www.ncbi.nlm.nih.gov/books/NBK557634/|title=Biochemistry, Dihydrotestosterone|publisher=StatPearls|year=2022}}</ref><ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |pages=431 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|pages=868|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|pages=66|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=441 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref><ref name="pmid23386646" /> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid29277706" /><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405" /> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-), therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the conversion of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1.<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRDA51 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid18923939"/><ref name="Nishiyama2011"/><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels do not decline with age as T does. While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === Biological Role of 11-Oxygenated Androgens === 11-oxygenated androgens are produced in physiological quantities in healthy primate organisms (including humans).<ref name="pmid30959151" /><ref name="pmid30753518" /><ref name="pmid32629108" /> Humans have two isozymes with 11β-hydroxylase activity, encoded by the genes ''CYP11B1'' (regulated by the adrenocorticotropic hormone (ACTH)) and ''CYP11B2'' (regulated by angiotensin II).<ref name="pmid22217826">{{cite journal|name-list-style=vanc|title=Molecular biology of 11β-hydroxylase and 11β-hydroxysteroid dehydrogenase enzymes |journal=J Steroid Biochem Mol Biol |volume=43 |issue=8 |pages=827–35 |pmid=22217826 |doi=10.1016/0960-0760(92)90309-7 |s2cid=19379671 |last1=White |first1=Perrin C. |last2=Pascoe |first2=Leigh |last3=Curnow |first3=Kathleen M. |last4=Tannin |first4=Grace |last5=Rösler |first5=Ariel |year=1992 }}</ref> Since the first step in the biosynthesis of 11-oxygenated androgens involves 11β-hydroxylation of a steroid substrate by CYP11B1/2 isozymes that are generally associated with their expression in the adrenal gland, 11-oxygenated androgens are considered androgens of adrenal origin. They follow the circadian rhythm of cortisol but correlate very weakly with T.<ref name="pmid34867794">{{cite journal |title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=751191 |pmid=34867794 |pmc=8636728 |doi=10.3389/fendo.2021.751191 |doi-access=free |last1=Turcu |first1=Adina F. |last2=Mallappa |first2=Ashwini |last3=Nella |first3=Aikaterini A. |last4=Chen |first4=Xuan |last5=Zhao |first5=Lili |last6=Nanba |first6=Aya T. |last7=Byrd |first7=James Brian |last8=Auchus |first8=Richard J. |last9=Merke |first9=Deborah P. |year=2021 }}</ref><ref name="pmid34324429">{{cite journal|title=Circadian rhythms of 11-oxygenated C19 steroids and ∆5-steroid sulfates in healthy men |journal=Eur J Endocrinol |volume=185 |issue=4 |pages=K1–K6 |pmid=34324429 |doi=10.1530/EJE-21-0348 |pmc=8826489 |pmc-embargo-date=August 27, 2022 |last1=Turcu |first1=Adina F. |last2=Zhao |first2=Lili |last3=Chen |first3=Xuan |last4=Yang |first4=Rebecca |last5=Rege |first5=Juilee |last6=Rainey |first6=William E. |last7=Veldhuis |first7=Johannes D. |last8=Auchus |first8=Richard J. |year=2021 }}</ref> The levels of 11-oxygenated androgens raise after ACTH stimulation<ref name="pmid23386646"/><ref name="pmid13211802">{{cite journal |vauthors=DOBRINER K, KAPPAS A, GALLAGHER TF |title=Studies in steroid metabolism. XXVI. Steroid isolation studies in human leukemia |journal=J Clin Invest |volume=33 |issue=11 |pages=1481–6 |date=November 1954 |pmid=13211802 |pmc=1072573 |doi=10.1172/JCI103026 |url=}}</ref> that further supports their adrenal origin. However, in addition to the adrenal glands, CYP11B1 is also expressed in Leydig cells and ovarian theca cells, albeit at far lower levels, so the production of 11KT precursors may be one of the most important functions of 11β-hydroxylase activity in the gonads.<ref name="pmid27428878">{{cite journal|title=11-Ketotestosterone Is a Major Androgen Produced in Human Gonads |journal=J Clin Endocrinol Metab |volume=101 |issue=10 |pages=3582–3591 |pmid=27428878 |doi=10.1210/jc.2016-2311 |last1=Imamichi |first1=Yoshitaka |last2=Yuhki |first2=Koh-Ichi |last3=Orisaka |first3=Makoto |last4=Kitano |first4=Takeshi |last5=Mukai |first5=Kuniaki |last6=Ushikubi |first6=Fumitaka |last7=Taniguchi |first7=Takanobu |last8=Umezawa |first8=Akihiro |last9=Miyamoto |first9=Kaoru |last10=Yazawa |first10=Takashi |year=2016 }}</ref> In an in vitro study by Strushkevich et al. published in 2013, both isozymes have been shown to convert Δ⁴ steroids (P4, 17-OHP, A4 and T), but they are very specific to the configuration of the A-ring (carbon positions 1 to 5) of steroids, i.e., they cannot convert Δ⁵ steroids with a hydroxy group at the carbon position 3.<ref name="pmid23322723">{{cite journal |pmc=5417327|year=2013|last1=Strushkevich|first1=N.|last2=Gilep|first2=A. A.|last3=Shen|first3=L.|last4=Arrowsmith|first4=C. H.|last5=Edwards|first5=A. M.|last6=Usanov|first6=S. A.|last7=Park|first7=H. W.|title=Structural Insights into Aldosterone Synthase Substrate Specificity and Targeted Inhibition|journal=Molecular Endocrinology (Baltimore, Md.)|volume=27|issue=2|pages=315–324|doi=10.1210/me.2012-1287|pmid=23322723}}</ref> The relative biological significance of the 11-oxygenated androgens has been subject to some debate. There have been enough studies to establish that 11OHA4 itself does not have any androgenic activity but an important precursor to 11KT and 11KDHT.{{cn}} Androgen activity has been established for 11KT, 11KDHT<ref name="pmid27442248" /> as well as 11OHT and 11OHDHT but circulating levels of all of these androgens have not been firmly established. 11KT may serve as a primary androgen for healthy women,<ref name="pmid32629108">{{cite journal|last1=Barnard|first1=Lise|last2=Nikolaou|first2=Nikolaos|last3=Louw|first3=Carla|last4=Schiffer|first4=Lina|last5=Gibson|first5=Hylton|last6=Gilligan|first6=Lorna C.|last7=Gangitano|first7=Elena|last8=Snoep|first8=Jacky|last9=Arlt|first9=Wiebke|year=2020|title=The A-ring reduction of 11-ketotestosterone is efficiently catalysed by AKR1D1 and SRD5A2 but not SRD5A1|url=|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=202|pages=105724|doi=10.1016/j.jsbmb.2020.105724|pmid=32629108|s2cid=220323715|last10=Tomlinson|first10=Jeremy W.|last11=Storbeck|first11=Karl-Heinz}}</ref><ref name="pmid30753518" /> as it circulates at similar levels to T, but unlike T, the levels of 11KT are stable across the menstrual cycle.<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref> There are conflicting reports on whether 11-oxygenated androgens decline in women with age, and whether the relative contribution of 11KT as compared with T is higher in postmenopausal women than in younger ones — Nanba et al. (2019)<ref name="pmid30753518" /> and Davio et al. (2020)<ref name="pmid32498089">{{cite journal|last1=Davio|first1=Angela|last2=Woolcock|first2=Helen|last3=Nanba|first3=Aya T.|last4=Rege|first4=Juilee|last5=o'Day|first5=Patrick|last6=Ren|first6=Jianwei|last7=Zhao|first7=Lili|last8=Ebina|first8=Hiroki|last9=Auchus|first9=Richard|year=2020|title=Sex Differences in 11-Oxygenated Androgen Patterns Across Adulthood|journal=J Clin Endocrinol Metab|volume=105|issue=8|pages=e2921–e2929|doi=10.1210/clinem/dgaa343|pmc=7340191|pmid=32498089|last10=Rainey|first10=William E.|last11=Turcu|first11=Adina F.}}</ref> found that 11KT do not decline with age in women, however, Skiba et al. (2019)<ref name="pmid31390028" /> reported that the levels do decline. The decline of circulating 11-androgens with age may be associated with declining levels of DHEA and A4, which serve as precursors. It should also be noted that about half of circulating A4 quantities and almost all DHEA quantities are of adrenal origin.<ref name="pmid25428847">{{cite journal |vauthors=Turcu A, Smith JM, Auchus R, Rainey WE |title=Adrenal androgens and androgen precursors-definition, synthesis, regulation and physiologic actions |journal=Compr Physiol |volume=4 |issue=4 |pages=1369–81 |date=October 2014 |pmid=25428847 |pmc=4437668 |doi=10.1002/cphy.c140006 |url=}}</ref> Therefore the decline of 11-oxygenated androgens - steroids of adrenal origin - may be in line with a decline with their adrenal androgen precursors - DHEA and A4. Studies to date also strongly suggest that 11KT is the primary active 11-oxygenated androgen. Notably, 11KDHT has been found to circulate at substantially lower levels than DHT at least in prostate cancer<ref name="pmid30472582"/> suggesting that the 5α-reduction of 11KT is not as physiologically relevant in that condition. While early studies did show that 11KT and other 11-oxygenated androgens are substrates for SRD5A1 and SRD5A2, these did not perform full kinetic analyses.<ref name="pmid23856005"/>. Barnard et al., in a study published in 2020 showed that SRD5A1 does not efficiently catalyze the 5α-reduction of 11KT or 11KA4,<ref name="pmid32629108"/> confirming that 11KT may be the more relevant active 11-oxygenated androgen given the abundant peripheral expression of SRD5A2. While this does not rule out the potential for 11KDHT to be produced by SRD5A2 (or to a lesser degree by SRD5A1) in specific tissues, current evidence does suggest that the emphasis should be taken off the production of 11KDHT and rather placed on 11KT. As for the measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal |title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report |journal=Int J Mol Sci |year=2020 |volume=21 |issue=13 |pmid=32610579 |pmc=7369945 |doi=10.3390/ijms21134622 |doi-access=free |last1=Sumińska |first1=Marta |last2=Bogusz-Górna |first2=Klaudia |last3=Wegner |first3=Dominika |last4=Fichna |first4=Marta |page=4622 }}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal | last1=Yildiz | first1=Bulent O. | title=Diagnosis of hyperandrogenism: clinical criteria | journal=Best Practice & Research. Clinical Endocrinology & Metabolism | publisher=Elsevier BV | volume=20 | issue=2 | year=2006 | issn=1521-690X | pmid=16772149 | doi=10.1016/j.beem.2006.02.004 | pages=167–176}}</ref><ref name="pmid24184282">{{cite journal | last1=Peigné | first1=Maëliss | last2=Villers-Capelle | first2=Anne | last3=Robin | first3=Geoffroy | last4=Dewailly | first4=Didier | title=Hyperandrogénie féminine | journal=Presse Medicale (Paris, France) | publisher=Elsevier BV | volume=42 | issue=11 | year=2013 | issn=0755-4982 | pmid=24184282 | doi=10.1016/j.lpm.2013.07.016 | pages=1487–1499 | s2cid=28921380 | language=fr}}</ref> Relying on T levels alone in conditions associated with hyperandrogenism may read to diagnostic pitfalls and confusion.<ref name="pmid32610579"/> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|title=Clinical significance of 11-oxygenated androgens |journal=Curr Opin Endocrinol Diabetes Obes |volume=24 |issue=3 |pages=252–259 |pmid=28234803 |pmc=5819755 |doi=10.1097/MED.0000000000000334 |last1=Turcu |first1=Adina F. |last2=Auchus |first2=Richard J. |year=2017 }}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome |journal=Reprod Biomed Online |volume=41 |issue=4 |pages=734–742 |pmid=32912651 |doi=10.1016/j.rbmo.2020.07.013 |s2cid=221625488 |last1=Yang |first1=Yabo |last2=Ouyang |first2=Nengyong |last3=Ye |first3=Yang |last4=Hu |first4=Qin |last5=Du |first5=Tao |last6=Di |first6=Na |last7=Xu |first7=Wenming |last8=Azziz |first8=Ricardo |last9=Yang |first9=Dongzi |last10=Zhao |first10=Xiaomiao |year=2020 }}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803"/> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by very potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal | last1=Balsamo | first1=Antonio | last2=Baronio | first2=Federico | last3=Ortolano | first3=Rita | last4=Menabo | first4=Soara | last5=Baldazzi | first5=Lilia | last6=Di Natale | first6=Valeria | last7=Vissani | first7=Sofia | last8=Cassio | first8=Alessandra | title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant | journal=Frontiers in Pediatrics | year=2020 | publisher=Frontiers Media SA | volume=8 | page=593315 | issn=2296-2360 | pmid=33415088 | pmc=7783414 | doi=10.3389/fped.2020.593315| doi-access=free }}</ref><ref name="pmid29277706">{{cite journal | last1=Kamrath | first1=Clemens | last2=Wettstaedt | first2=Lisa | last3=Boettcher | first3=Claudia | last4=Hartmann | first4=Michaela F. | last5=Wudy | first5=Stefan A. | title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia | journal=The Journal of Steroid Biochemistry and Molecular Biology | publisher=Elsevier BV | volume=178 | year=2018 | issn=0960-0760 | pmid=29277706 | doi=10.1016/j.jsbmb.2017.12.016 | pages=221–228| s2cid=3709499 }}</ref> Another issue with the use of T as a biomarker of androgen excess is the low circulating levels in women and the specificity and sensitivity of the assays used.<ref name="pmid29306916">{{cite journal |title=Falsely elevated plasma testosterone concentrations in neonates: importance of LC-MS/MS measurements |journal=Clin Chem Lab Med |volume=56 |issue=6 |pages=e141–e143 |pmid=29306916 |doi=10.1515/cclm-2017-1028 |last1=Hamer |first1=Henrike M. |last2=Finken |first2=Martijn J.J. |last3=Van Herwaarden |first3=Antonius E. |last4=Du Toit |first4=Therina |last5=Swart |first5=Amanda C. |last6=Heijboer |first6=Annemieke C. |year=2018 |hdl=10019.1/106715 |s2cid=13917408 }}</ref><ref name="pmid32912651" /><ref name="pmid30753518">{{cite journal|last1=Nanba|first1=Aya T.|last2=Rege|first2=Juilee|last3=Ren|first3=Jianwei|last4=Auchus|first4=Richard J.|last5=Rainey|first5=William E.|last6=Turcu|first6=Adina F.|year=2019|title=11-Oxygenated C19 Steroids Do Not Decline With Age in Women|journal=J Clin Endocrinol Metab|volume=104|issue=7|pages=2615–2622|doi=10.1210/jc.2018-02527|pmc=6525564|pmid=30753518}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterised, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women |journal=Fertil Steril |volume=58 |issue=1 |pages=148–52 |pmid=1623996 |doi=10.1016/s0015-0282(16)55152-8 |last1=Carmina |first1=E. |last2=Stanczyk |first2=F. Z. |last3=Chang |first3=L. |last4=Miles |first4=R. A. |last5=Lobo |first5=R. A. |year=1992 }}</ref><ref name="pmid14417423">{{cite journal |title=Urinary ketosteroids and pregnanetriol in hirsutism |journal=J Clin Endocrinol Metab |volume=20 |issue= 2|pages=180–6 |pmid=14417423 |doi=10.1210/jcem-20-2-180|last1=Lipsett |first1=Mortimer B. |last2=Riter |first2=Barbara |year=1960 }}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries |journal=J Clin Endocrinol Metab |volume=66 |issue=5 |pages=946–50 |pmid=3129451 |doi=10.1210/jcem-66-5-946 |last1=Polson |first1=D. W. |last2=Reed |first2=M. J. |last3=Franks |first3=S. |last4=Scanlon |first4=M. J. |last5=James |first5=V. H. T. |year=1988 }}</ref> However, due to to conflicting reports ratios did not find a firm footing in the clinical as a diagnostic tool. === On The Aromatization of Androgens === Unlike T and A4, 11-oxygenated androgens are unlikely to be converted by aromatase into estrogens ''in vivo'',<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> that was first predicted in 2016 by Imamichi at al. in an ''in vitro'' study.<ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women when not taking into account DHEA. However, DHEA has a very low affinity for the androgen receptor and thus should not be an important contributor, if at all, for receptor activation under normal conditions.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, Barnard et al., incubating ''in vitro'' three different aromatase-expressing cell cultures and ''ex vivo'' human placenta explant cultures with normal and radiolabeled steroids, detected conversion of 11-oxygenated and conventional androgens into 11-oxygenated estrogens; however, 11-oxygenated estrogens were not detected ''in vivo'': neither in pregnant women who have high aromatase expression nor in patients who have high 11-androgens levels due to CAH or adrenocortical carcinoma, probably due to relatively low aromatase activity towards 11-oxygenated androgens compared to conventional androgens.<ref name="pmid33340399"/> However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Therefore, the role of DHT and 11-oxygenated androgens should be seriously considered in women patients. === Disorders of Sex Development === Since both the canonical and backdoor pathways of androgen biosynthesis towards DHT lead to early male sexual differentiation<ref name="pmid30763313" /><ref name="pmid30943210">{{cite journal|title = The "backdoor pathway" of androgen synthesis in human male sexual development | journal = PLOS Biology | volume = 17 | issue = 4 | pages = e3000198 | pmid = 30943210 | pmc = 6464227 | doi = 10.1371/journal.pbio.3000198 | last1 = Miller | first1 = Walter L. | last2 = Auchus | first2 = Richard J. |year = 2019 }}</ref><ref name="pmid11035809" /><ref name="pmid15249131" /> and are required for normal human male genital development,<ref name="pmid30943210" /><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of male fetuses,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 acts as a substrate during the biosynthesis of DHT in the backdoor pathway.<ref name="pmid30763313"/> Flück et al. described in 2011 a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. These findings confirmed that DHT produced in a backdoor pathway, while not necessary for the sexual development of females, is important for that of males. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were enough to disrupt it.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). As such, this syndrome leads to DSD in both sexes, while affected girls go usually unrecognized until puberty, when they show amenorrhea. This syndrome is also rare with only a few cases reported.<ref name="pmid34711511"/> Besides that, 11-oxygenated androgens may play previously overlooked role in DSD.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> === Congenital Adrenal Hyperplasia === Another cause of androgen excess is congenital adrenal hyperplasia (CAH), a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal |vauthors=El-Maouche D, Arlt W, Merke DP |title=Congenital adrenal hyperplasia |journal=Lancet |volume=390 |issue=10108 |pages=2194–2210 |date=November 2017 |pmid=28576284 |doi=10.1016/S0140-6736(17)31431-9 |url=}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal |vauthors=Speiser PW, White PC |title=Congenital adrenal hyperplasia |journal=N Engl J Med |volume=349 |issue=8 |pages=776–88 |date=August 2003 |pmid=12930931 |doi=10.1056/NEJMra021561 |url=}}</ref><ref name="pmid30272171">{{cite journal | title = Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline | journal = The Journal of Clinical Endocrinology and Metabolism | volume = 103 | issue = 11 | pages = 4043–4088 | year = 2018 | pmid = 30272171 | pmc = 6456929 | doi = 10.1210/jc.2018-01865 }}</ref> Such deficiency leads to an excessive accumulation of a respective cortisol precursor, that becomes to serve as a substrate to androgens. In CYP21A2 deficiency<ref name="pmid22170725" /> including the mild forms (which are not always diagnosed)<ref name="pmid32966723">{{cite journal |vauthors=Merke DP, Auchus RJ |title=Congenital Adrenal Hyperplasia Due to 21-Hydroxylase Deficiency |journal=N Engl J Med |volume=383 |issue=13 |pages=1248–1261 |date=September 2020 |pmid=32966723 |doi=10.1056/NEJMra1909786 |url=}}</ref><ref name="pmid31499506">{{cite book|title=Hyperandrogenism in Women|last1=Pignatelli|first1=Duarte|last2=Pereira|first2=Sofia S.|last3=Pasquali|first3=Renato|year=2019|isbn=978-3-318-06470-4|series=Frontiers of Hormone Research|volume=53|pages=65–76|chapter=Androgens in Congenital Adrenal Hyperplasia|doi=10.1159/000494903|pmid=31499506|s2cid=202412336}}</ref> or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> elevated 17-OHP levels starts the backdoor pathway to DHT. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> The reason why 17-OHP serves as a prerequisite substrate for DHT within the backdoor pathway roundabout of T rather then an immediate substrate within the Δ⁴ pathway for A4, and then T, is because the catalytic activity 17,20-lyase reaction (which cleaves a side-chain from the steroid nucleus converting a pregnane to an androstane (androgen), i.e., from 17OPH5 to DHEA; from 17-OHP to A4) performed by CYP17A1 in humans is approximately 100 times more efficient in the Δ⁵ pathway than in the Δ⁴ pathway. Therefore, the catalytic efficiency of CYP17A1 for 17-OHP is about 100 times lower than for 17OHP5, resulting in negligible A4 being produced from 17-OHP in the Δ⁴ reaction pathway in humans.<ref name="pmid8325965">{{cite journal|last1=Swart|first1=P.|last2=Swart|first2=A. C.|last3=Waterman|first3=M. R.|last4=Estabrook|first4=R. W.|last5=Mason|first5=J. I.|year=1993|title=Progesterone 16 alpha-hydroxylase activity is catalyzed by human cytochrome P450 17 alpha-hydroxylase|journal=J Clin Endocrinol Metab|volume=77|issue=1|pages=98–102|doi=10.1210/jcem.77.1.8325965|pmid=8325965}}</ref><ref name="pmid12915666">{{cite journal|last1=Flück|first1=Christa E.|last2=Miller|first2=Walter L.|last3=Auchus|first3=Richard J.|year=2003|title=The 17, 20-lyase activity of cytochrome CYP17A1 from human fetal testis favors the delta5 steroidogenic pathway|url=https://pubmed.ncbi.nlm.nih.gov/12915666|journal=The Journal of Clinical Endocrinology and Metabolism|volume=88|issue=8|pages=3762–3766|doi=10.1210/jc.2003-030143|issn=0021-972X|pmid=12915666}}</ref><ref name="pmid15774560">{{cite journal|last1=Miller|first1=Walter L.|year=2005|title=Minireview: regulation of steroidogenesis by electron transfer|url=https://pubmed.ncbi.nlm.nih.gov/15774560|journal=Endocrinology|volume=146|issue=6|pages=2544–2550|doi=10.1210/en.2005-0096|issn=0013-7227|pmid=15774560}}</ref><ref name="pmid32007561"/> The accumulation of 17-OHP in CYP21A2 deficiency in CAH can be attributed to the fact that the primary enzyme for 17-OHP in normal conditions is CYP21A2, that is expressed in the adrenal and not the gonads.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|year=2019 |volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086 }}</ref><ref name="pmid26038201"/> In a 1998 study, Auchus et al. demonstrated that human CYP17A1 efficiently catalyzed the conversion of P4 to 17-OHP, but the conversion of 17-OHP to A4 was much less efficient than the corresponding conversion of 17OHP5 to DHEA.<ref name="pmid9452426"/> In rodents, quite contrary, the conversion of 17-OHP to A4 is very efficient.<ref name="pmid9452426">{{cite journal | last1=Auchus | first1=Richard J. | last2=Lee | first2=Tim C. | last3=Miller | first3=Walter L. | title=Cytochrome b 5 Augments the 17,20-Lyase Activity of Human P450c17 without Direct Electron Transfer | journal=The Journal of Biological Chemistry | year=1998 | publisher=Elsevier BV | volume=273 | issue=6 | issn=0021-9258 | pmid=9452426 | doi=10.1074/jbc.273.6.3158 | pages=3158–3165| doi-access=free }}</ref> This explains significant accumulation of 17-OHP in CYP21A2 deficiency or POR deficiency in humans, so that 17-OHP, while not 21-hydroxylated in sufficient quantities, and being better a substrate for 5α-reductase than for CYP17A1, is 5α-reduced serving as the prerequisite for this backdoor pathway. Hence, fetal excess of 17-OHP in CAH may provoke activation of this pathway to DHT and lead to external genital virilization in newborn girls, thus explaining DSD in girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025"/><ref name="pmid31505456">{{cite journal |vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE |title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia |journal=Eur J Endocrinol |volume=181 |issue=5 |pages=481–488 |date=November 2019 |pmid=31505456 |doi=10.1530/EJE-19-0379 |url=}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal |vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M |title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia |journal=J Obstet Gynaecol Res |volume=43 |issue=5 |pages=946–950 |date=May 2017 |pmid=28188961 |doi=10.1111/jog.13288 |url=}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal | last1=Turcu | first1=Adina F | last2=Mallappa | first2=Ashwini | last3=Elman | first3=Meredith S | last4=Avila | first4=Nilo A | last5=Marko | first5=Jamie | last6=Rao | first6=Hamsini | last7=Tsodikov | first7=Alexander | last8=Auchus | first8=Richard J | last9=Merke | first9=Deborah P | title = 11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency | journal = The Journal of Clinical Endocrinology and Metabolism | year=2017 | volume = 102 | issue = 8 | pages = 2701–2710 | pmid = 28472487 | pmc = 5546849 | doi = 10.1210/jc.2016-3989}}</ref><ref name="pmid26865584">{{cite journal|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency |journal=Eur J Endocrinol |volume=174 |issue=5 |pages=601–9 |pmid=26865584 |pmc=4874183 |doi=10.1530/EJE-15-1181 |last1=Turcu |first1=Adina F. |last2=Nanba |first2=Aya T. |last3=Chomic |first3=Robert |last4=Upadhyay |first4=Sunil K. |last5=Giordano |first5=Thomas J. |last6=Shields |first6=James J. |last7=Merke |first7=Deborah P. |last8=Rainey |first8=William E. |last9=Auchus |first9=Richard J. |year=2016 }}</ref><ref name="pmid29718004">{{cite journal|title = Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency | journal = Current Opinion in Endocrinology, Diabetes, and Obesity | volume = 25 | issue = 3 | pages = 178–184 | pmid = 29718004 | doi = 10.1097/MED.0000000000000402 | s2cid = 26072848 |last1 = White |first1 = Perrin C. |year = 2018 }}</ref><ref name="pmid34867794"/> In males with CAH, 11-oxygenated androgens may lead to devlopment of testicular adrenal rest tumors.<ref name="pmid25850025">{{cite journal|pmc=4454804|year=2015|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|last10=Auchus|first10=R. J.|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmid=25850025}}</ref><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors |journal=J Clin Endocrinol Metab |volume=107 |issue=1 |pages=e272–e280 |pmid=34390337 |pmc=8684463 |doi=10.1210/clinem/dgab598 |last1=Schröder |first1=Mariska A M. |last2=Turcu |first2=Adina F. |last3=o'Day |first3=Patrick |last4=Van Herwaarden |first4=Antonius E. |last5=Span |first5=Paul N. |last6=Auchus |first6=Richard J. |last7=Sweep |first7=Fred C G J. |last8=Claahsen-Van Der Grinten |first8=Hedi L. |year=2022 }}</ref> The biosynthesis of 11OHP4 from P4 and 21dF from 17-OHP by CYP11B1/2 in CAH may be attributed to CYP21A2 deficiency resulting in increased P4 and 17-OHP concentrations and, together with the unavailability of CYP11B1/2's main substrates, 11-deoxycortisol (11dF) and 11-deoxycorticosterone (DOC), drive the production of 11-oxygenated pregnanes.<ref name="pmid3546944" /> We have reasons to believe that this may be aggravated by elevated ACTH due to a feedback loop in hypothalamic-pituitary-adrenal axis caused by impaired cortisol synthesis associated with CYP21A2 deficiency; higher ACTH causes higher CYP11B1 expression. Multiple studies demonstrated that in CAH due to CYP21A2 deficiency, both 21dF levels<ref name="pmid4372245">{{cite journal |title=Plasma 17-hydroxyprogesterone, 21-deoxycortisol and cortisol in congenital adrenal hyperplasia |journal=J Clin Endocrinol Metab |volume=39 |issue=6 |pages=1099–102 |year=1974 |pmid=4372245 |doi=10.1210/jcem-39-6-1099 |last1=Franks |first1=Robert C. }}</ref><ref name="pmid476971">{{cite journal |title=Rapid assay of plasma 21-deoxycortisol and 11-deoxycortisol in congenital adrenal hyperplasia |journal=Clin Endocrinol (Oxf) |volume=10 |issue=4 |pages=367–75 |year=1979 |pmid=476971 |doi=10.1111/j.1365-2265.1979.tb02091.x |url=|last1=Fukushima |first1=D. K. |last2=Nishina |first2=T. |last3=Wu |first3=R. H. K. |last4=Hellman |first4=L. |last5=Finkelstein |first5=J. W. |s2cid=2979354 }}</ref><ref name="pmid6090811">{{cite journal |title=Development of plasma 21-deoxycortisol radioimmunoassay and application to the diagnosis of patients with 21-hydroxylase deficiency |journal=J Steroid Biochem |volume=21 |issue=2 |pages=185–91 |year=1984 |pmid=6090811 |doi=10.1016/0022-4731(84)90382-0 |last1=Milewicz |first1=A. |last2=Vecsei |first2=P. |last3=Korth-Schütz |first3=S. |last4=Haack |first4=D. |last5=Rösler |first5=A. |last6=Lichtwald |first6=K. |last7=Lewicka |first7=S. |last8=Mittelstaedt |first8=G.v. }}</ref><ref name="pmid2986404">{{cite journal |title=Radioimmunoassay for 21-deoxycortisol: clinical applications |journal=Acta Endocrinol (Copenh) |volume=108 |issue=4 |pages=537–44 |year=1985 |pmid=2986404 |doi=10.1530/acta.0.1080537 |last1=Gueux |first1=B. |last2=Fiet |first2=J. |last3=Pham-Huu-Trung |first3=M. T. |last4=Villette |first4=J. M. |last5=Gourmelen |first5=M. |last6=Galons |first6=H. |last7=Brerault |first7=J. L. |last8=Vexiau |first8=P. |last9=Julien |first9=R. }}</ref><ref name="pmid25850025" /> and 11OPH4 levels<ref name="pmid3546944">{{cite journal |last1=Gueux |first1=Bernard |last2=Fiet |first2=Jean |last3=Galons |first3=Hervé |last4=Boneté |first4=Rémi |last5=Villette |first5=Jean-Marie |last6=Vexiau |first6=Patrick |last7=Pham-Huu-Trung |first7=Marie-Thérèse |last8=Raux-Eurin |first8=Marie-Charles |last9=Gourmelen |first9=Micheline |last10=Brérault |first10=Jean-Louis |last11=Julien |first11=René |last12=Dreux |first12=Claude |title=The measurement of 11β-hydroxy-4-pregnene-3,20-dione (21-Deoxycorticosterone) by radioimmunoassay in human plasma |journal=Journal of Steroid Biochemistry |year=1987 |volume=26 |issue=1 |pages=145–150 |doi=10.1016/0022-4731(87)90043-4 |pmid=3546944 }}</ref><ref name="pmid2537337">{{cite journal |last1=Fiet |first1=Jean |last2=Gueux |first2=Bernard |last3=Rauxdemay |first3=Marie-Charles |last4=Kuttenn |first4=Frederique |last5=Vexiau |first5=Patrick |last6=Brerault |first6=Jeanlouis |last7=Couillin |first7=Philippe |last8=Galons |first8=Herve |last9=Villette |first9=Jeanmarie |last10=Julien |first10=Rene |last11=Dreux |first11=Claude |title=Increased Plasma 21-Deoxycorticosterone (21-DB) Levels in Late-Onset Adrenal 21-Hydroxylase Deficiency Suggest a Mild Defect of the Mineralocorticoid Pathway |journal=The Journal of Clinical Endocrinology & Metabolism |year=1989 |volume=68 |issue=3 |pages=542–547 |doi=10.1210/jcem-68-3-542 |pmid=2537337 }}</ref><ref name="pmid29264476">{{cite journal |last1=Fiet |first1=Jean |last2=Le Bouc |first2=Yves |last3=Guéchot |first3=Jérôme |last4=Hélin |first4=Nicolas |last5=Maubert |first5=Marie-Anne |last6=Farabos |first6=Dominique |last7=Lamazière |first7=Antonin |title=A Liquid Chromatography/Tandem Mass Spectometry Profile of 16 Serum Steroids, Including 21-Deoxycortisol and 21-Deoxycorticosterone, for Management of Congenital Adrenal Hyperplasia |journal=Journal of the Endocrine Society |year=2017 |volume=1 |issue=3 |pages=186–201 |doi=10.1210/js.2016-1048 |pmid=29264476 |pmc=5686660 }}</ref><ref name="pmid31821037">{{cite journal |title=Interaction between accumulated 21-deoxysteroids and mineralocorticoid signaling in 21-hydroxylase deficiency |journal=Am J Physiol Endocrinol Metab |volume=318 |issue=2 |pages=E102–E110 |year=2020 |pmid=31821037 |doi=10.1152/ajpendo.00368.2019 |last1=Travers |first1=Simon |last2=Bouvattier |first2=Claire |last3=Fagart |first3=Jérôme |last4=Martinerie |first4=Laetitia |last5=Viengchareun |first5=Say |last6=Pussard |first6=Eric |last7=Lombès |first7=Marc |s2cid=209314028 }}</ref> are increased. It was Robert Franks in who first published a study, in 1974, that compared 21dF levels of CAH patients with those of healthy controls. He measured 21dF plasma levels in twelve CAH patients before treatment, three after treatment, and four healthy controls following ACTH administration. Mean values of 21dF in CAH patients was 88 ng/ml while in healthy controls it was not detected. In untreated patients, values decreased after therapy. Even that, there were earlier reports about unique cases where 21dF was detected in CAH patients, but without direct comparison to healthy controls.<ref name="pmid5845501">{{cite journal |title=Detection of 21-deoxycortisol in blood from a patient with congenital adrenal hyperplasia |journal=Metabolism |year=1965 |volume=14 |issue=12 |pages=1276–81 |pmid=5845501 |doi=10.1016/s0026-0495(65)80008-7|last1=Wieland |first1=Ralph G. |last2=Maynard |first2=Donald E. |last3=Riley |first3=Thomas R. |last4=Hamwi |first4=George J. }}</ref><ref name="pmid13271547">{{cite journal|title=17alpha-hydroxyprogesterone and 21-desoxyhydrocortisone; their metabolism and possible role in congenital adrenal virilism |journal=J Clin Invest |volume=34 |issue=11 |pages=1639–46 |year=1955 |pmid=13271547 |pmc=438744 |doi=10.1172/JCI103217|last1=Jailer |first1=Joseph W. |last2=Gold |first2=Jay J. |last3=Vande Wiele |first3=Raymond |last4=Lieberman |first4=Seymour }}</ref> As for 11OHP4, it were Gueux et al. who first demonstrated, in 1987, elevated plasma levels of 11OHP4 in CAH. In that study, in treated classical CAH patients, some of which had salt-wasting form, mean levels of 11OHP4 (5908.7 pmol/l) were 332 times higher than in healthy controls (17.8 pmol/l). There was no difference in 11OHP4 in healthy controls depending on sex or phase of a menstrual cycle; ACTH stimulation in those control increased 11OHP4 four- to six-fold, while dexamethasone 1 mg at midnight decreased 11OHP4 to almost undetectable levels 12 hours later. Therefore, the authors hypothesized that at least in healthy people 11OHP4 is biosythesized exclusively in the adrenal, while gonads are not involved.<ref name="pmid3546944" /> Nevertheless, in studies focusing on CAH caused by CYP21A2 deficiency, 11OHP4 received less attention than 21dF.<ref name="pmid29277707"/> However, it was not until 2017 when 11OHP4 or 21dF were viewed as potential substrates in pathways towards potent 11-ogygenated androgens in ''in vitro'' studies.<ref name="pmid32007561"/><ref name="pmid29277707"/> In a 2016 study, Turcu et al. showed that in classic CAH due to CYP21A2 deficiency, in male and female patients who received glucocorticoid therapy, both conventional and 11-oxygenated androgens were elevated 3-4 fold compared to healthy controls. The exceptions were dehydroepiandrosterone (DHEA), dehydroepiandrosterone sulfate (DHEA-S), and androstenediol sulfate (A5-S), whose levels were 6.0, 7.5, and 9.4 times lower, respectively, in the patients with the condition compared to healthy controls, due to suppression by glucocorticoid treatment. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in a backdoor pathway from 17-OHP or P4 as consequence of abnormally upregulated SRD5A1.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Besides that, the levels of 11OHA4 and 11KA4 correlated with the levels of markers of insulin resistance; therefore, the study suggests that androgen excess precedes androgen-driven insulin resistance in PCOS.<ref name="pmid27901631" /> While earlier studies had commonly only measured 11OHA4 or 11OHAST and 11β-hydroxyetiocholanolone (11OHEt), urinary metabolites of 11OHA,<ref name="pmid33539964" /> while 11OHEt is also a metabolite of cortisol,<ref name="pmid31362062">{{cite journal |title=Human steroid biosynthesis, metabolism and excretion are differentially reflected by serum and urine steroid metabolomes: A comprehensive review |journal=J Steroid Biochem Mol Biol |volume=194 |issue= |pages=105439 |year=2019 |pmid=31362062 |pmc=6857441 |doi=10.1016/j.jsbmb.2019.105439 |url=|last1=Schiffer |first1=Lina |last2=Barnard |first2=Lise |last3=Baranowski |first3=Elizabeth S. |last4=Gilligan |first4=Lorna C. |last5=Taylor |first5=Angela E. |last6=Arlt |first6=Wiebke |last7=Shackleton |first7=Cedric H.L. |last8=Storbeck |first8=Karl-Heinz }}</ref><ref name="pmid27845856">{{cite journal |title=Modified-Release and Conventional Glucocorticoids and Diurnal Androgen Excretion in Congenital Adrenal Hyperplasia |journal=J Clin Endocrinol Metab |volume=102 |issue=6 |pages=1797–1806 |year=2017 |pmid=27845856 |pmc=5470768 |doi=10.1210/jc.2016-2855|last1=Jones |first1=Christopher M. |last2=Mallappa |first2=Ashwini |last3=Reisch |first3=Nicole |last4=Nikolaou |first4=Nikolaos |last5=Krone |first5=Nils |last6=Hughes |first6=Beverly A. |last7=o'Neil |first7=Donna M. |last8=Whitaker |first8=Martin J. |last9=Tomlinson |first9=Jeremy W. |last10=Storbeck |first10=Karl-Heinz |last11=Merke |first11=Deborah P. |last12=Ross |first12=Richard J. |last13=Arlt |first13=Wiebke }}</ref> more recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Another study reported that 11OHT was the only significantly elevated 11-oxygeated androgen in PCOS and together with 11KT, correlated with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227"/> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Premature Adrenarche === In a 2018 study, Rege et al. demonstrated that levels of 11KT in girls aged between 4 and 7 years during normal adrenarche (healthy controls) exceeded those of T by 2.43 times, and in those with premature adrenarche by 3.48 times. However, the levels of T in girls with premature adrenarche were higher by just 13% compared to age-matched healthy controls.<ref name="pmid30137510">{{cite journal | last1=Rege | first1=Juilee | last2=Turcu | first2=Adina | last3=Kasa-Vubu | first3=Josephine Z | last4=Lerario | first4=Antonio M | last5=Auchus | first5=Gabriela C | last6=Auchus | first6=Richard J | last7=Smith | first7=Joshua M | last8=White | first8=Perrin C | last9=Rainey | first9=William E | title=11KT is the dominant circulating bioactive androgen during normal and premature adrenarche | journal=The Journal of Clinical Endocrinology and Metabolism | year=2018 | publisher=The Endocrine Society | volume=103 | issue=12 | pages=4589–4598 | issn=0021-972X | pmid=30137510 | pmc=6226603 | doi=10.1210/jc.2018-00736 }}</ref> === Prostate Cancer === In some cases of advanced prostate cancer, androgen deprivation therapy related to gonadal T depletion does not produce long-term effects, and metastatic tumors may develop into castration-resistant prostate cancer (CRPC). The development of CRPC depends on adrenal precursor steroids to produce DHT in the tumor in a metabolic pathway called the "5α-dione" pathway - the pathway in which T is not involved. SRD5A1, the expression of which increases in CRPC, reduces A4 to 5α-dione, which is then converted to DHT.<ref name="pmid21795608"/><ref name="pmid31900912"/> Therefore, the DHT produced within the "5α-dione" pathway hampers the androgen deprivation therapy. Although blood levels of T are decreased by 90-95% in men whose testicles have been removed, DHT in the prostate is only decreased by 50%, thus indicating the presence of a metabolic pathway in the prostate that does not require testicular T to produce DHT.<ref name="pmid18471780">{{cite journal | last1=Luu-The | first1=Van | last2=Bélanger | first2=Alain | last3=Labrie | first3=Fernand | title=Androgen biosynthetic pathways in the human prostate | journal=Best Practice & Research. Clinical Endocrinology & Metabolism | publisher=Elsevier BV | volume=22 | issue=2 | year=2008 | issn=1521-690X | pmid=18471780 | doi=10.1016/j.beem.2008.01.008 | pages=207–221}}</ref> Chang et al., incubating six established human prostate cancer cell lines from patients with CRPC in the presence of radiolabeled A4, showed in their experiment published in 2011<ref name="pmid21795608">{{cite journal|last1=Chang | first1=K.-H. | last2=Li | first2=R. | last3=Papari-Zareei | first3=M. | last4=Watumull | first4=L. | last5=Zhao | first5=Y. D. | last6=Auchus | first6=R. J. | last7=Sharifi | first7=N. | title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer | journal=Proceedings of the National Academy of Sciences of the United States of America |year=2011 | publisher=Proceedings of the National Academy of Sciences | volume=108 | issue=33 | issn=0027-8424 | pmid=21795608 | pmc=3158152 | doi=10.1073/pnas.1107898108 | pages=13728–13733|bibcode=2011PNAS..10813728C |doi-access=free }}</ref> the presence of this pathway to DHT which bypasses T and they called this the "alternative" pathway, that became later commonly called as the "5α-dione" pathway.<ref name="pmid23856005"/> The authors demonstrated that this was the dominant pathway in prostate cancer (over the direct conversion of A4 to T) with SRD5A1 (which is upregulated in prostate cancer) first converting A4 to androstanedione (5α-dione), also known as 5α-androstane-3,17-dione, and then HSD17B3 / AKR1C3 converting 5α-dione to DHT (not necessarily via AST and 3α-diol). The study also found that the SRD5A2 is not involved in this "alternative" pathway.<ref name="pmid31900912"/> Therefore, the study showed the importance of taking into consideration this "alternative" pathway in selecting drugs that inhibit 5α-reductase activity.<ref name="pmid21901017">{{cite journal |title=Prostate cancer: DHT bypasses testosterone to drive progression to castration resistance |journal=Nat Rev Urol |volume=8 |issue=9 |pages=470 |year=September 2011 |pmid=21901017 |doi=10.1038/nrurol.2011.122 }}</ref><ref name="pmid22064602" /><ref name="pmid22336886">{{cite journal |title=Dihydrotestosterone synthesis from adrenal precursors does not involve testosterone in castration-resistant prostate cancer |journal=Cancer Biol Ther |volume=13 |issue=5 |pages=237–8 |year=2012 |pmid=22336886 |doi=10.4161/cbt.19608}}</ref> Another pathway that may be activated in CRPC, which may also hamper the androgen deprivation therapy, is the backdoor pathway from P4 to DHT. Chen et al. in a study published in 2014<ref name="pmid25320358">{{cite journal |vauthors=Chen EJ, Sowalsky AG, Gao S, Cai C, Voznesensky O, Schaefer R, Loda M, True LD, Ye H, Troncoso P, Lis RL, Kantoff PW, Montgomery RB, Nelson PS, Bubley GJ, Balk SP, Taplin ME |title=Abiraterone treatment in castration-resistant prostate cancer selects for progesterone responsive mutant androgen receptors |journal=Clin Cancer Res |volume=21 |issue=6 |pages=1273–80 |date=March 2015 |pmid=25320358 |pmc=4359958 |doi=10.1158/1078-0432.CCR-14-1220 |url=}}</ref> predicted that abiraterone, a CYP17A1 inhibitor, with about 6-fold more selective for inhibition of 17α-hydroxylase over 17,20-lyase,<ref name="pmid28890368">{{cite journal |vauthors=de Mello Martins AGG, Allegretta G, Unteregger G, Haupenthal J, Eberhard J, Hoffmann M, van der Zee JA, Junker K, Stöckle M, Müller R, Hartmann RW, Ohlmann CH |title=CYP17A1-independent production of the neurosteroid-derived 5α-pregnan-3β,6α-diol-20-one in androgen-responsive prostate cancer cell lines under serum starvation and inhibition by Abiraterone |journal=J Steroid Biochem Mol Biol |volume=174 |issue= |pages=183–191 |date=November 2017 |pmid=28890368 |doi=10.1016/j.jsbmb.2017.09.006 |url=}}</ref><ref name="pmid28373265">{{cite journal |vauthors=Petrunak EM, Rogers SA, Aubé J, Scott EE |title=Structural and Functional Evaluation of Clinically Relevant Inhibitors of Steroidogenic Cytochrome P450 17A1 |journal=Drug Metab Dispos |volume=45 |issue=6 |pages=635–645 |date=June 2017 |pmid=28373265 |pmc=5438109 |doi=10.1124/dmd.117.075317 |url=}}</ref><ref name="pmid29710837">{{cite journal |vauthors=Fernández-Cancio M, Camats N, Flück CE, Zalewski A, Dick B, Frey BM, Monné R, Torán N, Audí L, Pandey AV |title=Mechanism of the Dual Activities of Human CYP17A1 and Binding to Anti-Prostate Cancer Drug Abiraterone Revealed by a Novel V366M Mutation Causing 17,20 Lyase Deficiency |journal=Pharmaceuticals (Basel) |volume=11 |issue=2 |pages= |date=April 2018 |pmid=29710837 |pmc=6027421 |doi=10.3390/ph11020037 |url=}}</ref> although disrupting canonical androgen biosynthesis, while lowering levels of T, causes elevation of P4, that can be 5α-reduced hence start a backdoor pathway from P4 to DHT with roundabout of T.<ref name="pmid25320358"/> Besides that, in CRPC, 11-oxygenated androgens contribute significantly to the androgen pool.<ref name="pmid23856005"/><ref name="pmid31900912"/> 11-oxygenated androgens play a previously overlooked role in the reactivation of androgen signaling in CRPC,<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> because after eliminating testicular T biosynthesis by chemical or physical castration, CRPC has been shown to develop the ability to convert inactive circulating adrenal androgen precursors, DHEA and A4, to potent 11-oxygenated androgens in the 11-oxygenated pathway in addition to the 5α-dione pathway.<ref name="pmid31672619">{{cite journal |title=The role of adrenal derived androgens in castration resistant prostate cancer |journal=J Steroid Biochem Mol Biol |volume=197 |issue= |pages=105506 |year=2020 |pmid=31672619 |doi=10.1016/j.jsbmb.2019.105506|last1=Barnard |first1=Monique |last2=Mostaghel |first2=Elahe A. |last3=Auchus |first3=Richard J. |last4=Storbeck |first4=Karl-Heinz |pmc=7883395 }}</ref><ref name="pmid33974560" /> In a 2021 study, Snaterse et al. demonstrated that 11KT is the most circulating active androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool. They also demonstrated that 11KT levels are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> In a 2018 study by du Toit et al., the full range of androgen pathway metabolites have been shown in normal prostate and various prostate cancer cell models. 11OHA4 and 11OHT were both converted to potent androgens, 11KT and 11KDHT. Compared to T and DHT, 11-oxygenated androgens were the most predominant androgens. High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant. Furthermore, glucuronidation of the 11-oxygenated androgens is hampered by the presence of an oxo- or a hydroxy- group at position 11 of androgens in prostate cancer cell models while in prostate cancer patients' plasma 11KDHT was present only in the unconjugated form, with 11KT also predominantly unconjugated.<ref name="pmid28939401"/> Of all 11-oxygenated androgens in prostate cancer, it may be that 11KT is the primary active 11-oxygenated androgen, rather than 11KDHT, as observed in experiments on prostate tissue homogenates by Häkkinen et al. published in 2018<ref name="pmid30472582">{{cite journal|last1=Häkkinen|first1=Merja R.|last2=Murtola|first2=Teemu|last3=Voutilainen|first3=Raimo|last4=Poutanen|first4=Matti|last5=Linnanen|first5=Tero|last6=Koskivuori|first6=Johanna|last7=Lakka|first7=Timo|last8=Jääskeläinen|first8=Jarmo|last9=Auriola|first9=Seppo|year=2019|title=Simultaneous analysis by LC-MS/MS of 22 ketosteroids with hydroxylamine derivatization and underivatized estradiol from human plasma, serum and prostate tissue|journal=J Pharm Biomed Anal|volume=164|issue=|pages=642–652|doi=10.1016/j.jpba.2018.11.035|pmid=30472582|s2cid=53729550}}</ref> Ventura-Bahena et al., in a 2021 study, describing results of epidemiological studies related to androgens and prostate cancer that focused on specific androgen concentrations (such as T, A4, and DHEA) as inconsistent, hypothesized that the differences in androgen biosynthetic pathways rather than differences in specific androgen levels are associated with prostate cancer development.<ref name="pmid34520388"/> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but also in clinical conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> The contribution of the 11-oxygenated androgens, as well as the biosynthesis of 11-oxygenated pregnanes to active androgens via a backdoor pathway, have also been demonstrated in BPH cell models showing the conversion of 11OHP4 and 11KP4 in the backdoor pathway resulting in the production of 11KDHT. Backdoor pathway intermediates were also detected in BPH tissue as well as in circulation in BPH patients.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> In a paper published in 2008, Dimitrakov et al. hypothesized that CP/CPPS may be associated with a mild CYP21A2 deficiency, a cause of non-classic CAH that leads to androgen excesses.<ref name="pmid18308097"/> Non-classic CAH was generally thought to be asymptomatic in men.<ref name="pmid28582566">{{cite journal |title=Non-classic congenital adrenal hyperplasia due to 21-hydroxylase deficiency revisited: an update with a special focus on adolescent and adult women |journal=Hum Reprod Update |volume=23 |issue=5 |pages=580–599 |year=2017 |pmid=28582566 |doi=10.1093/humupd/dmx014 |last1=Carmina |first1=Enrico |last2=Dewailly |first2=Didier |last3=Escobar-Morreale |first3=Héctor F. |last4=Kelestimur |first4=Fahrettin |last5=Moran |first5=Carlos |last6=Oberfield |first6=Sharon |last7=Witchel |first7=Selma F. |last8=Azziz |first8=Ricardo }}</ref><ref name="pmid20671993">{{cite journal |title=Nonclassic congenital adrenal hyperplasia |journal=Int J Pediatr Endocrinol |volume=2010 |pages=625105 |year=2010 |pmid=20671993 |pmc=2910408 |doi=10.1155/2010/625105|doi-access=free |last1=Witchel |first1=Selma Feldman |last2=Azziz |first2=Ricardo }}</ref> The authors of that 2008 paper, therefore, concluded that CP/CPPS may be a consequence of a systemic condition of androgen excess but not a disease that originates in the prostate such as a localized prostate infection, inflammation, or dysfunction. We hypothesize that CYP21A2 deficiency in CP/CPPS may be associated with elevated androgens produced by pathways activated by such deficiency, i.e. backdoor pathway from P4 or 17-OHP to DHT and the pathways towards 11-oxygenated androgens. ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} 43r7scj3n71m1hn4auk1ihnmeqid6z6 0 271693 2410857 2410739 2022-08-02T00:40:52Z 2001:B042:4005:6BDE:D1CB:22BD:C6D8:F0A8 /* Condensed table */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b'' |- |2||1||11||2||3 |- |3||3||111||3||13 |- |4||5||221||3||41 |- |5||22||10₉₃13||96||5⁹⁵+8 |- |6||11||40041||5||5209 |- |7||71||3₁₆1||17||(7¹⁷−5)/2 |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7 |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10 |- |10||77||50₂₈27||31||5×10³⁰+27 |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10 |- |12||106||40₃₉77||42||4×12⁴¹+91 |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1 |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2 |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15 |- |17^*||10408~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13 |- |21^*||13375~13396||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21 |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49 |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1 |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁰⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 ===Base 10=== 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 ===Base 11=== 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ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====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} (If these three families contain primes (and they are excepted to contain primes), then the smallest prime in families 9{5} and A{3}A in base ''b'' = 13 will be index 3196 and 3197 quasi-minimal prime in base ''b'' = 13, and the smallest prime in families {3}AF in base ''b'' = 16 will be index 2347 quasi-minimal prime in base ''b'' = 16) === Base 17 === All searched to length 29000 * 1{7} * 1F{0}7 * 4{7}A * 51{0}D * 70F{0}D * 8{B}9 * 9{5}9 * 95{F} * A{D}F * B{0}B3 * B{0}DB * {B}2BE * {B}2E * {B}E9 * {B}EE * D0G{D} * E9{B} * F1{9} * FD0{D} * G{7}F === Base 21 === All searched to length 20000 * 2{7}9D * 2F{C}A * 4{3}B * 5{0}DJ * {5}FEK * {7}ID * 99{0}99H * {9}0D * {9}D * B0{H}6H * B3{0}EB * B9{0}E5 * B{D}B * B{H}6H * DH{D} * F{9}D * {F}35 * G{0}FK * H{0}7771 * H{D} * {J}6J === Base 36 === All searched to length 20000 * 7{K}Z * B{0}EUV * HM{0}N * N{0}YYN * O{L}Z * S{0}8H ==Primality certificates for the proven primes > 10²⁹⁹== See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] c9tuf3jd9f87f9rjqomvfn392tw6dfc 2410858 2410857 2022-08-02T01:14:50Z 2001:B011:A004:1F5E:3CA4:6371:FEAD:BB0 /* Unsolved families */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b'' |- |2||1||11||2||3 |- |3||3||111||3||13 |- |4||5||221||3||41 |- |5||22||10₉₃13||96||5⁹⁵+8 |- |6||11||40041||5||5209 |- |7||71||3₁₆1||17||(7¹⁷−5)/2 |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7 |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10 |- |10||77||50₂₈27||31||5×10³⁰+27 |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10 |- |12||106||40₃₉77||42||4×12⁴¹+91 |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1 |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2 |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15 |- |17^*||10408~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13 |- |21^*||13375~13396||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21 |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49 |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1 |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁰⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 ===Base 10=== 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 ===Base 11=== 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ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====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} (If these three families contain primes (and they are excepted to contain primes), then the smallest prime in families 9{5} and A{3}A in base ''b'' = 13 will be index 3196 and 3197 quasi-minimal prime in base ''b'' = 13, and the smallest prime in families {3}AF in base ''b'' = 16 will be index 2347 quasi-minimal prime in base ''b'' = 16) === Base 17 === All searched to length 29000 * 1{7} * 1F{0}7 * 4{7}A * 51{0}D * 70F{0}D * 8{B}9 * 9{5}9 * 95{F} * A{D}F * B{0}B3 * B{0}DB * {B}2BE * {B}2E * {B}E9 * {B}EE * D0G{D} * E9{B} * F1{9} * FD0{D} * G{7}F === Base 21 === All searched to length 20000 * 2{7}9D * 2F{C}A * 4{3}B * 5{0}DJ * {5}FEK * {7}ID * 99{0}99H * {9}0D * {9}D * B0{H}6H * B3{0}EB * B9{0}E5 * B{D}B * B{H}6H * DH{D} * F{9}D * {F}35 * G{0}FK * H{0}7771 * H{D} * {J}6J === Base 36 === All searched to length 20000 * 7{K}Z * B{0}EUV * HM{0}N * N{0}YYN * O{L}Z * S{0}8H ==Primality certificates for the proven primes > 10²⁹⁹== See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] cy569vrrxv4tzmoeehyyr0jb1iukpj0 2410859 2410858 2022-08-02T01:28:57Z 2001:B011:A004:1F5E:3CA4:6371:FEAD:BB0 /* Unsolved families */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b'' |- |2||1||11||2||3 |- |3||3||111||3||13 |- |4||5||221||3||41 |- |5||22||10₉₃13||96||5⁹⁵+8 |- |6||11||40041||5||5209 |- |7||71||3₁₆1||17||(7¹⁷−5)/2 |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7 |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10 |- |10||77||50₂₈27||31||5×10³⁰+27 |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10 |- |12||106||40₃₉77||42||4×12⁴¹+91 |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1 |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2 |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15 |- |17^*||10408~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13 |- |21^*||13375~13396||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21 |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49 |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1 |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁰⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 ===Base 10=== 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 ===Base 11=== 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ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====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} (If these three families contain primes (and they are excepted to contain primes), then the smallest prime in families 9{5} and A{3}A in base ''b'' = 13 will be index 3196 and 3197 quasi-minimal prime in base ''b'' = 13, and the smallest prime in families {3}AF in base ''b'' = 16 will be index 2347 quasi-minimal prime in base ''b'' = 16) === Base 17 === All searched to length 29000 * 1{7} * 1F{0}7 * 4{7}A * 51{0}D * 70F{0}D * 8{B}9 * 9{5}9 * 95{F} * A{D}F * B{0}B3 * B{0}DB * {B}2BE * {B}2E * {B}E9 * {B}EE * D0G{D} * E9{B} * F1{9} * FD0{D} * G{7}F === Base 21 === All searched to length 20000 * 2{7}9D * 2F{C}A * 4{3}B * 5{0}DJ * {5}FEK * {7}ID * 99{0}99H * {9}0D * {9}D * B0{H}6H * B3{0}EB * B9{0}E5 * B{D}B * B{H}6H * DH{D} * F{9}D * {F}35 * G{0}FK * H{0}7771 * H{D} * {J}6J === Base 36 === All searched to length 20000 * 7{K}Z * B{0}EUV * HM{0}N * N{0}YYN * O{L}Z * S{0}8H ==Primality certificates for the proven primes > 10²⁹⁹== See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] nruz5sumtgga5bdycg4b1po3x4nu9j3 2410860 2410859 2022-08-02T01:42:44Z 2001:B011:A004:1F5E:3CA4:6371:FEAD:BB0 /* Unsolved families */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b'' |- |2||1||11||2||3 |- |3||3||111||3||13 |- |4||5||221||3||41 |- |5||22||10₉₃13||96||5⁹⁵+8 |- |6||11||40041||5||5209 |- |7||71||3₁₆1||17||(7¹⁷−5)/2 |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7 |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10 |- |10||77||50₂₈27||31||5×10³⁰+27 |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10 |- |12||106||40₃₉77||42||4×12⁴¹+91 |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1 |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2 |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15 |- |17^*||10408~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13 |- |21^*||13375~13396||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21 |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49 |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1 |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁰⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 ===Base 10=== 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 ===Base 11=== 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ ====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} (If these three families contain primes (and they are excepted to contain primes), then the smallest prime in families 9{5} and A{3}A in base ''b'' = 13 will be index 3196 and 3197 quasi-minimal prime in base ''b'' = 13, and the smallest prime in families {3}AF in base ''b'' = 16 will be index 2347 quasi-minimal prime in base ''b'' = 16) === Base 17 === All searched to length 29000 * 1{7} * 1F{0}7 * 4{7}A * 51{0}D * 70F{0}D * 8{B}9 * 9{5}9 * 95{F} * A{D}F * B{0}B3 * B{0}DB * {B}2BE * {B}2E * {B}E9 * {B}EE * D0G{D} * E9{B} * F1{9} * FD0{D} * G{7}F === Base 21 === All searched to length 20000 * 2{7}9D * 2F{C}A * 4{3}B * 5{0}DJ * {5}FEK * {7}ID * 99{0}99H * {9}0D * {9}D * B0{H}6H * B3{0}EB * B9{0}E5 * B{D}B * B{H}6H * DH{D} * F{9}D * {F}35 * G{0}FK * H{0}7771 * H{D} * {J}6J === Base 36 === All searched to length 20000 * 7{K}Z * B{0}EUV * HM{0}N * N{0}YYN * O{L}Z * S{0}8H ==Primality certificates for the proven primes > 10²⁹⁹== See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] 2zvz2rkyl8gp6xr1sey9fjs6i09vh16 2410863 2410860 2022-08-02T02:04:21Z 2001:B011:A004:1F5E:3CA4:6371:FEAD:BB0 /* Unsolved families */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b'' |- |2||1||11||2||3 |- |3||3||111||3||13 |- |4||5||221||3||41 |- |5||22||10₉₃13||96||5⁹⁵+8 |- |6||11||40041||5||5209 |- |7||71||3₁₆1||17||(7¹⁷−5)/2 |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7 |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10 |- |10||77||50₂₈27||31||5×10³⁰+27 |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10 |- |12||106||40₃₉77||42||4×12⁴¹+91 |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1 |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2 |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15 |- |17^*||10408~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13 |- |21^*||13375~13396||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21 |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49 |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1 |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁰⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 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===Base 10=== 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 ===Base 11=== 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} (If these three families contain primes (and they are excepted to contain primes), then the smallest prime in families 9{5} and A{3}A in base ''b'' = 13 will be index 3196 and 3197 quasi-minimal prime in base ''b'' = 13, and the smallest prime in families {3}AF in base ''b'' = 16 will be index 2347 quasi-minimal prime in base ''b'' = 16) === Base 17 === All searched to length 29000 * 1{7} * 1F{0}7 * 4{7}A * 51{0}D * 70F{0}D * 8{B}9 * 9{5}9 * 95{F} * A{D}F * B{0}B3 * B{0}DB * {B}2BE * {B}2E * {B}E9 * {B}EE * D0G{D} * E9{B} * F1{9} * FD0{D} * G{7}F === Base 21 === All searched to length 20000 * 2{7}9D * 2F{C}A * 4{3}B * 5{0}DJ * {5}FEK * {7}ID * 99{0}99H * {9}0D * {9}D * B0{H}6H * B3{0}EB * B9{0}E5 * B{D}B * B{H}6H * DH{D} * F{9}D * {F}35 * G{0}FK * H{0}7771 * H{D} * {J}6J === Base 36 === All searched to length 20000 * 7{K}Z * B{0}EUV * HM{0}N * N{0}YYN * O{L}Z * S{0}8H ==Primality certificates for the proven primes > 10²⁹⁹== See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] k7nruml0f8a2nm9kwdlxvj1uv69vi4r 2410911 2410863 2022-08-02T05:48:39Z 118.170.41.232 /* Unsolved families */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b'' |- |2||1||11||2||3 |- |3||3||111||3||13 |- |4||5||221||3||41 |- |5||22||10₉₃13||96||5⁹⁵+8 |- |6||11||40041||5||5209 |- |7||71||3₁₆1||17||(7¹⁷−5)/2 |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7 |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10 |- |10||77||50₂₈27||31||5×10³⁰+27 |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10 |- |12||106||40₃₉77||42||4×12⁴¹+91 |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1 |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2 |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15 |- |17^*||10408~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13 |- |21^*||13375~13396||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21 |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49 |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1 |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁰⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 ===Base 10=== 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 ===Base 11=== 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ ====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||95{F}||(2543×17^''n''−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||A{D}F||(173×17^''n''+1+19)/16||29000|[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} (If these three families contain primes (and they are excepted to contain primes), then the smallest prime in families 9{5} and A{3}A in base ''b'' = 13 will be index 3196 and 3197 quasi-minimal prime in base ''b'' = 13, and the smallest prime in families {3}AF in base ''b'' = 16 will be index 2347 quasi-minimal prime in base ''b'' = 16) === Base 17 === All searched to length 29000 * 1{7} * 1F{0}7 * 4{7}A * 51{0}D * 70F{0}D * 8{B}9 * 9{5}9 * 95{F} * A{D}F * B{0}B3 * B{0}DB * {B}2BE * {B}2E * {B}E9 * {B}EE * D0G{D} * E9{B} * F1{9} * FD0{D} * G{7}F === Base 21 === All searched to length 20000 * 2{7}9D * 2F{C}A * 4{3}B * 5{0}DJ * {5}FEK * {7}ID * 99{0}99H * {9}0D * {9}D * B0{H}6H * B3{0}EB * B9{0}E5 * B{D}B * B{H}6H * DH{D} * F{9}D * {F}35 * G{0}FK * H{0}7771 * H{D} * {J}6J === Base 36 === All searched to length 20000 * 7{K}Z * B{0}EUV * HM{0}N * N{0}YYN * O{L}Z * S{0}8H ==Primality certificates for the proven primes > 10²⁹⁹== See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] of75jnxeab614kb43u301vwb0pi2nmz 2410912 2410911 2022-08-02T05:48:59Z 118.170.41.232 /* Unsolved families */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b'' |- |2||1||11||2||3 |- |3||3||111||3||13 |- |4||5||221||3||41 |- |5||22||10₉₃13||96||5⁹⁵+8 |- |6||11||40041||5||5209 |- |7||71||3₁₆1||17||(7¹⁷−5)/2 |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7 |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10 |- |10||77||50₂₈27||31||5×10³⁰+27 |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10 |- |12||106||40₃₉77||42||4×12⁴¹+91 |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1 |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2 |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15 |- |17^*||10408~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13 |- |21^*||13375~13396||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21 |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49 |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1 |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁰⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ ====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||95{F}||(2543×17^''n''−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||A{D}F||(173×17^''n''+1+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} (If these three families contain primes (and they are excepted to contain primes), then the smallest prime in families 9{5} and A{3}A in base ''b'' = 13 will be index 3196 and 3197 quasi-minimal prime in base ''b'' = 13, and the smallest prime in families {3}AF in base ''b'' = 16 will be index 2347 quasi-minimal prime in base ''b'' = 16) === Base 17 === All searched to length 29000 * 1{7} * 1F{0}7 * 4{7}A * 51{0}D * 70F{0}D * 8{B}9 * 9{5}9 * 95{F} * A{D}F * B{0}B3 * B{0}DB * {B}2BE * {B}2E * {B}E9 * {B}EE * D0G{D} * E9{B} * F1{9} * FD0{D} * G{7}F === Base 21 === All searched to length 20000 * 2{7}9D * 2F{C}A * 4{3}B * 5{0}DJ * {5}FEK * {7}ID * 99{0}99H * {9}0D * {9}D * B0{H}6H * B3{0}EB * B9{0}E5 * B{D}B * B{H}6H * DH{D} * F{9}D * {F}35 * G{0}FK * H{0}7771 * H{D} * {J}6J === Base 36 === All searched to length 20000 * 7{K}Z * B{0}EUV * HM{0}N * N{0}YYN * O{L}Z * S{0}8H ==Primality certificates for the proven primes > 10²⁹⁹== See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] hs2y4825sqpswd6ebxdhaibwedsqr5h 2410913 2410912 2022-08-02T05:52:13Z 118.170.41.232 /* Unsolved families */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b'' |- |2||1||11||2||3 |- |3||3||111||3||13 |- |4||5||221||3||41 |- |5||22||10₉₃13||96||5⁹⁵+8 |- |6||11||40041||5||5209 |- |7||71||3₁₆1||17||(7¹⁷−5)/2 |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7 |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10 |- |10||77||50₂₈27||31||5×10³⁰+27 |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10 |- |12||106||40₃₉77||42||4×12⁴¹+91 |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1 |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2 |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15 |- |17^*||10408~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13 |- |21^*||13375~13396||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21 |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49 |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1 |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁰⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ ====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||95{F}||(2543×17^''n''−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||A{D}F||(173×17^''n''+1+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}B3||11×17^''n''+2+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}DB||11×17^''n''+2+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} (If these three families contain primes (and they are excepted to contain primes), then the smallest prime in families 9{5} and A{3}A in base ''b'' = 13 will be index 3196 and 3197 quasi-minimal prime in base ''b'' = 13, and the smallest prime in families {3}AF in base ''b'' = 16 will be index 2347 quasi-minimal prime in base ''b'' = 16) === Base 17 === All searched to length 29000 * 1{7} * 1F{0}7 * 4{7}A * 51{0}D * 70F{0}D * 8{B}9 * 9{5}9 * 95{F} * A{D}F * B{0}B3 * B{0}DB * {B}2BE * {B}2E * {B}E9 * {B}EE * D0G{D} * E9{B} * F1{9} * FD0{D} * G{7}F === Base 21 === All searched to length 20000 * 2{7}9D * 2F{C}A * 4{3}B * 5{0}DJ * {5}FEK * {7}ID * 99{0}99H * {9}0D * {9}D * B0{H}6H * B3{0}EB * B9{0}E5 * B{D}B * B{H}6H * DH{D} * F{9}D * {F}35 * G{0}FK * H{0}7771 * H{D} * {J}6J === Base 36 === All searched to length 20000 * 7{K}Z * B{0}EUV * HM{0}N * N{0}YYN * O{L}Z * S{0}8H ==Primality certificates for the proven primes > 10²⁹⁹== See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] 1itybn2rs8n8va702qpf5uyfkpc6xmx 2410920 2410913 2022-08-02T05:58:43Z 118.170.41.232 /* Unsolved families */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b'' |- |2||1||11||2||3 |- |3||3||111||3||13 |- |4||5||221||3||41 |- |5||22||10₉₃13||96||5⁹⁵+8 |- |6||11||40041||5||5209 |- |7||71||3₁₆1||17||(7¹⁷−5)/2 |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7 |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10 |- |10||77||50₂₈27||31||5×10³⁰+27 |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10 |- |12||106||40₃₉77||42||4×12⁴¹+91 |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1 |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2 |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15 |- |17^*||10408~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13 |- |21^*||13375~13396||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21 |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49 |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1 |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁰⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ ====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||95{F}||(2543×17^''n''−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||A{D}F||(173×17^''n''+1+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}B3||11×17^''n''+2+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}DB||11×17^''n''+2+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2BE||(11×17^''n''+3−41579)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2E||(11×17^''n''+2−2411)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} (If these three families contain primes (and they are excepted to contain primes), then the smallest prime in families 9{5} and A{3}A in base ''b'' = 13 will be index 3196 and 3197 quasi-minimal prime in base ''b'' = 13, and the smallest prime in families {3}AF in base ''b'' = 16 will be index 2347 quasi-minimal prime in base ''b'' = 16) === Base 17 === All searched to length 29000 * 1{7} * 1F{0}7 * 4{7}A * 51{0}D * 70F{0}D * 8{B}9 * 9{5}9 * 95{F} * A{D}F * B{0}B3 * B{0}DB * {B}2BE * {B}2E * {B}E9 * {B}EE * D0G{D} * E9{B} * F1{9} * FD0{D} * G{7}F === Base 21 === All searched to length 20000 * 2{7}9D * 2F{C}A * 4{3}B * 5{0}DJ * {5}FEK * {7}ID * 99{0}99H * {9}0D * {9}D * B0{H}6H * B3{0}EB * B9{0}E5 * B{D}B * B{H}6H * DH{D} * F{9}D * {F}35 * G{0}FK * H{0}7771 * H{D} * {J}6J === Base 36 === All searched to length 20000 * 7{K}Z * B{0}EUV * HM{0}N * N{0}YYN * O{L}Z * S{0}8H ==Primality certificates for the proven primes > 10²⁹⁹== See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] g8ami29v4cbppiwnreo93w5yrdt4quw 2410928 2410920 2022-08-02T06:07:44Z 118.170.69.213 /* Unsolved families */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b'' |- |2||1||11||2||3 |- |3||3||111||3||13 |- |4||5||221||3||41 |- |5||22||10₉₃13||96||5⁹⁵+8 |- |6||11||40041||5||5209 |- |7||71||3₁₆1||17||(7¹⁷−5)/2 |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7 |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10 |- |10||77||50₂₈27||31||5×10³⁰+27 |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10 |- |12||106||40₃₉77||42||4×12⁴¹+91 |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1 |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2 |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15 |- |17^*||10408~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13 |- |21^*||13375~13396||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21 |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49 |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1 |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁰⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJLJ ====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||95{F}||(2543×17^''n''−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||A{D}F||(173×17^''n''+1+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}B3||11×17^''n''+2+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}DB||11×17^''n''+2+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2BE||(11×17^''n''+3−41579)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2E||(11×17^''n''+2−2411)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D} === Base 17 === All searched to length 29000 * 1{7} * 1F{0}7 * 4{7}A * 51{0}D * 70F{0}D * 8{B}9 * 9{5}9 * 95{F} * A{D}F * B{0}B3 * B{0}DB * {B}2BE * {B}2E * {B}E9 * {B}EE * D0G{D} * E9{B} * F1{9} * FD0{D} * G{7}F === Base 21 === All searched to length 20000 * 2{7}9D * 2F{C}A * 4{3}B * 5{0}DJ * {5}FEK * {7}ID * 99{0}99H * {9}0D * {9}D * B0{H}6H * B3{0}EB * B9{0}E5 * B{D}B * B{H}6H * DH{D} * F{9}D * {F}35 * G{0}FK * H{0}7771 * H{D} * {J}6J === Base 36 === All searched to length 20000 * 7{K}Z * B{0}EUV * HM{0}N * N{0}YYN * O{L}Z * S{0}8H ==Primality certificates for the proven primes > 10²⁹⁹== See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] 26amswl2b8ualg8jemp6qt6aua64t89 2410929 2410928 2022-08-02T06:12:16Z 118.170.69.213 /* Unsolved families */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b'' |- |2||1||11||2||3 |- |3||3||111||3||13 |- |4||5||221||3||41 |- |5||22||10₉₃13||96||5⁹⁵+8 |- |6||11||40041||5||5209 |- |7||71||3₁₆1||17||(7¹⁷−5)/2 |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7 |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10 |- |10||77||50₂₈27||31||5×10³⁰+27 |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10 |- |12||106||40₃₉77||42||4×12⁴¹+91 |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1 |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2 |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15 |- |17^*||10408~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13 |- |21^*||13375~13396||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21 |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49 |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1 |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁰⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')) by covering congruence, algebraic factorization, or combine of them. {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||95{F}||(2543×17^''n''−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||A{D}F||(173×17^''n''+1+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}B3||11×17^''n''+2+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}DB||11×17^''n''+2+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2BE||(11×17^''n''+3−41579)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2E||(11×17^''n''+2−2411)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D} === Base 17 === All searched to length 29000 * 1{7} * 1F{0}7 * 4{7}A * 51{0}D * 70F{0}D * 8{B}9 * 9{5}9 * 95{F} * A{D}F * B{0}B3 * B{0}DB * {B}2BE * {B}2E * {B}E9 * {B}EE * D0G{D} * E9{B} * F1{9} * FD0{D} * G{7}F === Base 21 === All searched to length 20000 * 2{7}9D * 2F{C}A * 4{3}B * 5{0}DJ * {5}FEK * {7}ID * 99{0}99H * {9}0D * {9}D * B0{H}6H * B3{0}EB * B9{0}E5 * B{D}B * B{H}6H * DH{D} * F{9}D * {F}35 * G{0}FK * H{0}7771 * H{D} * {J}6J === Base 36 === All searched to length 20000 * 7{K}Z * B{0}EUV * HM{0}N * N{0}YYN * O{L}Z * S{0}8H ==Primality certificates for the proven primes > 10²⁹⁹== See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] rq2wo4agc5lrsl7x6ivm3gy502vh9mx 2410930 2410929 2022-08-02T06:15:14Z 118.170.69.213 /* Unsolved families */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b'' |- |2||1||11||2||3 |- |3||3||111||3||13 |- |4||5||221||3||41 |- |5||22||10₉₃13||96||5⁹⁵+8 |- |6||11||40041||5||5209 |- |7||71||3₁₆1||17||(7¹⁷−5)/2 |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7 |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10 |- |10||77||50₂₈27||31||5×10³⁰+27 |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10 |- |12||106||40₃₉77||42||4×12⁴¹+91 |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1 |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2 |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15 |- |17^*||10408~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13 |- |21^*||13375~13396||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21 |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49 |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1 |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁰⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')) by covering congruence, algebraic factorization, or combine of them. {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||29000||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||29000||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||29000||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||29000||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||29000||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||29000||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||29000||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||95{F}||(2543×17^''n''−15)/16||29000||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||A{D}F||(173×17^''n''+1+19)/16||29000||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}B3||11×17^''n''+2+190||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}DB||11×17^''n''+2+232||29000||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2BE||(11×17^''n''+3−41579)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2E||(11×17^''n''+2−2411)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}E9||(11×17^''n''+2+773)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}EE||(11×17^''n''+2+853)/16||29000||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D} === Base 17 === All searched to length 29000 * 1{7} * 1F{0}7 * 4{7}A * 51{0}D * 70F{0}D * 8{B}9 * 9{5}9 * 95{F} * A{D}F * B{0}B3 * B{0}DB * {B}2BE * {B}2E * {B}E9 * {B}EE * D0G{D} * E9{B} * F1{9} * FD0{D} * G{7}F === Base 21 === All searched to length 20000 * 2{7}9D * 2F{C}A * 4{3}B * 5{0}DJ * {5}FEK * {7}ID * 99{0}99H * {9}0D * {9}D * B0{H}6H * B3{0}EB * B9{0}E5 * B{D}B * B{H}6H * DH{D} * F{9}D * {F}35 * G{0}FK * H{0}7771 * H{D} * {J}6J === Base 36 === All searched to length 20000 * 7{K}Z * B{0}EUV * HM{0}N * N{0}YYN * O{L}Z * S{0}8H ==Primality certificates for the proven primes > 10²⁹⁹== See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||[http://factordb.com/cert.php?id=1100000003590502516] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] ev70x82fh74xbdrsr5jlmdkch0eombm 3 276205 2410847 2410429 2022-08-01T22:51:23Z Dave Braunschweig 426084 /* Subpages */ Reply wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Jacsaw!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 22:10, 1 August 2021 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} == Subpages == Please use subpages for course pages. See [https://en.wikiversity.org/w/index.php?title=User%3AJacsaw%2Fsandbox&type=revision&diff=2384897&oldid=2384799] for how to implement this. When you are ready to move this project out of your sandbox and into main space, let me or one of the other custodians know. We can move all of the pages for you at once vs. moving them individually. [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 18:23, 22 March 2022 (UTC) :Hi, I would like to move this content out of sandbox to a main page, and would love to move the subpages at the same time. Are you able to assist? thank you! [[User:Jacsaw|Jacsaw]] ([[User talk:Jacsaw|discuss]] • [[Special:Contributions/Jacsaw|contribs]]) 00:56, 29 July 2022 (UTC) ::Please include my name in the post so I know you've reached out to me. I can do the move. What name do you want for the base page (title for the learning project)? -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:35, 30 July 2022 (UTC) :::@[[User:Dave Braunschweig|Dave Braunschweig]] the title of the page is Living Syllabus - Advocacy in Tech and Society. Do I link it from the main Wikiversity site? or do you need it direct somewhere? I am planning on linking it to a school and resource on the main wikiversity site. Thanks for your help. [[User:Jacsaw|Jacsaw]] ([[User talk:Jacsaw|discuss]] • [[Special:Contributions/Jacsaw|contribs]]) 18:39, 30 July 2022 (UTC) ::::this is the main page [[User:Jacsaw/sandbox#Advocacy in Technology and Society]] [[User:Jacsaw|Jacsaw]] ([[User talk:Jacsaw|discuss]] • [[Special:Contributions/Jacsaw|contribs]]) 18:39, 30 July 2022 (UTC) :::::Sorry, been off-wiki. See [[Advocacy in Technology and Society]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 22:51, 1 August 2022 (UTC) If students will be editing this content, it is time to move it out of your user space. It can either be a Draft: space resource, if you prefer, or a regular main space resource. Please let me know which you prefer so it can be moved before additional students work on the content. Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:12, 28 March 2022 (UTC) :Hi, yes, the students have started adding content to the modules. I was planning on moving it out of sandbox once content was added and reviewed. Does marking it as draft prevent it from being public while it's being built out? thanks! [[Special:Contributions/24.187.47.101|24.187.47.101]] ([[User talk:24.187.47.101|discuss]]) 01:23, 28 March 2022 (UTC) ::The only real difference between Draft: space and main space is whether or not the content appears in local (Wikiversity) search results. Almost all classes with student-developed content are in main space. See [[Solidarity Economy in Latin America]], [[Digital Media Concepts]], and [[Federal Writers' Project – Life Histories]] for current examples. If you tell me what title you want, I can move the pages with one click rather than you having to move each page individually. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:10, 28 March 2022 (UTC) bhkaqfepdim7kjn5uyie2v73s3lunh2 2410851 2410847 2022-08-01T23:38:29Z 24.187.47.101 /* Subpages */ Reply wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Jacsaw!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 22:10, 1 August 2021 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} == Subpages == Please use subpages for course pages. See [https://en.wikiversity.org/w/index.php?title=User%3AJacsaw%2Fsandbox&type=revision&diff=2384897&oldid=2384799] for how to implement this. When you are ready to move this project out of your sandbox and into main space, let me or one of the other custodians know. We can move all of the pages for you at once vs. moving them individually. [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 18:23, 22 March 2022 (UTC) :Hi, I would like to move this content out of sandbox to a main page, and would love to move the subpages at the same time. Are you able to assist? thank you! [[User:Jacsaw|Jacsaw]] ([[User talk:Jacsaw|discuss]] • [[Special:Contributions/Jacsaw|contribs]]) 00:56, 29 July 2022 (UTC) ::Please include my name in the post so I know you've reached out to me. I can do the move. What name do you want for the base page (title for the learning project)? -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:35, 30 July 2022 (UTC) :::@[[User:Dave Braunschweig|Dave Braunschweig]] the title of the page is Living Syllabus - Advocacy in Tech and Society. Do I link it from the main Wikiversity site? or do you need it direct somewhere? I am planning on linking it to a school and resource on the main wikiversity site. Thanks for your help. [[User:Jacsaw|Jacsaw]] ([[User talk:Jacsaw|discuss]] • [[Special:Contributions/Jacsaw|contribs]]) 18:39, 30 July 2022 (UTC) ::::this is the main page [[User:Jacsaw/sandbox#Advocacy in Technology and Society]] [[User:Jacsaw|Jacsaw]] ([[User talk:Jacsaw|discuss]] • [[Special:Contributions/Jacsaw|contribs]]) 18:39, 30 July 2022 (UTC) :::::Sorry, been off-wiki. See [[Advocacy in Technology and Society]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 22:51, 1 August 2022 (UTC) :::thank you so much for your help! [[Special:Contributions/24.187.47.101|24.187.47.101]] ([[User talk:24.187.47.101|discuss]]) 23:38, 1 August 2022 (UTC) If students will be editing this content, it is time to move it out of your user space. It can either be a Draft: space resource, if you prefer, or a regular main space resource. Please let me know which you prefer so it can be moved before additional students work on the content. Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:12, 28 March 2022 (UTC) :Hi, yes, the students have started adding content to the modules. I was planning on moving it out of sandbox once content was added and reviewed. Does marking it as draft prevent it from being public while it's being built out? thanks! [[Special:Contributions/24.187.47.101|24.187.47.101]] ([[User talk:24.187.47.101|discuss]]) 01:23, 28 March 2022 (UTC) ::The only real difference between Draft: space and main space is whether or not the content appears in local (Wikiversity) search results. Almost all classes with student-developed content are in main space. See [[Solidarity Economy in Latin America]], [[Digital Media Concepts]], and [[Federal Writers' Project – Life Histories]] for current examples. If you tell me what title you want, I can move the pages with one click rather than you having to move each page individually. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:10, 28 March 2022 (UTC) q6mpachxyo8p1jkzukzqycxflm5d7ep 0 276564 2410856 2409198 2022-08-02T00:32:01Z Jtneill 10242 {{METE}} wikitext text/x-wiki {{METE}} {{title|Psilocybin assisted psychotherapy:<br>How can psilocybin be used to assist psychotherapy?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== Overview of protocols for psilocybin assisted psychotherapy across patient cohorts. case studies supplementation with omega 3 for enhanced neurogenesis and lasting neurological change. integration strategies {{RoundBoxTop|theme=3}} '''Focus questions:''' * Why use psylocybin in therapy? * How to get the most out of a psylocybin assisted session? * What is integration? {{RoundBoxBottom}} == What does psilocybin do? == *How does psilocybin impact neurological connections *what is the theraputic impact of this change *why is it helpful in a clinical setting [[File:Your_brain_on_psilocybin.jpg|center|thumb|550x550px|Communication between brain networks in people given psilocybin (right) or a non-psychedelic compound (left).]] ''Psilocybin is a serotonergic psychedelic with untapped therapeutic potential. There are hints that the use of psychedelics can produce neural adaptations, although the extent and timescale of the impact in a mammalian brain are unknown. In this study, we used chronic two-photon microscopy to image longitudinally the apical dendritic spines of layer 5 pyramidal neurons in the mouse medial frontal cortex. We found that a single dose of psilocybin led to ∼10% increases in spine size and density, driven by an elevated spine formation rate. The structural remodeling occurred quickly within 24 h and was persistent 1 month later. Psilocybin also ameliorated stress-related behavioral deficit and elevated excitatory neurotransmission. Overall, the results demonstrate that psilocybin-evoked synaptic rewiring in the cortex is fast and enduring, potentially providing a structural trace for long-term integration of experiences and lasting beneficial actions.'' == Psilocybin assisted therapy protocols == * Describe hopkins protocols after screening for contraindications like a family or personal history of psychosis and medications like lithium, which can increase the risk of seasures, participants can be given a 250mg dose of niacin in a therapuic environment to acclimatise themselves to the somatic state changes that can accompany a psychadelic experience. == How to prepare for a psilocybin assisted session == * Two weeks before the session *one week before the sesion *Day of the session ''The reduction in stroke-induced neurological deficits with n-3 PUFA post-treatment was associated with enhanced angiogenesis, oligodendrogenesis, neuron survival and white matter restoration. Together, these results indicate that the neurological benefits of n-3 PUFA administration after stroke extend to older animals and are associated with improved neuronal survival and brain remodeling, therefore suggesting that post-stroke administration of n-3 PUFAs is a viable clinically relevant treatment option against stroke.'' == What is integration and why is it important? == *What is post session integration *why is integration important *Integration activities ==Conclusion== ==See also== * [[Motivation and emotion/Book/2019/Psilocybin and emotion|Psilocybin and emotion]] (Book chapter, 2019)]] ==References== {{Hanging indent|1= Aday, J.S., Davis, A.K., Mitzkovitz, C.M., Bloesch, E.M., & Davoli, C.C. (2021). Predicting reactions to psychedelic drugs: a systematic review of states and traits related to acute drug effects. ''ACS Pharmacology & Translational Science, 4''(2), 424–435. https://pubs.acs.org/doi/10.1021/acsptsci.1c00014 Fleischmann, C., Shohami, E., Trembovler, V., Heled, Y,. & Horowitz, M. (2020) Cognitive Effects of Astaxanthin Pretreatment on Recovery From Traumatic Brain Injury. ''Frontiers in Neurology''. (11) https://doi.org/10.3389/fneur.2020.00999 Garcia-Romeu, A., Barrett, F.S., Carbonaro, T.M., Johnson, M.W., & Griffiths, R.R. (2021). Optimal dosing for psilocybin pharmacotherapy: Considering weight-adjusted and fixed dosing approaches. ''Journal of Psychopharmacology, 35''(4), 353-361.https://journals.sagepub.com/doi/full/10.1177/0269881121991822 Jiang X, Suenaga J, Pu H, Wei Z, Smith AD, Hu X, Shi Y, Chen J. Post-stroke administration of omega-3 polyunsaturated fatty acids promotes neurovascular restoration after ischemic stroke in mice: Efficacy declines with aging. Neurobiol Dis. 2019 Jun;126:62-75. doi: 10.1016/j.nbd.2018.09.012. Epub 2018 Sep 12. PMID: 30218758. Petri, G.,&nbsp;Expert, P.&nbsp;, Turkheimer, F.&nbsp;, Carhart-Harris, R.,&nbsp;Nutt, D.&nbsp;, Hellyer, P. J.&nbsp; & Vaccarino, F. (2014). Homological scaffolds of brain functional networks. ''Journal of The Royal Society Interface,'' https://royalsocietypublishing.org/doi/abs/10.1098/rsif.2014.0873 Strickland, J.C., Garcia-Romeu, A., & Johnson, M.W. (2021). Set and setting: a randomized study of different musical genres in supporting psychedelic therapy. ''ACS Pharmacology & Translational Science, 4''(2), 472-478. https://pubs.acs.org/doi/full/10.1021/acsptsci.0c00187 }} Shao LX, Liao C, Gregg I, Davoudian PA, Savalia NK, Delagarza K, Kwan AC. Psilocybin induces rapid and persistent growth of dendritic spines in frontal cortex in vivo. Neuron. 2021 Aug 18;109(16):2535-2544.e4. doi: 10.1016/j.neuron.2021.06.008. Epub 2021 Jul 5. PMID: 34228959; PMCID: PMC8376772. In this section, list the cited references in [[w:APA style|APA style]] (7th ed.). For example: ==External links== [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Drugs/Psilocybin]] [[Category:Motivation and emotion/Book/Psychotherapy]] rcm2wcl0fn3rvqve79gugrj9sj8cv8g 0 277657 2410790 2410756 2022-08-01T13:56:07Z 2001:8003:1D4D:1900:2D6D:58D7:278E:A177 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Academic help-seeking/]] - What are the barriers and enablers of AHS and how can AHS be fostered? - [[User:MyUserName|MyUserName]] # [[/Academic self-regulation/]] - What is academic self-regulation, why does it matter, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? - [[User:MyUserName|MyUserName]] # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it? - [[User:MyUserName|MyUserName]] # [[/Antidepressants and motivation/]] - What are the effects of popular antidepressants on motivation? - [[User:MyUserName|MyUserName]] # [[/Approach motivation/]] - What is approach motivation and how does it lead to behaviour? - [[User:U3189370|U3189370]] # [[/Behavioural economics and motivation/]] - What aspects of motivation theory are useful in behavioural economics? - [[User:U3141987|U3141987]] # [[/Behavioural model of health services/]] - What is the BMHS and how can it be used? - [[UserFV:MyUserName|MyUserName]] # [[/Beneficence as a psychological need/]] - What is beneficence and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention? - [[User:MyUserName|MyUserName]] # [[/Choice overload/]] - How much choose is too much? How much choice is enough? - [[User:MyUserName|MyUserName]] # [[/Chunking and goal pursuit/]] - How does chunking affect goal pursuit? - [[User:MyUserName|MyUserName]] # [[/Cognitive entrenchment/]] - What is cognitive entrenchment and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[/Climate change helplessness/]] - How does learned helpless impact motivation to engage in behaviours to limit climate change? - [[User:MyUserName|MyUserName]] # [[/Closeness communication bias/]] - What is the CCB, why does it occur, and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirably outcomes? - [[User:MyUserName|MyUserName]] # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories? - [[User:KingMob221|KingMob221]] # [[/Construal level theory/]] - What is construal level theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Courage motivation/]] - What is courage, what motivates courage, and how can courage be enhanced? -[[User:U3213871|U3213871] # [[/Death drive/]] - What is the death drive and how can it be negotiated? - [[User:U3086459|U3086459]] # [[/Drugs-violence nexus and motivation/]] - What is the role of motivation in the drugs-violence nexus? - [[User:MyUserName|MyUserName]] # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD? - [[User:MyUserName|MyUserName]] # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? - [[User:MyUserName|MyUserName]] # [[/Equity theory/]] - What is equity theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/ERG theory/]] - What is Alderfer's ERG theory? - [[User:MyUserName|MyUserName]] # [[/Frame of reference and motivation/]] - How does frame of reference affect motivation? - [[User:MyUserName|MyUserName]] # [[/Freedom and motivation/]] - What is the effect of freedom on motivation? - [[User:MyUserName|MyUserName]] # [[/Fully functioning person/]] - What is a FFP and how can full functioning be developed? - [[User:U3217727|U3217727]] # [[/Functional fixedness/]] - What is functional fixedness and how can it be overcome? - [[User:U3214117]] # [[/Functional imagery training/]] - What is FIT and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Gamification and work motivation/]] - How can gamification enhance work motivation? - [[User:MyUserName|MyUserName]] # [[/Giving up goals/]] - When should we give up goals and when should we persist? - [[User:MyUserName|MyUserName]] # [[/Green prescription motivation/]] - What motivates green prescription compliance? - [[User:MyUserName|MyUserName]] # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour? - [[User:MyUserName|MyUserName]] # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis, what is the evidence, and how does it help to understand drug addiction? - [[U3218292|U3218292]] # [[/Honesty motivation/]] - What motivates honesty? - [[User:MyUserName|MyUserName]] # [[/Humour and work/]] - What is the role of humour in the workplace? - [[User:MyUserName|MyUserName]] # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? - [[User:MyUserName|MyUserName]] # [[/Kindness motivation/]] - What motivates kindness? - [[User:MyUserName|MyUserName]] # [[/Motivational music and exercise/]] - How can music be used to help motivate exercise? - [[User:MyUserName|MyUserName]] # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Nucleus accumbens and motivation/]] - What role does the nucleus accumbens play in motivation? - [[User:MyUserName|MyUserName]] # [[/Physiological needs/]] - How do human's physiological needs affect motivation? - [[U3203655]] # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19? - [[User:MyUserName|MyUserName]] # [[/Relative deprivation and motivation/]] - What is the effect of relative deprivation on motivation? - [[User:MyUserName|MyUserName]] # [[/Retrospective regret/]] - What is the motivational role of retrospective regret? - [[User:MyUserName|MyUserName]] # [[/Revenge motivation/]] - What motivates revenge and how does it affect us? - [[User:MyUserName|MyUserName]] # [[/Self-efficacy and academic achievement/]] - What role does self-efficacy play in academic achievement? - [[User:U943292|U943292]] # [[/Self-efficacy and achievement/]] - What role does self-efficacy play in achievement outcomes? - [[User:MyUserName|MyUserName]] # [[/Sexual harassment at work motivation/]] - What motivates sexual harassment at work and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Signature strengths/]] - What are signature strengths and how can they be applied? - [[User:MyUserName|MyUserName]] # [[/Social cure/]] - What is the social cure and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Stretch goals/]] - What are stretch goals? Do they work? - [[User:MyUserName|MyUserName]] # [[/Sublimation/]] - What is sublimation and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Survival needs and motivation/]] - What are survival needs and how do they influence motivation? - [[User:MyUserName|MyUserName]] # [[/Task initiation/]] - What are the challenges with task initiation and how to get get started? - [[User:MyUserName|MyUserName]] # [[/Theoretical domains framework/]] - What is the TDF and how can be used to guide behaviour change? - [[User:MyUserName|MyUserName]] # [[/Time and motivation/]] - What is the effect of time on motivation? - [[User:MyUserName|MyUserName]] # [[/Time management/]] - How can one's time be managed effectively? - [[User:MyUserName|MyUserName]] # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - [[User:MyUserName|MyUserName]] # [[/Uncertainty avoidance/]] - What is uncertainty avoidance, why does it occur, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Urgency bias and productivity/]] - What is the impact of urgency bias on productivity and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Vocational identity/]] - What is vocational identity and how does it develop? - [[User:MyUserName|MyUserName]] # [[/Volunteer tourism motivation/]] - What motivates volunteer tourism? - [[User:Efost|Efost]] # [[/Wanting and liking/]] - What are the similarities and differences between wanting and liking, and what are the implications? - [[User:MyUserName|MyUserName]] # [[/Work breaks, well-being, and productivity/]] - How do work breaks affect well-being and productivity? - [[User:MyUserName|MyUserName]] # [[/Work and flow/]] - What characteristics of work can produce flow and how can flow at work be fostered? - [[User:MyUserName|MyUserName]] ==Emotion== # [[/Animal emotion/]] - What is the emotional experience of animals? - [[User:MyUserName|MyUserName]] # [[/Attributions and emotion/]] - How do attributions affect emotion? - [[User:MyUserName|MyUserName]] # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? - [[User:MyUserName|MyUserName]] # [[/Benzodiazepines and emotion/]] - What are the effects of benzodiazepines on emotion? - FulaAjeo22 # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? - [[User:MyUserName|MyUserName]] # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? - [[User:U3211603|U3211603]] # [[/Cognitive dissonance reduction/]] - What strategies do people use to reduce cognitive dissonance and how effective are they? - [[User:MyUserName|MyUserName]] # [[/Colonisation and emotion in Australia/]] - What are the emotional responses to colonisation in Australia? - [[User:MyUserName|MyUserName]] # [[/Compassion/]] - What is compassion, what are its pros and cons, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? - [[User:MyUserName|MyUserName]] # [[/Contempt/]] - What is contempt, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Core emotions/]] - What are the core emotions and what is their function? MyUserName # [[/Creative arts and trauma/]] - How can creative arts help in dealing with trauma? - [[User:MyUserName|MyUserName]] # [[/Cultural influences on shame, guilt, and pride/]] - How does culture influence shame, guilt, and pride? - [[User:MyUserName|MyUserName]] # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? - [[User:MyUserName|MyUserName]] # [[/Difficult conversations and emotion/]] - What communication and emotional skills are needed to successfully negotiate difficult conversations? - [[User:MyUserName|MyUserName]] # [[/Disappointment/]] - What is disappointment, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/DMT and spirituality/]] - How can DMT facilitate spiritual experiences? - [[User:MyUserName|MyUserName]] # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? - [[User:MyUserName|MyUserName]] # [[/Ecological grief/]] - What is ecological grief and what can be done about it? - [[User:Brewerjr|Brewerjr]] # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress? - [[User:MyUserName|MyUserName]] # [[/Embarrassment/]] - What is embarrassment, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? - [[User:MyUserName|MyUserName]] # [[/Emotion knowledge/]] - What is emotion knowledge and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? - [[User:MyUserName|MyUserName]] # [[/Endocannabinoid system and emotion/]] - What is the role of the endocannabinoid system in emotion? - [[User:MyUserName|MyUserName]] # [[/Environmental grief/]] - What is eco-grief, its causes and consequences, and what can be done? - [[User:MyUserName|MyUserName]] # [[/Exercise and endocannabinoids/]] - What is the relationship between exercise and the endocannabinoid system? - [[User:MyUserName|MyUserName]] # [[/Expressive suppression and emotion regulation/]] - What is the role of expressive suppression in emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Fairness and emotion/]] - What is the relation between fairness and emotion? - [[User:MyUserName|MyUserName]] # [[/Fatigue and emotion/]] - What is the effect of fatigue on emotion and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Fear/]] - What is fear, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Fear of working out/]] - What is FOWO and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Fundamental attribution error and emotion/]] - What is the relationship between the FAE and emotion? - [[User:MyUserName|MyUserName]] # [[/Gratitude and subjective wellbeing/]] - What is the relationship between gratitude and subjective wellbeing? - [[User:MyUserName|MyUserName]] # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Heart rate variability and emotion regulation/]] - What is the relationship between HRV and emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Humility/]] - What is humility, what causes it, and is it desirable? - [[User:MyUserName|MyUserName]] # [[/Hypomania and emotion/]] - What are the emotional characteristics of hypomania? - [[User:MyUserName|MyUserName]] # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[Indigenous Australian emotionality]] - In what ways is emotionality experienced by Indigenous Australian people? - [[User:MyUserName|MyUserName]] # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness? - [[User:MyUserName|MyUserName]] # [[/Inspiration/]] - What is inspiration, what causes it, what are its consequences, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Insular cortex and emotion/]] - What role does the insular cortex play in emotion? - [[User:MyUserName|MyUserName]] # [[/Interoception and emotion/]] - What is the relationship between interoception and emotion? - [[User:MyUserName|MyUserName]] # [[/Kama muta/]] - What is kama muta, what are its effects, and how can it be fostered? - [[User:U3183521|U3183521]] # [[/Linguistic relativism and emotion/]] - What is the role of linguistic relativism in emotion? - [[User:MyUserName|MyUserName]] # [[/Menstrual cycle mood disorders/]] - What causes menstrual cycle mood disorders and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and creativity/]] - How can mindfulness enhance creativity? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and driving/]] - How can mindfulness affect driving? - [[User:MyUserName|MyUserName]] # [[/Mindful self-care/]] - What is mindful self-care, why does it matter, and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Mixed emotions/]] - What are mixed emotions, what causes them, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mudita/]] - What is mudita and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Natural disasters and emotion/]] - How do people respond emotionally to natural disasters and how can they be supported? - [[User:MyUserName|MyUserName]] # [[/Nature therapy/]] - What is nature therapy and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Narcissism and emotion/]] - What is the relationship between narcissism and emotion? - [[User:MyUserName|MyUserName]] # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy? - [[User:MyUserName|MyUserName]] # [[/Needle fear/]] - How does needle fear develop, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? - [[User:JorjaFive|JorjaFive]] # [[/Psychological distress/]] - What is PD, what are the main types, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Psychological trauma/]] - What causes psychological trauma, what are the consequences, and how can people recover from psychological trauma? - [[User:U3210431|U3210431]] # [[/Psilocybin assisted psychotherapy/]] - How can psilocybin be used to assist psychotherapy? - [[User:MyUserName|MyUserName]] # [[/Rational compassion/]] - What is rational compassion and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? - [[User:MyUserName|MyUserName]] # [[/Religiosity and coping/]] - What is the relationship between religiosity and coping? - [[User:MyUserName|MyUserName]] # [[/Resentment/]] - What is resentment, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Risk-as-feelings/]] - What is the emotional experience of risk and how does it influence decision-making and behaviour? - [[User:MyUserName|MyUserName]] # [[/Self-esteem and culture/]] - What are the cultural influences on self-esteem? - [[User:MyUserName|MyUserName]] # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? - [[User:MyUserName|MyUserName]] # [[/Social media and suicide prevention/]] - How can social media be used to help prevent suicide? - [[User:MyUserName|MyUserName]] # [[/Sorry business/]] - What is sorry business and what role does it play in Indigenous communities in Australia? - [[User:MyUserName|MyUserName]] # [[/Stress control mindset/]] - What is a SCM, why does it matter, and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Suffering as emotion/]] - What is the emotional experience of suffering and how can people cope with suffering? - [[User:MyUserName|MyUserName]] # [[/Telemental health/]] - What are the pros and cons of TMH and what are the key ingredients for effective TMH practices? - [[User:MyUserName|MyUserName]] # [[/Topophilia/]] - What is topophilia, how does it develop, and what are the psychological impacts? - [[User:MyUserName|MyUserName]] # [[/Triumph/]] - What is triumph, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Unemployment and mental health/]]: What is the relationship between unemployment and mental health? - [[User:MyUserName|MyUserName]] # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? - [[User:MyUserName|MyUserName]] # [[/Wave metaphor for emotion/]] - In what respects is an ocean wave a helpful metaphor for understanding human emotions? - [[User:MyUserName|MyUserName]] # [[/Window of tolerance/]] - What is the window of tolerance and how this concept be used? - [[User:MyUserName|MyUserName]] # [[/Workplace mental health training/]] - What is WMHT, what techniques are used, and what are the impacts? - [[User:MyUserName|MyUserName]] # [[/Zoom fatigue/]] - What is Zoom fatigue, what causes it, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] ==Motivation and emotion== # [[/Financial investing, motivation, and emotion/]] - What role does motivation and emotion play in financial investing? - [[User:MyUserName|MyUserName]] # [[/Hostage negotiation, motivation, and emotion/]] - What role does motivation and emotion play in hostage negotiation? - [[User:U3213549|U3213549]] # [[/Money priming, motivation, and emotion/]] - What is the effect of money priming on motivation and emotion? - [[User:MyUserName|MyUserName]] # [[/Motivational dimensional model of affect/]] - What is the motivational dimensional model of affect and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Napping, motivation, and emotion/]] - What are the motivational and emotional effects of napping? - [[User:MyUserName|MyUserName]] # [[/Overchoice, emotion, and motivation/]] - What are the emotional and motivational effects of overchoice? - [[User:MyUserName|MyUserName]] # [[/Patience and impatience/]] - What are the psychological causes and consequences of patience and impatience? - [[User:MyUserName|MyUserName]] # [[/Reward system, motivation, and emotion/]] - What role does the reward system play in motivation and emotion? - [[User:MyUserName|MyUserName]] [[Category:Motivation and emotion/Book/2022]] ggas3drxqioy4s3ejmyhavlqhj69sy3 2410831 2410790 2022-08-01T21:49:55Z U3216256 2942574 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Academic help-seeking/]] - What are the barriers and enablers of AHS and how can AHS be fostered? - [[User:MyUserName|MyUserName]] # [[/Academic self-regulation/]] - What is academic self-regulation, why does it matter, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? - [[User:MyUserName|MyUserName]] # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it? - [[User:MyUserName|MyUserName]] # [[/Antidepressants and motivation/]] - What are the effects of popular antidepressants on motivation? - [[User:MyUserName|MyUserName]] # [[/Approach motivation/]] - What is approach motivation and how does it lead to behaviour? - [[User:U3189370|U3189370]] # [[/Behavioural economics and motivation/]] - What aspects of motivation theory are useful in behavioural economics? - [[User:U3141987|U3141987]] # [[/Behavioural model of health services/]] - What is the BMHS and how can it be used? - [[UserFV:MyUserName|MyUserName]] # [[/Beneficence as a psychological need/]] - What is beneficence and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention? - [[User:MyUserName|MyUserName]] # [[/Choice overload/]] - How much choose is too much? How much choice is enough? - [[User:MyUserName|MyUserName]] # [[/Chunking and goal pursuit/]] - How does chunking affect goal pursuit? - [[User:MyUserName|MyUserName]] # [[/Cognitive entrenchment/]] - What is cognitive entrenchment and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[/Climate change helplessness/]] - How does learned helpless impact motivation to engage in behaviours to limit climate change? - [[User:MyUserName|MyUserName]] # [[/Closeness communication bias/]] - What is the CCB, why does it occur, and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirably outcomes? - [[User:MyUserName|MyUserName]] # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories? - [[User:KingMob221|KingMob221]] # [[/Construal level theory/]] - What is construal level theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Courage motivation/]] - What is courage, what motivates courage, and how can courage be enhanced? -[[User:U3213871|U3213871] # [[/Death drive/]] - What is the death drive and how can it be negotiated? - [[User:U3086459|U3086459]] # [[/Drugs-violence nexus and motivation/]] - What is the role of motivation in the drugs-violence nexus? - [[User:MyUserName|MyUserName]] # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD? - [[User:MyUserName|MyUserName]] # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? - [[User:MyUserName|MyUserName]] # [[/Equity theory/]] - What is equity theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/ERG theory/]] - What is Alderfer's ERG theory? - [[User:MyUserName|MyUserName]] # [[/Frame of reference and motivation/]] - How does frame of reference affect motivation? - [[User:MyUserName|MyUserName]] # [[/Freedom and motivation/]] - What is the effect of freedom on motivation? - [[User:MyUserName|MyUserName]] # [[/Fully functioning person/]] - What is a FFP and how can full functioning be developed? - [[User:U3217727|U3217727]] # [[/Functional fixedness/]] - What is functional fixedness and how can it be overcome? - [[User:U3214117]] # [[/Functional imagery training/]] - What is FIT and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Gamification and work motivation/]] - How can gamification enhance work motivation? - [[User:MyUserName|MyUserName]] # [[/Giving up goals/]] - When should we give up goals and when should we persist? - [[User:MyUserName|MyUserName]] # [[/Green prescription motivation/]] - What motivates green prescription compliance? - [[User:MyUserName|MyUserName]] # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour? - [[User:MyUserName|MyUserName]] # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis, what is the evidence, and how does it help to understand drug addiction? - [[U3218292|U3218292]] # [[/Honesty motivation/]] - What motivates honesty? - [[User:MyUserName|MyUserName]] # [[/Humour and work/]] - What is the role of humour in the workplace? - [[User:MyUserName|MyUserName]] # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? - [[User:MyUserName|MyUserName]] # [[/Kindness motivation/]] - What motivates kindness? - [[User:MyUserName|MyUserName]] # [[/Motivational music and exercise/]] - How can music be used to help motivate exercise? - [[User:MyUserName|MyUserName]] # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Nucleus accumbens and motivation/]] - What role does the nucleus accumbens play in motivation? - [[User:MyUserName|MyUserName]] # [[/Physiological needs/]] - How do human's physiological needs affect motivation? - [[U3203655]] # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19? - [[User:MyUserName|MyUserName]] # [[/Relative deprivation and motivation/]] - What is the effect of relative deprivation on motivation? - [[User:MyUserName|MyUserName]] # [[/Retrospective regret/]] - What is the motivational role of retrospective regret? - [[User:MyUserName|MyUserName]] # [[/Revenge motivation/]] - What motivates revenge and how does it affect us? - [[User:MyUserName|MyUserName]] # [[/Self-efficacy and academic achievement/]] - What role does self-efficacy play in academic achievement? - [[User:U943292|U943292]] # [[/Self-efficacy and achievement/]] - What role does self-efficacy play in achievement outcomes? - [[User:MyUserName|MyUserName]] # [[/Sexual harassment at work motivation/]] - What motivates sexual harassment at work and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Signature strengths/]] - What are signature strengths and how can they be applied? - [[User:MyUserName|MyUserName]] # [[/Social cure/]] - What is the social cure and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Stretch goals/]] - What are stretch goals? Do they work? - [[User:MyUserName|MyUserName]] # [[/Sublimation/]] - What is sublimation and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Survival needs and motivation/]] - What are survival needs and how do they influence motivation? - [[User:MyUserName|MyUserName]] # [[/Task initiation/]] - What are the challenges with task initiation and how to get get started? - [[User:MyUserName|MyUserName]] # [[/Theoretical domains framework/]] - What is the TDF and how can be used to guide behaviour change? - [[User:MyUserName|MyUserName]] # [[/Time and motivation/]] - What is the effect of time on motivation? - [[User:MyUserName|MyUserName]] # [[/Time management/]] - How can one's time be managed effectively? - [[User:MyUserName|MyUserName]] # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - [[User:MyUserName|MyUserName]] # [[/Uncertainty avoidance/]] - What is uncertainty avoidance, why does it occur, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Urgency bias and productivity/]] - What is the impact of urgency bias on productivity and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Vocational identity/]] - What is vocational identity and how does it develop? - [[User:MyUserName|MyUserName]] # [[/Volunteer tourism motivation/]] - What motivates volunteer tourism? - [[User:Efost|Efost]] # [[/Wanting and liking/]] - What are the similarities and differences between wanting and liking, and what are the implications? - [[User:MyUserName|MyUserName]] # [[/Work breaks, well-being, and productivity/]] - How do work breaks affect well-being and productivity? - [[User:MyUserName|MyUserName]] # [[/Work and flow/]] - What characteristics of work can produce flow and how can flow at work be fostered? - [[User:MyUserName|MyUserName]] ==Emotion== # [[/Animal emotion/]] - What is the emotional experience of animals? - [[User:MyUserName|MyUserName]] # [[/Attributions and emotion/]] - How do attributions affect emotion? - [[User:MyUserName|MyUserName]] # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? - [[User:MyUserName|MyUserName]] # [[/Benzodiazepines and emotion/]] - What are the effects of benzodiazepines on emotion? - FulaAjeo22 # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? - [[User:MyUserName|MyUserName]] # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? - [[User:U3211603|U3211603]] # [[/Cognitive dissonance reduction/]] - What strategies do people use to reduce cognitive dissonance and how effective are they? - [[User:MyUserName|MyUserName]] # [[/Colonisation and emotion in Australia/]] - What are the emotional responses to colonisation in Australia? - [[User:MyUserName|MyUserName]] # [[/Compassion/]] - What is compassion, what are its pros and cons, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? - [[User:MyUserName|MyUserName]] # [[/Contempt/]] - What is contempt, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Core emotions/]] - What are the core emotions and what is their function? MyUserName # [[/Creative arts and trauma/]] - How can creative arts help in dealing with trauma? - [[User:MyUserName|MyUserName]] # [[/Cultural influences on shame, guilt, and pride/]] - How does culture influence shame, guilt, and pride? - [[User:MyUserName|MyUserName]] # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? - [[User:MyUserName|MyUserName]] # [[/Difficult conversations and emotion/]] - What communication and emotional skills are needed to successfully negotiate difficult conversations? - [[User:MyUserName|MyUserName]] # [[/Disappointment/]] - What is disappointment, what causes it, and how can it be managed? - [[User:U3216256|U3216256]] # [[/DMT and spirituality/]] - How can DMT facilitate spiritual experiences? - [[User:MyUserName|MyUserName]] # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? - [[User:MyUserName|MyUserName]] # [[/Ecological grief/]] - What is ecological grief and what can be done about it? - [[User:Brewerjr|Brewerjr]] # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress? - [[User:MyUserName|MyUserName]] # [[/Embarrassment/]] - What is embarrassment, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? - [[User:MyUserName|MyUserName]] # [[/Emotion knowledge/]] - What is emotion knowledge and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? - [[User:MyUserName|MyUserName]] # [[/Endocannabinoid system and emotion/]] - What is the role of the endocannabinoid system in emotion? - [[User:MyUserName|MyUserName]] # [[/Environmental grief/]] - What is eco-grief, its causes and consequences, and what can be done? - [[User:MyUserName|MyUserName]] # [[/Exercise and endocannabinoids/]] - What is the relationship between exercise and the endocannabinoid system? - [[User:MyUserName|MyUserName]] # [[/Expressive suppression and emotion regulation/]] - What is the role of expressive suppression in emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Fairness and emotion/]] - What is the relation between fairness and emotion? - [[User:MyUserName|MyUserName]] # [[/Fatigue and emotion/]] - What is the effect of fatigue on emotion and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Fear/]] - What is fear, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Fear of working out/]] - What is FOWO and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Fundamental attribution error and emotion/]] - What is the relationship between the FAE and emotion? - [[User:MyUserName|MyUserName]] # [[/Gratitude and subjective wellbeing/]] - What is the relationship between gratitude and subjective wellbeing? - [[User:MyUserName|MyUserName]] # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Heart rate variability and emotion regulation/]] - What is the relationship between HRV and emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Humility/]] - What is humility, what causes it, and is it desirable? - [[User:MyUserName|MyUserName]] # [[/Hypomania and emotion/]] - What are the emotional characteristics of hypomania? - [[User:MyUserName|MyUserName]] # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[Indigenous Australian emotionality]] - In what ways is emotionality experienced by Indigenous Australian people? - [[User:MyUserName|MyUserName]] # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness? - [[User:MyUserName|MyUserName]] # [[/Inspiration/]] - What is inspiration, what causes it, what are its consequences, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Insular cortex and emotion/]] - What role does the insular cortex play in emotion? - [[User:MyUserName|MyUserName]] # [[/Interoception and emotion/]] - What is the relationship between interoception and emotion? - [[User:MyUserName|MyUserName]] # [[/Kama muta/]] - What is kama muta, what are its effects, and how can it be fostered? - [[User:U3183521|U3183521]] # [[/Linguistic relativism and emotion/]] - What is the role of linguistic relativism in emotion? - [[User:MyUserName|MyUserName]] # [[/Menstrual cycle mood disorders/]] - What causes menstrual cycle mood disorders and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and creativity/]] - How can mindfulness enhance creativity? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and driving/]] - How can mindfulness affect driving? - [[User:MyUserName|MyUserName]] # [[/Mindful self-care/]] - What is mindful self-care, why does it matter, and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Mixed emotions/]] - What are mixed emotions, what causes them, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mudita/]] - What is mudita and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Natural disasters and emotion/]] - How do people respond emotionally to natural disasters and how can they be supported? - [[User:MyUserName|MyUserName]] # [[/Nature therapy/]] - What is nature therapy and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Narcissism and emotion/]] - What is the relationship between narcissism and emotion? - [[User:MyUserName|MyUserName]] # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy? - [[User:MyUserName|MyUserName]] # [[/Needle fear/]] - How does needle fear develop, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? - [[User:JorjaFive|JorjaFive]] # [[/Psychological distress/]] - What is PD, what are the main types, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Psychological trauma/]] - What causes psychological trauma, what are the consequences, and how can people recover from psychological trauma? - [[User:U3210431|U3210431]] # [[/Psilocybin assisted psychotherapy/]] - How can psilocybin be used to assist psychotherapy? - [[User:MyUserName|MyUserName]] # [[/Rational compassion/]] - What is rational compassion and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? - [[User:MyUserName|MyUserName]] # [[/Religiosity and coping/]] - What is the relationship between religiosity and coping? - [[User:MyUserName|MyUserName]] # [[/Resentment/]] - What is resentment, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Risk-as-feelings/]] - What is the emotional experience of risk and how does it influence decision-making and behaviour? - [[User:MyUserName|MyUserName]] # [[/Self-esteem and culture/]] - What are the cultural influences on self-esteem? - [[User:MyUserName|MyUserName]] # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? - [[User:MyUserName|MyUserName]] # [[/Social media and suicide prevention/]] - How can social media be used to help prevent suicide? - [[User:MyUserName|MyUserName]] # [[/Sorry business/]] - What is sorry business and what role does it play in Indigenous communities in Australia? - [[User:MyUserName|MyUserName]] # [[/Stress control mindset/]] - What is a SCM, why does it matter, and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Suffering as emotion/]] - What is the emotional experience of suffering and how can people cope with suffering? - [[User:MyUserName|MyUserName]] # [[/Telemental health/]] - What are the pros and cons of TMH and what are the key ingredients for effective TMH practices? - [[User:MyUserName|MyUserName]] # [[/Topophilia/]] - What is topophilia, how does it develop, and what are the psychological impacts? - [[User:MyUserName|MyUserName]] # [[/Triumph/]] - What is triumph, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Unemployment and mental health/]]: What is the relationship between unemployment and mental health? - [[User:MyUserName|MyUserName]] # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? - [[User:MyUserName|MyUserName]] # [[/Wave metaphor for emotion/]] - In what respects is an ocean wave a helpful metaphor for understanding human emotions? - [[User:MyUserName|MyUserName]] # [[/Window of tolerance/]] - What is the window of tolerance and how this concept be used? - [[User:MyUserName|MyUserName]] # [[/Workplace mental health training/]] - What is WMHT, what techniques are used, and what are the impacts? - [[User:MyUserName|MyUserName]] # [[/Zoom fatigue/]] - What is Zoom fatigue, what causes it, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] ==Motivation and emotion== # [[/Financial investing, motivation, and emotion/]] - What role does motivation and emotion play in financial investing? - [[User:MyUserName|MyUserName]] # [[/Hostage negotiation, motivation, and emotion/]] - What role does motivation and emotion play in hostage negotiation? - [[User:U3213549|U3213549]] # [[/Money priming, motivation, and emotion/]] - What is the effect of money priming on motivation and emotion? - [[User:MyUserName|MyUserName]] # [[/Motivational dimensional model of affect/]] - What is the motivational dimensional model of affect and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Napping, motivation, and emotion/]] - What are the motivational and emotional effects of napping? - [[User:MyUserName|MyUserName]] # [[/Overchoice, emotion, and motivation/]] - What are the emotional and motivational effects of overchoice? - [[User:MyUserName|MyUserName]] # [[/Patience and impatience/]] - What are the psychological causes and consequences of patience and impatience? - [[User:MyUserName|MyUserName]] # [[/Reward system, motivation, and emotion/]] - What role does the reward system play in motivation and emotion? - [[User:MyUserName|MyUserName]] [[Category:Motivation and emotion/Book/2022]] shru7mbrf65j7xv29kx7hfdyonimd52 2410853 2410831 2022-08-02T00:07:02Z U3083720 2947468 /* Motivation */ added username to Psilocybin assisted psychotherapy wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Academic help-seeking/]] - What are the barriers and enablers of AHS and how can AHS be fostered? - [[User:MyUserName|MyUserName]] # [[/Academic self-regulation/]] - What is academic self-regulation, why does it matter, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? - [[User:MyUserName|MyUserName]] # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it? - [[User:MyUserName|MyUserName]] # [[/Antidepressants and motivation/]] - What are the effects of popular antidepressants on motivation? - [[User:MyUserName|MyUserName]] # [[/Approach motivation/]] - What is approach motivation and how does it lead to behaviour? - [[User:U3189370|U3189370]] # [[/Behavioural economics and motivation/]] - What aspects of motivation theory are useful in behavioural economics? - [[User:U3141987|U3141987]] # [[/Behavioural model of health services/]] - What is the BMHS and how can it be used? - [[UserFV:MyUserName|MyUserName]] # [[/Beneficence as a psychological need/]] - What is beneficence and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention? - [[User:MyUserName|MyUserName]] # [[/Choice overload/]] - How much choose is too much? How much choice is enough? - [[User:MyUserName|MyUserName]] # [[/Chunking and goal pursuit/]] - How does chunking affect goal pursuit? - [[User:MyUserName|MyUserName]] # [[/Cognitive entrenchment/]] - What is cognitive entrenchment and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[/Climate change helplessness/]] - How does learned helpless impact motivation to engage in behaviours to limit climate change? - [[User:MyUserName|MyUserName]] # [[/Closeness communication bias/]] - What is the CCB, why does it occur, and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirably outcomes? - [[User:MyUserName|MyUserName]] # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories? - [[User:KingMob221|KingMob221]] # [[/Construal level theory/]] - What is construal level theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Courage motivation/]] - What is courage, what motivates courage, and how can courage be enhanced? -[[User:U3213871|U3213871] # [[/Death drive/]] - What is the death drive and how can it be negotiated? - [[User:U3086459|U3086459]] # [[/Drugs-violence nexus and motivation/]] - What is the role of motivation in the drugs-violence nexus? - [[User:MyUserName|MyUserName]] # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD? - [[User:MyUserName|MyUserName]] # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? - [[User:MyUserName|MyUserName]] # [[/Equity theory/]] - What is equity theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/ERG theory/]] - What is Alderfer's ERG theory? - [[User:MyUserName|MyUserName]] # [[/Frame of reference and motivation/]] - How does frame of reference affect motivation? - [[User:MyUserName|MyUserName]] # [[/Freedom and motivation/]] - What is the effect of freedom on motivation? - [[User:MyUserName|MyUserName]] # [[/Fully functioning person/]] - What is a FFP and how can full functioning be developed? - [[User:U3217727|U3217727]] # [[/Functional fixedness/]] - What is functional fixedness and how can it be overcome? - [[User:U3214117]] # [[/Functional imagery training/]] - What is FIT and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Gamification and work motivation/]] - How can gamification enhance work motivation? - [[User:MyUserName|MyUserName]] # [[/Giving up goals/]] - When should we give up goals and when should we persist? - [[User:MyUserName|MyUserName]] # [[/Green prescription motivation/]] - What motivates green prescription compliance? - [[User:MyUserName|MyUserName]] # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour? - [[User:MyUserName|MyUserName]] # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis, what is the evidence, and how does it help to understand drug addiction? - [[U3218292|U3218292]] # [[/Honesty motivation/]] - What motivates honesty? - [[User:MyUserName|MyUserName]] # [[/Humour and work/]] - What is the role of humour in the workplace? - [[User:MyUserName|MyUserName]] # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? - [[User:MyUserName|MyUserName]] # [[/Kindness motivation/]] - What motivates kindness? - [[User:MyUserName|MyUserName]] # [[/Motivational music and exercise/]] - How can music be used to help motivate exercise? - [[User:MyUserName|MyUserName]] # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Nucleus accumbens and motivation/]] - What role does the nucleus accumbens play in motivation? - [[User:MyUserName|MyUserName]] # [[/Physiological needs/]] - How do human's physiological needs affect motivation? - [[U3203655]] # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19? - [[User:MyUserName|MyUserName]] # [[/Relative deprivation and motivation/]] - What is the effect of relative deprivation on motivation? - [[User:MyUserName|MyUserName]] # [[/Retrospective regret/]] - What is the motivational role of retrospective regret? - [[User:MyUserName|MyUserName]] # [[/Revenge motivation/]] - What motivates revenge and how does it affect us? - [[User:MyUserName|MyUserName]] # [[/Self-efficacy and academic achievement/]] - What role does self-efficacy play in academic achievement? - [[User:U943292|U943292]] # [[/Self-efficacy and achievement/]] - What role does self-efficacy play in achievement outcomes? - [[User:MyUserName|MyUserName]] # [[/Sexual harassment at work motivation/]] - What motivates sexual harassment at work and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Signature strengths/]] - What are signature strengths and how can they be applied? - [[User:MyUserName|MyUserName]] # [[/Social cure/]] - What is the social cure and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Stretch goals/]] - What are stretch goals? Do they work? - [[User:MyUserName|MyUserName]] # [[/Sublimation/]] - What is sublimation and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Survival needs and motivation/]] - What are survival needs and how do they influence motivation? - [[User:MyUserName|MyUserName]] # [[/Task initiation/]] - What are the challenges with task initiation and how to get get started? - [[User:MyUserName|MyUserName]] # [[/Theoretical domains framework/]] - What is the TDF and how can be used to guide behaviour change? - [[User:MyUserName|MyUserName]] # [[/Time and motivation/]] - What is the effect of time on motivation? - [[User:MyUserName|MyUserName]] # [[/Time management/]] - How can one's time be managed effectively? - [[User:MyUserName|MyUserName]] # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - [[User:MyUserName|MyUserName]] # [[/Uncertainty avoidance/]] - What is uncertainty avoidance, why does it occur, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Urgency bias and productivity/]] - What is the impact of urgency bias on productivity and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Vocational identity/]] - What is vocational identity and how does it develop? - [[User:MyUserName|MyUserName]] # [[/Volunteer tourism motivation/]] - What motivates volunteer tourism? - [[User:Efost|Efost]] # [[/Wanting and liking/]] - What are the similarities and differences between wanting and liking, and what are the implications? - [[User:MyUserName|MyUserName]] # [[/Work breaks, well-being, and productivity/]] - How do work breaks affect well-being and productivity? - [[User:MyUserName|MyUserName]] # [[/Work and flow/]] - What characteristics of work can produce flow and how can flow at work be fostered? - [[User:MyUserName|MyUserName]] ==Emotion== # [[/Animal emotion/]] - What is the emotional experience of animals? - [[User:MyUserName|MyUserName]] # [[/Attributions and emotion/]] - How do attributions affect emotion? - [[User:MyUserName|MyUserName]] # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? - [[User:MyUserName|MyUserName]] # [[/Benzodiazepines and emotion/]] - What are the effects of benzodiazepines on emotion? - FulaAjeo22 # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? - [[User:MyUserName|MyUserName]] # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? - [[User:U3211603|U3211603]] # [[/Cognitive dissonance reduction/]] - What strategies do people use to reduce cognitive dissonance and how effective are they? - [[User:MyUserName|MyUserName]] # [[/Colonisation and emotion in Australia/]] - What are the emotional responses to colonisation in Australia? - [[User:MyUserName|MyUserName]] # [[/Compassion/]] - What is compassion, what are its pros and cons, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? - [[User:MyUserName|MyUserName]] # [[/Contempt/]] - What is contempt, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Core emotions/]] - What are the core emotions and what is their function? MyUserName # [[/Creative arts and trauma/]] - How can creative arts help in dealing with trauma? - [[User:MyUserName|MyUserName]] # [[/Cultural influences on shame, guilt, and pride/]] - How does culture influence shame, guilt, and pride? - [[User:MyUserName|MyUserName]] # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? - [[User:MyUserName|MyUserName]] # [[/Difficult conversations and emotion/]] - What communication and emotional skills are needed to successfully negotiate difficult conversations? - [[User:MyUserName|MyUserName]] # [[/Disappointment/]] - What is disappointment, what causes it, and how can it be managed? - [[User:U3216256|U3216256]] # [[/DMT and spirituality/]] - How can DMT facilitate spiritual experiences? - [[User:MyUserName|MyUserName]] # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? - [[User:MyUserName|MyUserName]] # [[/Ecological grief/]] - What is ecological grief and what can be done about it? - [[User:Brewerjr|Brewerjr]] # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress? - [[User:MyUserName|MyUserName]] # [[/Embarrassment/]] - What is embarrassment, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? - [[User:MyUserName|MyUserName]] # [[/Emotion knowledge/]] - What is emotion knowledge and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? - [[User:MyUserName|MyUserName]] # [[/Endocannabinoid system and emotion/]] - What is the role of the endocannabinoid system in emotion? - [[User:MyUserName|MyUserName]] # [[/Environmental grief/]] - What is eco-grief, its causes and consequences, and what can be done? - [[User:MyUserName|MyUserName]] # [[/Exercise and endocannabinoids/]] - What is the relationship between exercise and the endocannabinoid system? - [[User:MyUserName|MyUserName]] # [[/Expressive suppression and emotion regulation/]] - What is the role of expressive suppression in emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Fairness and emotion/]] - What is the relation between fairness and emotion? - [[User:MyUserName|MyUserName]] # [[/Fatigue and emotion/]] - What is the effect of fatigue on emotion and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Fear/]] - What is fear, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Fear of working out/]] - What is FOWO and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Fundamental attribution error and emotion/]] - What is the relationship between the FAE and emotion? - [[User:MyUserName|MyUserName]] # [[/Gratitude and subjective wellbeing/]] - What is the relationship between gratitude and subjective wellbeing? - [[User:MyUserName|MyUserName]] # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Heart rate variability and emotion regulation/]] - What is the relationship between HRV and emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Humility/]] - What is humility, what causes it, and is it desirable? - [[User:MyUserName|MyUserName]] # [[/Hypomania and emotion/]] - What are the emotional characteristics of hypomania? - [[User:MyUserName|MyUserName]] # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[Indigenous Australian emotionality]] - In what ways is emotionality experienced by Indigenous Australian people? - [[User:MyUserName|MyUserName]] # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness? - [[User:MyUserName|MyUserName]] # [[/Inspiration/]] - What is inspiration, what causes it, what are its consequences, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Insular cortex and emotion/]] - What role does the insular cortex play in emotion? - [[User:MyUserName|MyUserName]] # [[/Interoception and emotion/]] - What is the relationship between interoception and emotion? - [[User:MyUserName|MyUserName]] # [[/Kama muta/]] - What is kama muta, what are its effects, and how can it be fostered? - [[User:U3183521|U3183521]] # [[/Linguistic relativism and emotion/]] - What is the role of linguistic relativism in emotion? - [[User:MyUserName|MyUserName]] # [[/Menstrual cycle mood disorders/]] - What causes menstrual cycle mood disorders and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and creativity/]] - How can mindfulness enhance creativity? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and driving/]] - How can mindfulness affect driving? - [[User:MyUserName|MyUserName]] # [[/Mindful self-care/]] - What is mindful self-care, why does it matter, and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Mixed emotions/]] - What are mixed emotions, what causes them, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mudita/]] - What is mudita and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Natural disasters and emotion/]] - How do people respond emotionally to natural disasters and how can they be supported? - [[User:MyUserName|MyUserName]] # [[/Nature therapy/]] - What is nature therapy and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Narcissism and emotion/]] - What is the relationship between narcissism and emotion? - [[User:MyUserName|MyUserName]] # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy? - [[User:MyUserName|MyUserName]] # [[/Needle fear/]] - How does needle fear develop, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? - [[User:JorjaFive|JorjaFive]] # [[/Psychological distress/]] - What is PD, what are the main types, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Psychological trauma/]] - What causes psychological trauma, what are the consequences, and how can people recover from psychological trauma? - [[User:U3210431|U3210431]] # [[/Psilocybin assisted psychotherapy/]] - How can psilocybin be used to assist psychotherapy? - [[User:U3083720|MyUserName]] # [[/Rational compassion/]] - What is rational compassion and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? - [[User:MyUserName|MyUserName]] # [[/Religiosity and coping/]] - What is the relationship between religiosity and coping? - [[User:MyUserName|MyUserName]] # [[/Resentment/]] - What is resentment, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Risk-as-feelings/]] - What is the emotional experience of risk and how does it influence decision-making and behaviour? - [[User:MyUserName|MyUserName]] # [[/Self-esteem and culture/]] - What are the cultural influences on self-esteem? - [[User:MyUserName|MyUserName]] # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? - [[User:MyUserName|MyUserName]] # [[/Social media and suicide prevention/]] - How can social media be used to help prevent suicide? - [[User:MyUserName|MyUserName]] # [[/Sorry business/]] - What is sorry business and what role does it play in Indigenous communities in Australia? - [[User:MyUserName|MyUserName]] # [[/Stress control mindset/]] - What is a SCM, why does it matter, and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Suffering as emotion/]] - What is the emotional experience of suffering and how can people cope with suffering? - [[User:MyUserName|MyUserName]] # [[/Telemental health/]] - What are the pros and cons of TMH and what are the key ingredients for effective TMH practices? - [[User:MyUserName|MyUserName]] # [[/Topophilia/]] - What is topophilia, how does it develop, and what are the psychological impacts? - [[User:MyUserName|MyUserName]] # [[/Triumph/]] - What is triumph, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Unemployment and mental health/]]: What is the relationship between unemployment and mental health? - [[User:MyUserName|MyUserName]] # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? - [[User:MyUserName|MyUserName]] # [[/Wave metaphor for emotion/]] - In what respects is an ocean wave a helpful metaphor for understanding human emotions? - [[User:MyUserName|MyUserName]] # [[/Window of tolerance/]] - What is the window of tolerance and how this concept be used? - [[User:MyUserName|MyUserName]] # [[/Workplace mental health training/]] - What is WMHT, what techniques are used, and what are the impacts? - [[User:MyUserName|MyUserName]] # [[/Zoom fatigue/]] - What is Zoom fatigue, what causes it, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] ==Motivation and emotion== # [[/Financial investing, motivation, and emotion/]] - What role does motivation and emotion play in financial investing? - [[User:MyUserName|MyUserName]] # [[/Hostage negotiation, motivation, and emotion/]] - What role does motivation and emotion play in hostage negotiation? - [[User:U3213549|U3213549]] # [[/Money priming, motivation, and emotion/]] - What is the effect of money priming on motivation and emotion? - [[User:MyUserName|MyUserName]] # [[/Motivational dimensional model of affect/]] - What is the motivational dimensional model of affect and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Napping, motivation, and emotion/]] - What are the motivational and emotional effects of napping? - [[User:MyUserName|MyUserName]] # [[/Overchoice, emotion, and motivation/]] - What are the emotional and motivational effects of overchoice? - [[User:MyUserName|MyUserName]] # [[/Patience and impatience/]] - What are the psychological causes and consequences of patience and impatience? - [[User:MyUserName|MyUserName]] # [[/Reward system, motivation, and emotion/]] - What role does the reward system play in motivation and emotion? - [[User:MyUserName|MyUserName]] [[Category:Motivation and emotion/Book/2022]] a3qteshdsd8ki2t44mfpb67ae0zi0mq 2410855 2410853 2022-08-02T00:10:12Z U3083720 2947468 /* Motivation */ wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Academic help-seeking/]] - What are the barriers and enablers of AHS and how can AHS be fostered? - [[User:MyUserName|MyUserName]] # [[/Academic self-regulation/]] - What is academic self-regulation, why does it matter, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? - [[User:MyUserName|MyUserName]] # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it? - [[User:MyUserName|MyUserName]] # [[/Antidepressants and motivation/]] - What are the effects of popular antidepressants on motivation? - [[User:MyUserName|MyUserName]] # [[/Approach motivation/]] - What is approach motivation and how does it lead to behaviour? - [[User:U3189370|U3189370]] # [[/Behavioural economics and motivation/]] - What aspects of motivation theory are useful in behavioural economics? - [[User:U3141987|U3141987]] # [[/Behavioural model of health services/]] - What is the BMHS and how can it be used? - [[UserFV:MyUserName|MyUserName]] # [[/Beneficence as a psychological need/]] - What is beneficence and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention? - [[User:MyUserName|MyUserName]] # [[/Choice overload/]] - How much choose is too much? How much choice is enough? - [[User:MyUserName|MyUserName]] # [[/Chunking and goal pursuit/]] - How does chunking affect goal pursuit? - [[User:MyUserName|MyUserName]] # [[/Cognitive entrenchment/]] - What is cognitive entrenchment and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[/Climate change helplessness/]] - How does learned helpless impact motivation to engage in behaviours to limit climate change? - [[User:MyUserName|MyUserName]] # [[/Closeness communication bias/]] - What is the CCB, why does it occur, and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirably outcomes? - [[User:MyUserName|MyUserName]] # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories? - [[User:KingMob221|KingMob221]] # [[/Construal level theory/]] - What is construal level theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Courage motivation/]] - What is courage, what motivates courage, and how can courage be enhanced? -[[User:U3213871|U3213871] # [[/Death drive/]] - What is the death drive and how can it be negotiated? - [[User:U3086459|U3086459]] # [[/Drugs-violence nexus and motivation/]] - What is the role of motivation in the drugs-violence nexus? - [[User:MyUserName|MyUserName]] # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD? - [[User:MyUserName|MyUserName]] # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? - [[User:MyUserName|MyUserName]] # [[/Equity theory/]] - What is equity theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/ERG theory/]] - What is Alderfer's ERG theory? - [[User:MyUserName|MyUserName]] # [[/Frame of reference and motivation/]] - How does frame of reference affect motivation? - [[User:MyUserName|MyUserName]] # [[/Freedom and motivation/]] - What is the effect of freedom on motivation? - [[User:MyUserName|MyUserName]] # [[/Fully functioning person/]] - What is a FFP and how can full functioning be developed? - [[User:U3217727|U3217727]] # [[/Functional fixedness/]] - What is functional fixedness and how can it be overcome? - [[User:U3214117]] # [[/Functional imagery training/]] - What is FIT and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Gamification and work motivation/]] - How can gamification enhance work motivation? - [[User:MyUserName|MyUserName]] # [[/Giving up goals/]] - When should we give up goals and when should we persist? - [[User:MyUserName|MyUserName]] # [[/Green prescription motivation/]] - What motivates green prescription compliance? - [[User:MyUserName|MyUserName]] # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour? - [[User:MyUserName|MyUserName]] # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis, what is the evidence, and how does it help to understand drug addiction? - [[U3218292|U3218292]] # [[/Honesty motivation/]] - What motivates honesty? - [[User:MyUserName|MyUserName]] # [[/Humour and work/]] - What is the role of humour in the workplace? - [[User:MyUserName|MyUserName]] # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? - [[User:MyUserName|MyUserName]] # [[/Kindness motivation/]] - What motivates kindness? - [[User:MyUserName|MyUserName]] # [[/Motivational music and exercise/]] - How can music be used to help motivate exercise? - [[User:MyUserName|MyUserName]] # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Nucleus accumbens and motivation/]] - What role does the nucleus accumbens play in motivation? - [[User:MyUserName|MyUserName]] # [[/Physiological needs/]] - How do human's physiological needs affect motivation? - [[U3203655]] # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19? - [[User:MyUserName|MyUserName]] # [[/Relative deprivation and motivation/]] - What is the effect of relative deprivation on motivation? - [[User:MyUserName|MyUserName]] # [[/Retrospective regret/]] - What is the motivational role of retrospective regret? - [[User:MyUserName|MyUserName]] # [[/Revenge motivation/]] - What motivates revenge and how does it affect us? - [[User:MyUserName|MyUserName]] # [[/Self-efficacy and academic achievement/]] - What role does self-efficacy play in academic achievement? - [[User:U943292|U943292]] # [[/Self-efficacy and achievement/]] - What role does self-efficacy play in achievement outcomes? - [[User:MyUserName|MyUserName]] # [[/Sexual harassment at work motivation/]] - What motivates sexual harassment at work and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Signature strengths/]] - What are signature strengths and how can they be applied? - [[User:MyUserName|MyUserName]] # [[/Social cure/]] - What is the social cure and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Stretch goals/]] - What are stretch goals? Do they work? - [[User:MyUserName|MyUserName]] # [[/Sublimation/]] - What is sublimation and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Survival needs and motivation/]] - What are survival needs and how do they influence motivation? - [[User:MyUserName|MyUserName]] # [[/Task initiation/]] - What are the challenges with task initiation and how to get get started? - [[User:MyUserName|MyUserName]] # [[/Theoretical domains framework/]] - What is the TDF and how can be used to guide behaviour change? - [[User:MyUserName|MyUserName]] # [[/Time and motivation/]] - What is the effect of time on motivation? - [[User:MyUserName|MyUserName]] # [[/Time management/]] - How can one's time be managed effectively? - [[User:MyUserName|MyUserName]] # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - [[User:MyUserName|MyUserName]] # [[/Uncertainty avoidance/]] - What is uncertainty avoidance, why does it occur, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Urgency bias and productivity/]] - What is the impact of urgency bias on productivity and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Vocational identity/]] - What is vocational identity and how does it develop? - [[User:MyUserName|MyUserName]] # [[/Volunteer tourism motivation/]] - What motivates volunteer tourism? - [[User:Efost|Efost]] # [[/Wanting and liking/]] - What are the similarities and differences between wanting and liking, and what are the implications? - [[User:MyUserName|MyUserName]] # [[/Work breaks, well-being, and productivity/]] - How do work breaks affect well-being and productivity? - [[User:MyUserName|MyUserName]] # [[/Work and flow/]] - What characteristics of work can produce flow and how can flow at work be fostered? - [[User:MyUserName|MyUserName]] ==Emotion== # [[/Animal emotion/]] - What is the emotional experience of animals? - [[User:MyUserName|MyUserName]] # [[/Attributions and emotion/]] - How do attributions affect emotion? - [[User:MyUserName|MyUserName]] # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? - [[User:MyUserName|MyUserName]] # [[/Benzodiazepines and emotion/]] - What are the effects of benzodiazepines on emotion? - FulaAjeo22 # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? - [[User:MyUserName|MyUserName]] # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? - [[User:U3211603|U3211603]] # [[/Cognitive dissonance reduction/]] - What strategies do people use to reduce cognitive dissonance and how effective are they? - [[User:MyUserName|MyUserName]] # [[/Colonisation and emotion in Australia/]] - What are the emotional responses to colonisation in Australia? - [[User:MyUserName|MyUserName]] # [[/Compassion/]] - What is compassion, what are its pros and cons, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? - [[User:MyUserName|MyUserName]] # [[/Contempt/]] - What is contempt, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Core emotions/]] - What are the core emotions and what is their function? MyUserName # [[/Creative arts and trauma/]] - How can creative arts help in dealing with trauma? - [[User:MyUserName|MyUserName]] # [[/Cultural influences on shame, guilt, and pride/]] - How does culture influence shame, guilt, and pride? - [[User:MyUserName|MyUserName]] # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? - [[User:MyUserName|MyUserName]] # [[/Difficult conversations and emotion/]] - What communication and emotional skills are needed to successfully negotiate difficult conversations? - [[User:MyUserName|MyUserName]] # [[/Disappointment/]] - What is disappointment, what causes it, and how can it be managed? - [[User:U3216256|U3216256]] # [[/DMT and spirituality/]] - How can DMT facilitate spiritual experiences? - [[User:MyUserName|MyUserName]] # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? - [[User:MyUserName|MyUserName]] # [[/Ecological grief/]] - What is ecological grief and what can be done about it? - [[User:Brewerjr|Brewerjr]] # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress? - [[User:MyUserName|MyUserName]] # [[/Embarrassment/]] - What is embarrassment, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? - [[User:MyUserName|MyUserName]] # [[/Emotion knowledge/]] - What is emotion knowledge and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? - [[User:MyUserName|MyUserName]] # [[/Endocannabinoid system and emotion/]] - What is the role of the endocannabinoid system in emotion? - [[User:MyUserName|MyUserName]] # [[/Environmental grief/]] - What is eco-grief, its causes and consequences, and what can be done? - [[User:MyUserName|MyUserName]] # [[/Exercise and endocannabinoids/]] - What is the relationship between exercise and the endocannabinoid system? - [[User:MyUserName|MyUserName]] # [[/Expressive suppression and emotion regulation/]] - What is the role of expressive suppression in emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Fairness and emotion/]] - What is the relation between fairness and emotion? - [[User:MyUserName|MyUserName]] # [[/Fatigue and emotion/]] - What is the effect of fatigue on emotion and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Fear/]] - What is fear, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Fear of working out/]] - What is FOWO and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Fundamental attribution error and emotion/]] - What is the relationship between the FAE and emotion? - [[User:MyUserName|MyUserName]] # [[/Gratitude and subjective wellbeing/]] - What is the relationship between gratitude and subjective wellbeing? - [[User:MyUserName|MyUserName]] # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Heart rate variability and emotion regulation/]] - What is the relationship between HRV and emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Humility/]] - What is humility, what causes it, and is it desirable? - [[User:MyUserName|MyUserName]] # [[/Hypomania and emotion/]] - What are the emotional characteristics of hypomania? - [[User:MyUserName|MyUserName]] # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[Indigenous Australian emotionality]] - In what ways is emotionality experienced by Indigenous Australian people? - [[User:MyUserName|MyUserName]] # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness? - [[User:MyUserName|MyUserName]] # [[/Inspiration/]] - What is inspiration, what causes it, what are its consequences, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Insular cortex and emotion/]] - What role does the insular cortex play in emotion? - [[User:MyUserName|MyUserName]] # [[/Interoception and emotion/]] - What is the relationship between interoception and emotion? - [[User:MyUserName|MyUserName]] # [[/Kama muta/]] - What is kama muta, what are its effects, and how can it be fostered? - [[User:U3183521|U3183521]] # [[/Linguistic relativism and emotion/]] - What is the role of linguistic relativism in emotion? - [[User:MyUserName|MyUserName]] # [[/Menstrual cycle mood disorders/]] - What causes menstrual cycle mood disorders and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and creativity/]] - How can mindfulness enhance creativity? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and driving/]] - How can mindfulness affect driving? - [[User:MyUserName|MyUserName]] # [[/Mindful self-care/]] - What is mindful self-care, why does it matter, and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Mixed emotions/]] - What are mixed emotions, what causes them, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mudita/]] - What is mudita and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Natural disasters and emotion/]] - How do people respond emotionally to natural disasters and how can they be supported? - [[User:MyUserName|MyUserName]] # [[/Nature therapy/]] - What is nature therapy and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Narcissism and emotion/]] - What is the relationship between narcissism and emotion? - [[User:MyUserName|MyUserName]] # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy? - [[User:MyUserName|MyUserName]] # [[/Needle fear/]] - How does needle fear develop, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? - [[User:JorjaFive|JorjaFive]] # [[/Psychological distress/]] - What is PD, what are the main types, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Psychological trauma/]] - What causes psychological trauma, what are the consequences, and how can people recover from psychological trauma? - [[User:U3210431|U3210431]] # [[/Psilocybin assisted psychotherapy/]] - How can psilocybin be used to assist psychotherapy? - [[User:U3083720|U3083720]] # [[/Rational compassion/]] - What is rational compassion and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? - [[User:MyUserName|MyUserName]] # [[/Religiosity and coping/]] - What is the relationship between religiosity and coping? - [[User:MyUserName|MyUserName]] # [[/Resentment/]] - What is resentment, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Risk-as-feelings/]] - What is the emotional experience of risk and how does it influence decision-making and behaviour? - [[User:MyUserName|MyUserName]] # [[/Self-esteem and culture/]] - What are the cultural influences on self-esteem? - [[User:MyUserName|MyUserName]] # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? - [[User:MyUserName|MyUserName]] # [[/Social media and suicide prevention/]] - How can social media be used to help prevent suicide? - [[User:MyUserName|MyUserName]] # [[/Sorry business/]] - What is sorry business and what role does it play in Indigenous communities in Australia? - [[User:MyUserName|MyUserName]] # [[/Stress control mindset/]] - What is a SCM, why does it matter, and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Suffering as emotion/]] - What is the emotional experience of suffering and how can people cope with suffering? - [[User:MyUserName|MyUserName]] # [[/Telemental health/]] - What are the pros and cons of TMH and what are the key ingredients for effective TMH practices? - [[User:MyUserName|MyUserName]] # [[/Topophilia/]] - What is topophilia, how does it develop, and what are the psychological impacts? - [[User:MyUserName|MyUserName]] # [[/Triumph/]] - What is triumph, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Unemployment and mental health/]]: What is the relationship between unemployment and mental health? - [[User:MyUserName|MyUserName]] # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? - [[User:MyUserName|MyUserName]] # [[/Wave metaphor for emotion/]] - In what respects is an ocean wave a helpful metaphor for understanding human emotions? - [[User:MyUserName|MyUserName]] # [[/Window of tolerance/]] - What is the window of tolerance and how this concept be used? - [[User:MyUserName|MyUserName]] # [[/Workplace mental health training/]] - What is WMHT, what techniques are used, and what are the impacts? - [[User:MyUserName|MyUserName]] # [[/Zoom fatigue/]] - What is Zoom fatigue, what causes it, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] ==Motivation and emotion== # [[/Financial investing, motivation, and emotion/]] - What role does motivation and emotion play in financial investing? - [[User:MyUserName|MyUserName]] # [[/Hostage negotiation, motivation, and emotion/]] - What role does motivation and emotion play in hostage negotiation? - [[User:U3213549|U3213549]] # [[/Money priming, motivation, and emotion/]] - What is the effect of money priming on motivation and emotion? - [[User:MyUserName|MyUserName]] # [[/Motivational dimensional model of affect/]] - What is the motivational dimensional model of affect and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Napping, motivation, and emotion/]] - What are the motivational and emotional effects of napping? - [[User:MyUserName|MyUserName]] # [[/Overchoice, emotion, and motivation/]] - What are the emotional and motivational effects of overchoice? - [[User:MyUserName|MyUserName]] # [[/Patience and impatience/]] - What are the psychological causes and consequences of patience and impatience? - [[User:MyUserName|MyUserName]] # [[/Reward system, motivation, and emotion/]] - What role does the reward system play in motivation and emotion? - [[User:MyUserName|MyUserName]] [[Category:Motivation and emotion/Book/2022]] bv5ljfonmd8waby06lhpfa5bemmoraw 2410926 2410855 2022-08-02T06:00:34Z Jtneill 10242 Fix user name wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Academic help-seeking/]] - What are the barriers and enablers of AHS and how can AHS be fostered? - [[User:MyUserName|MyUserName]] # [[/Academic self-regulation/]] - What is academic self-regulation, why does it matter, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? - [[User:MyUserName|MyUserName]] # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it? - [[User:MyUserName|MyUserName]] # [[/Antidepressants and motivation/]] - What are the effects of popular antidepressants on motivation? - [[User:MyUserName|MyUserName]] # [[/Approach motivation/]] - What is approach motivation and how does it lead to behaviour? - [[User:U3189370|U3189370]] # [[/Behavioural economics and motivation/]] - What aspects of motivation theory are useful in behavioural economics? - [[User:U3141987|U3141987]] # [[/Behavioural model of health services/]] - What is the BMHS and how can it be used? - [[UserFV:MyUserName|MyUserName]] # [[/Beneficence as a psychological need/]] - What is beneficence and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention? - [[User:MyUserName|MyUserName]] # [[/Choice overload/]] - How much choose is too much? How much choice is enough? - [[User:MyUserName|MyUserName]] # [[/Chunking and goal pursuit/]] - How does chunking affect goal pursuit? - [[User:MyUserName|MyUserName]] # [[/Cognitive entrenchment/]] - What is cognitive entrenchment and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[/Climate change helplessness/]] - How does learned helpless impact motivation to engage in behaviours to limit climate change? - [[User:MyUserName|MyUserName]] # [[/Closeness communication bias/]] - What is the CCB, why does it occur, and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirably outcomes? - [[User:MyUserName|MyUserName]] # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories? - [[User:KingMob221|KingMob221]] # [[/Construal level theory/]] - What is construal level theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Courage motivation/]] - What is courage, what motivates courage, and how can courage be enhanced? -[[User:U3213871|U3213871] # [[/Death drive/]] - What is the death drive and how can it be negotiated? - [[User:U3086459|U3086459]] # [[/Drugs-violence nexus and motivation/]] - What is the role of motivation in the drugs-violence nexus? - [[User:MyUserName|MyUserName]] # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD? - [[User:MyUserName|MyUserName]] # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? - [[User:MyUserName|MyUserName]] # [[/Equity theory/]] - What is equity theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/ERG theory/]] - What is Alderfer's ERG theory? - [[User:MyUserName|MyUserName]] # [[/Frame of reference and motivation/]] - How does frame of reference affect motivation? - [[User:MyUserName|MyUserName]] # [[/Freedom and motivation/]] - What is the effect of freedom on motivation? - [[User:MyUserName|MyUserName]] # [[/Fully functioning person/]] - What is a FFP and how can full functioning be developed? - [[User:U3217727|U3217727]] # [[/Functional fixedness/]] - What is functional fixedness and how can it be overcome? - [[User:U3214117]] # [[/Functional imagery training/]] - What is FIT and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Gamification and work motivation/]] - How can gamification enhance work motivation? - [[User:MyUserName|MyUserName]] # [[/Giving up goals/]] - When should we give up goals and when should we persist? - [[User:MyUserName|MyUserName]] # [[/Green prescription motivation/]] - What motivates green prescription compliance? - [[User:MyUserName|MyUserName]] # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour? - [[User:MyUserName|MyUserName]] # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis, what is the evidence, and how does it help to understand drug addiction? - [[U3218292|U3218292]] # [[/Honesty motivation/]] - What motivates honesty? - [[User:MyUserName|MyUserName]] # [[/Humour and work/]] - What is the role of humour in the workplace? - [[User:MyUserName|MyUserName]] # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? - [[User:MyUserName|MyUserName]] # [[/Kindness motivation/]] - What motivates kindness? - [[User:MyUserName|MyUserName]] # [[/Motivational music and exercise/]] - How can music be used to help motivate exercise? - [[User:MyUserName|MyUserName]] # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Nucleus accumbens and motivation/]] - What role does the nucleus accumbens play in motivation? - [[User:MyUserName|MyUserName]] # [[/Physiological needs/]] - How do human's physiological needs affect motivation? - [[U3203655]] # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19? - [[User:MyUserName|MyUserName]] # [[/Relative deprivation and motivation/]] - What is the effect of relative deprivation on motivation? - [[User:MyUserName|MyUserName]] # [[/Retrospective regret/]] - What is the motivational role of retrospective regret? - [[User:MyUserName|MyUserName]] # [[/Revenge motivation/]] - What motivates revenge and how does it affect us? - [[User:MyUserName|MyUserName]] # [[/Self-efficacy and academic achievement/]] - What role does self-efficacy play in academic achievement? - [[User:U943292|U943292]] # [[/Self-efficacy and achievement/]] - What role does self-efficacy play in achievement outcomes? - [[User:MyUserName|MyUserName]] # [[/Sexual harassment at work motivation/]] - What motivates sexual harassment at work and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Signature strengths/]] - What are signature strengths and how can they be applied? - [[User:MyUserName|MyUserName]] # [[/Social cure/]] - What is the social cure and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Stretch goals/]] - What are stretch goals? Do they work? - [[User:MyUserName|MyUserName]] # [[/Sublimation/]] - What is sublimation and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Survival needs and motivation/]] - What are survival needs and how do they influence motivation? - [[User:MyUserName|MyUserName]] # [[/Task initiation/]] - What are the challenges with task initiation and how to get get started? - [[User:MyUserName|MyUserName]] # [[/Theoretical domains framework/]] - What is the TDF and how can be used to guide behaviour change? - [[User:MyUserName|MyUserName]] # [[/Time and motivation/]] - What is the effect of time on motivation? - [[User:MyUserName|MyUserName]] # [[/Time management/]] - How can one's time be managed effectively? - [[User:MyUserName|MyUserName]] # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - [[User:MyUserName|MyUserName]] # [[/Uncertainty avoidance/]] - What is uncertainty avoidance, why does it occur, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Urgency bias and productivity/]] - What is the impact of urgency bias on productivity and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Vocational identity/]] - What is vocational identity and how does it develop? - [[User:MyUserName|MyUserName]] # [[/Volunteer tourism motivation/]] - What motivates volunteer tourism? - [[User:Efost|Efost]] # [[/Wanting and liking/]] - What are the similarities and differences between wanting and liking, and what are the implications? - [[User:MyUserName|MyUserName]] # [[/Work breaks, well-being, and productivity/]] - How do work breaks affect well-being and productivity? - [[User:MyUserName|MyUserName]] # [[/Work and flow/]] - What characteristics of work can produce flow and how can flow at work be fostered? - [[User:MyUserName|MyUserName]] ==Emotion== # [[/Animal emotion/]] - What is the emotional experience of animals? - [[User:MyUserName|MyUserName]] # [[/Attributions and emotion/]] - How do attributions affect emotion? - [[User:MyUserName|MyUserName]] # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? - [[User:MyUserName|MyUserName]] # [[/Benzodiazepines and emotion/]] - What are the effects of benzodiazepines on emotion? - [[User:FulaAjeo22|FulaAjeo22]] # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? - [[User:MyUserName|MyUserName]] # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? - [[User:U3211603|U3211603]] # [[/Cognitive dissonance reduction/]] - What strategies do people use to reduce cognitive dissonance and how effective are they? - [[User:MyUserName|MyUserName]] # [[/Colonisation and emotion in Australia/]] - What are the emotional responses to colonisation in Australia? - [[User:MyUserName|MyUserName]] # [[/Compassion/]] - What is compassion, what are its pros and cons, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? - [[User:MyUserName|MyUserName]] # [[/Contempt/]] - What is contempt, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Core emotions/]] - What are the core emotions and what is their function? MyUserName # [[/Creative arts and trauma/]] - How can creative arts help in dealing with trauma? - [[User:MyUserName|MyUserName]] # [[/Cultural influences on shame, guilt, and pride/]] - How does culture influence shame, guilt, and pride? - [[User:MyUserName|MyUserName]] # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? - [[User:MyUserName|MyUserName]] # [[/Difficult conversations and emotion/]] - What communication and emotional skills are needed to successfully negotiate difficult conversations? - [[User:MyUserName|MyUserName]] # [[/Disappointment/]] - What is disappointment, what causes it, and how can it be managed? - [[User:U3216256|U3216256]] # [[/DMT and spirituality/]] - How can DMT facilitate spiritual experiences? - [[User:MyUserName|MyUserName]] # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? - [[User:MyUserName|MyUserName]] # [[/Ecological grief/]] - What is ecological grief and what can be done about it? - [[User:Brewerjr|Brewerjr]] # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress? - [[User:MyUserName|MyUserName]] # [[/Embarrassment/]] - What is embarrassment, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? - [[User:MyUserName|MyUserName]] # [[/Emotion knowledge/]] - What is emotion knowledge and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? - [[User:MyUserName|MyUserName]] # [[/Endocannabinoid system and emotion/]] - What is the role of the endocannabinoid system in emotion? - [[User:MyUserName|MyUserName]] # [[/Environmental grief/]] - What is eco-grief, its causes and consequences, and what can be done? - [[User:MyUserName|MyUserName]] # [[/Exercise and endocannabinoids/]] - What is the relationship between exercise and the endocannabinoid system? - [[User:MyUserName|MyUserName]] # [[/Expressive suppression and emotion regulation/]] - What is the role of expressive suppression in emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Fairness and emotion/]] - What is the relation between fairness and emotion? - [[User:MyUserName|MyUserName]] # [[/Fatigue and emotion/]] - What is the effect of fatigue on emotion and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Fear/]] - What is fear, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Fear of working out/]] - What is FOWO and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Fundamental attribution error and emotion/]] - What is the relationship between the FAE and emotion? - [[User:MyUserName|MyUserName]] # [[/Gratitude and subjective wellbeing/]] - What is the relationship between gratitude and subjective wellbeing? - [[User:MyUserName|MyUserName]] # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Heart rate variability and emotion regulation/]] - What is the relationship between HRV and emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Humility/]] - What is humility, what causes it, and is it desirable? - [[User:MyUserName|MyUserName]] # [[/Hypomania and emotion/]] - What are the emotional characteristics of hypomania? - [[User:MyUserName|MyUserName]] # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[Indigenous Australian emotionality]] - In what ways is emotionality experienced by Indigenous Australian people? - [[User:MyUserName|MyUserName]] # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness? - [[User:MyUserName|MyUserName]] # [[/Inspiration/]] - What is inspiration, what causes it, what are its consequences, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Insular cortex and emotion/]] - What role does the insular cortex play in emotion? - [[User:MyUserName|MyUserName]] # [[/Interoception and emotion/]] - What is the relationship between interoception and emotion? - [[User:MyUserName|MyUserName]] # [[/Kama muta/]] - What is kama muta, what are its effects, and how can it be fostered? - [[User:U3183521|U3183521]] # [[/Linguistic relativism and emotion/]] - What is the role of linguistic relativism in emotion? - [[User:MyUserName|MyUserName]] # [[/Menstrual cycle mood disorders/]] - What causes menstrual cycle mood disorders and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and creativity/]] - How can mindfulness enhance creativity? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and driving/]] - How can mindfulness affect driving? - [[User:MyUserName|MyUserName]] # [[/Mindful self-care/]] - What is mindful self-care, why does it matter, and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Mixed emotions/]] - What are mixed emotions, what causes them, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mudita/]] - What is mudita and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Natural disasters and emotion/]] - How do people respond emotionally to natural disasters and how can they be supported? - [[User:MyUserName|MyUserName]] # [[/Nature therapy/]] - What is nature therapy and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Narcissism and emotion/]] - What is the relationship between narcissism and emotion? - [[User:MyUserName|MyUserName]] # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy? - [[User:MyUserName|MyUserName]] # [[/Needle fear/]] - How does needle fear develop, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? - [[User:JorjaFive|JorjaFive]] # [[/Psychological distress/]] - What is PD, what are the main types, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Psychological trauma/]] - What causes psychological trauma, what are the consequences, and how can people recover from psychological trauma? - [[User:U3210431|U3210431]] # [[/Psilocybin assisted psychotherapy/]] - How can psilocybin be used to assist psychotherapy? - [[User:U3083720|U3083720]] # [[/Rational compassion/]] - What is rational compassion and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? - [[User:MyUserName|MyUserName]] # [[/Religiosity and coping/]] - What is the relationship between religiosity and coping? - [[User:MyUserName|MyUserName]] # [[/Resentment/]] - What is resentment, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Risk-as-feelings/]] - What is the emotional experience of risk and how does it influence decision-making and behaviour? - [[User:MyUserName|MyUserName]] # [[/Self-esteem and culture/]] - What are the cultural influences on self-esteem? - [[User:MyUserName|MyUserName]] # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? - [[User:MyUserName|MyUserName]] # [[/Social media and suicide prevention/]] - How can social media be used to help prevent suicide? - [[User:MyUserName|MyUserName]] # [[/Sorry business/]] - What is sorry business and what role does it play in Indigenous communities in Australia? - [[User:MyUserName|MyUserName]] # [[/Stress control mindset/]] - What is a SCM, why does it matter, and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Suffering as emotion/]] - What is the emotional experience of suffering and how can people cope with suffering? - [[User:MyUserName|MyUserName]] # [[/Telemental health/]] - What are the pros and cons of TMH and what are the key ingredients for effective TMH practices? - [[User:MyUserName|MyUserName]] # [[/Topophilia/]] - What is topophilia, how does it develop, and what are the psychological impacts? - [[User:MyUserName|MyUserName]] # [[/Triumph/]] - What is triumph, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Unemployment and mental health/]]: What is the relationship between unemployment and mental health? - [[User:MyUserName|MyUserName]] # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? - [[User:MyUserName|MyUserName]] # [[/Wave metaphor for emotion/]] - In what respects is an ocean wave a helpful metaphor for understanding human emotions? - [[User:MyUserName|MyUserName]] # [[/Window of tolerance/]] - What is the window of tolerance and how this concept be used? - [[User:MyUserName|MyUserName]] # [[/Workplace mental health training/]] - What is WMHT, what techniques are used, and what are the impacts? - [[User:MyUserName|MyUserName]] # [[/Zoom fatigue/]] - What is Zoom fatigue, what causes it, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] ==Motivation and emotion== # [[/Financial investing, motivation, and emotion/]] - What role does motivation and emotion play in financial investing? - [[User:MyUserName|MyUserName]] # [[/Hostage negotiation, motivation, and emotion/]] - What role does motivation and emotion play in hostage negotiation? - [[User:U3213549|U3213549]] # [[/Money priming, motivation, and emotion/]] - What is the effect of money priming on motivation and emotion? - [[User:MyUserName|MyUserName]] # [[/Motivational dimensional model of affect/]] - What is the motivational dimensional model of affect and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Napping, motivation, and emotion/]] - What are the motivational and emotional effects of napping? - [[User:MyUserName|MyUserName]] # [[/Overchoice, emotion, and motivation/]] - What are the emotional and motivational effects of overchoice? - [[User:MyUserName|MyUserName]] # [[/Patience and impatience/]] - What are the psychological causes and consequences of patience and impatience? - [[User:MyUserName|MyUserName]] # [[/Reward system, motivation, and emotion/]] - What role does the reward system play in motivation and emotion? - [[User:MyUserName|MyUserName]] [[Category:Motivation and emotion/Book/2022]] 2wwk6asilbnput2ehv4g4mqxt43cjin 2410927 2410926 2022-08-02T06:01:57Z Jtneill 10242 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Academic help-seeking/]] - What are the barriers and enablers of AHS and how can AHS be fostered? - [[User:MyUserName|MyUserName]] # [[/Academic self-regulation/]] - What is academic self-regulation, why does it matter, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? - [[User:MyUserName|MyUserName]] # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it? - [[User:MyUserName|MyUserName]] # [[/Antidepressants and motivation/]] - What are the effects of popular antidepressants on motivation? - [[User:MyUserName|MyUserName]] # [[/Approach motivation/]] - What is approach motivation and how does it lead to behaviour? - [[User:U3189370|U3189370]] # [[/Behavioural economics and motivation/]] - What aspects of motivation theory are useful in behavioural economics? - [[User:U3141987|U3141987]] # [[/Behavioural model of health services/]] - What is the BMHS and how can it be used? - [[UserFV:MyUserName|MyUserName]] # [[/Beneficence as a psychological need/]] - What is beneficence and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention? - [[User:MyUserName|MyUserName]] # [[/Choice overload/]] - What is choice overload? What is the optimal amount of choice? - [[User:MyUserName|MyUserName]] # [[/Chunking and goal pursuit/]] - How does chunking affect goal pursuit? - [[User:MyUserName|MyUserName]] # [[/Cognitive entrenchment/]] - What is cognitive entrenchment and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[/Climate change helplessness/]] - How does learned helpless impact motivation to engage in behaviours to limit climate change? - [[User:MyUserName|MyUserName]] # [[/Closeness communication bias/]] - What is the CCB, why does it occur, and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirably outcomes? - [[User:MyUserName|MyUserName]] # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories? - [[User:KingMob221|KingMob221]] # [[/Construal level theory/]] - What is construal level theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Courage motivation/]] - What is courage, what motivates courage, and how can courage be enhanced? -[[User:U3213871|U3213871] # [[/Death drive/]] - What is the death drive and how can it be negotiated? - [[User:U3086459|U3086459]] # [[/Drugs-violence nexus and motivation/]] - What is the role of motivation in the drugs-violence nexus? - [[User:MyUserName|MyUserName]] # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD? - [[User:MyUserName|MyUserName]] # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? - [[User:MyUserName|MyUserName]] # [[/Equity theory/]] - What is equity theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/ERG theory/]] - What is Alderfer's ERG theory? - [[User:MyUserName|MyUserName]] # [[/Frame of reference and motivation/]] - How does frame of reference affect motivation? - [[User:MyUserName|MyUserName]] # [[/Freedom and motivation/]] - What is the effect of freedom on motivation? - [[User:MyUserName|MyUserName]] # [[/Fully functioning person/]] - What is a FFP and how can full functioning be developed? - [[User:U3217727|U3217727]] # [[/Functional fixedness/]] - What is functional fixedness and how can it be overcome? - [[User:U3214117]] # [[/Functional imagery training/]] - What is FIT and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Gamification and work motivation/]] - How can gamification enhance work motivation? - [[User:MyUserName|MyUserName]] # [[/Giving up goals/]] - When should we give up goals and when should we persist? - [[User:MyUserName|MyUserName]] # [[/Green prescription motivation/]] - What motivates green prescription compliance? - [[User:MyUserName|MyUserName]] # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour? - [[User:MyUserName|MyUserName]] # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis, what is the evidence, and how does it help to understand drug addiction? - [[U3218292|U3218292]] # [[/Honesty motivation/]] - What motivates honesty? - [[User:MyUserName|MyUserName]] # [[/Humour and work/]] - What is the role of humour in the workplace? - [[User:MyUserName|MyUserName]] # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? - [[User:MyUserName|MyUserName]] # [[/Kindness motivation/]] - What motivates kindness? - [[User:MyUserName|MyUserName]] # [[/Motivational music and exercise/]] - How can music be used to help motivate exercise? - [[User:MyUserName|MyUserName]] # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Nucleus accumbens and motivation/]] - What role does the nucleus accumbens play in motivation? - [[User:MyUserName|MyUserName]] # [[/Physiological needs/]] - How do human's physiological needs affect motivation? - [[U3203655]] # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19? - [[User:MyUserName|MyUserName]] # [[/Relative deprivation and motivation/]] - What is the effect of relative deprivation on motivation? - [[User:MyUserName|MyUserName]] # [[/Retrospective regret/]] - What is the motivational role of retrospective regret? - [[User:MyUserName|MyUserName]] # [[/Revenge motivation/]] - What motivates revenge and how does it affect us? - [[User:MyUserName|MyUserName]] # [[/Self-efficacy and academic achievement/]] - What role does self-efficacy play in academic achievement? - [[User:U943292|U943292]] # [[/Self-efficacy and achievement/]] - What role does self-efficacy play in achievement outcomes? - [[User:MyUserName|MyUserName]] # [[/Sexual harassment at work motivation/]] - What motivates sexual harassment at work and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Signature strengths/]] - What are signature strengths and how can they be applied? - [[User:MyUserName|MyUserName]] # [[/Social cure/]] - What is the social cure and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Stretch goals/]] - What are stretch goals? Do they work? - [[User:MyUserName|MyUserName]] # [[/Sublimation/]] - What is sublimation and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Survival needs and motivation/]] - What are survival needs and how do they influence motivation? - [[User:MyUserName|MyUserName]] # [[/Task initiation/]] - What are the challenges with task initiation and how to get get started? - [[User:MyUserName|MyUserName]] # [[/Theoretical domains framework/]] - What is the TDF and how can be used to guide behaviour change? - [[User:MyUserName|MyUserName]] # [[/Time and motivation/]] - What is the effect of time on motivation? - [[User:MyUserName|MyUserName]] # [[/Time management/]] - How can one's time be managed effectively? - [[User:MyUserName|MyUserName]] # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - [[User:MyUserName|MyUserName]] # [[/Uncertainty avoidance/]] - What is uncertainty avoidance, why does it occur, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Urgency bias and productivity/]] - What is the impact of urgency bias on productivity and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Vocational identity/]] - What is vocational identity and how does it develop? - [[User:MyUserName|MyUserName]] # [[/Volunteer tourism motivation/]] - What motivates volunteer tourism? - [[User:Efost|Efost]] # [[/Wanting and liking/]] - What are the similarities and differences between wanting and liking, and what are the implications? - [[User:MyUserName|MyUserName]] # [[/Work breaks, well-being, and productivity/]] - How do work breaks affect well-being and productivity? - [[User:MyUserName|MyUserName]] # [[/Work and flow/]] - What characteristics of work can produce flow and how can flow at work be fostered? - [[User:MyUserName|MyUserName]] ==Emotion== # [[/Animal emotion/]] - What is the emotional experience of animals? - [[User:MyUserName|MyUserName]] # [[/Attributions and emotion/]] - How do attributions affect emotion? - [[User:MyUserName|MyUserName]] # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? - [[User:MyUserName|MyUserName]] # [[/Benzodiazepines and emotion/]] - What are the effects of benzodiazepines on emotion? - [[User:FulaAjeo22|FulaAjeo22]] # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? - [[User:MyUserName|MyUserName]] # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? - [[User:U3211603|U3211603]] # [[/Cognitive dissonance reduction/]] - What strategies do people use to reduce cognitive dissonance and how effective are they? - [[User:MyUserName|MyUserName]] # [[/Colonisation and emotion in Australia/]] - What are the emotional responses to colonisation in Australia? - [[User:MyUserName|MyUserName]] # [[/Compassion/]] - What is compassion, what are its pros and cons, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? - [[User:MyUserName|MyUserName]] # [[/Contempt/]] - What is contempt, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Core emotions/]] - What are the core emotions and what is their function? MyUserName # [[/Creative arts and trauma/]] - How can creative arts help in dealing with trauma? - [[User:MyUserName|MyUserName]] # [[/Cultural influences on shame, guilt, and pride/]] - How does culture influence shame, guilt, and pride? - [[User:MyUserName|MyUserName]] # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? - [[User:MyUserName|MyUserName]] # [[/Difficult conversations and emotion/]] - What communication and emotional skills are needed to successfully negotiate difficult conversations? - [[User:MyUserName|MyUserName]] # [[/Disappointment/]] - What is disappointment, what causes it, and how can it be managed? - [[User:U3216256|U3216256]] # [[/DMT and spirituality/]] - How can DMT facilitate spiritual experiences? - [[User:MyUserName|MyUserName]] # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? - [[User:MyUserName|MyUserName]] # [[/Ecological grief/]] - What is ecological grief and what can be done about it? - [[User:Brewerjr|Brewerjr]] # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress? - [[User:MyUserName|MyUserName]] # [[/Embarrassment/]] - What is embarrassment, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? - [[User:MyUserName|MyUserName]] # [[/Emotion knowledge/]] - What is emotion knowledge and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? - [[User:MyUserName|MyUserName]] # [[/Endocannabinoid system and emotion/]] - What is the role of the endocannabinoid system in emotion? - [[User:MyUserName|MyUserName]] # [[/Environmental grief/]] - What is eco-grief, its causes and consequences, and what can be done? - [[User:MyUserName|MyUserName]] # [[/Exercise and endocannabinoids/]] - What is the relationship between exercise and the endocannabinoid system? - [[User:MyUserName|MyUserName]] # [[/Expressive suppression and emotion regulation/]] - What is the role of expressive suppression in emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Fairness and emotion/]] - What is the relation between fairness and emotion? - [[User:MyUserName|MyUserName]] # [[/Fatigue and emotion/]] - What is the effect of fatigue on emotion and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Fear/]] - What is fear, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Fear of working out/]] - What is FOWO and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Fundamental attribution error and emotion/]] - What is the relationship between the FAE and emotion? - [[User:MyUserName|MyUserName]] # [[/Gratitude and subjective wellbeing/]] - What is the relationship between gratitude and subjective wellbeing? - [[User:MyUserName|MyUserName]] # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Heart rate variability and emotion regulation/]] - What is the relationship between HRV and emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Humility/]] - What is humility, what causes it, and is it desirable? - [[User:MyUserName|MyUserName]] # [[/Hypomania and emotion/]] - What are the emotional characteristics of hypomania? - [[User:MyUserName|MyUserName]] # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[Indigenous Australian emotionality]] - In what ways is emotionality experienced by Indigenous Australian people? - [[User:MyUserName|MyUserName]] # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness? - [[User:MyUserName|MyUserName]] # [[/Inspiration/]] - What is inspiration, what causes it, what are its consequences, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Insular cortex and emotion/]] - What role does the insular cortex play in emotion? - [[User:MyUserName|MyUserName]] # [[/Interoception and emotion/]] - What is the relationship between interoception and emotion? - [[User:MyUserName|MyUserName]] # [[/Kama muta/]] - What is kama muta, what are its effects, and how can it be fostered? - [[User:U3183521|U3183521]] # [[/Linguistic relativism and emotion/]] - What is the role of linguistic relativism in emotion? - [[User:MyUserName|MyUserName]] # [[/Menstrual cycle mood disorders/]] - What causes menstrual cycle mood disorders and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and creativity/]] - How can mindfulness enhance creativity? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and driving/]] - How can mindfulness affect driving? - [[User:MyUserName|MyUserName]] # [[/Mindful self-care/]] - What is mindful self-care, why does it matter, and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Mixed emotions/]] - What are mixed emotions, what causes them, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mudita/]] - What is mudita and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Natural disasters and emotion/]] - How do people respond emotionally to natural disasters and how can they be supported? - [[User:MyUserName|MyUserName]] # [[/Nature therapy/]] - What is nature therapy and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Narcissism and emotion/]] - What is the relationship between narcissism and emotion? - [[User:MyUserName|MyUserName]] # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy? - [[User:MyUserName|MyUserName]] # [[/Needle fear/]] - How does needle fear develop, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? - [[User:JorjaFive|JorjaFive]] # [[/Psychological distress/]] - What is PD, what are the main types, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Psychological trauma/]] - What causes psychological trauma, what are the consequences, and how can people recover from psychological trauma? - [[User:U3210431|U3210431]] # [[/Psilocybin assisted psychotherapy/]] - How can psilocybin be used to assist psychotherapy? - [[User:U3083720|U3083720]] # [[/Rational compassion/]] - What is rational compassion and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? - [[User:MyUserName|MyUserName]] # [[/Religiosity and coping/]] - What is the relationship between religiosity and coping? - [[User:MyUserName|MyUserName]] # [[/Resentment/]] - What is resentment, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Risk-as-feelings/]] - What is the emotional experience of risk and how does it influence decision-making and behaviour? - [[User:MyUserName|MyUserName]] # [[/Self-esteem and culture/]] - What are the cultural influences on self-esteem? - [[User:MyUserName|MyUserName]] # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? - [[User:MyUserName|MyUserName]] # [[/Social media and suicide prevention/]] - How can social media be used to help prevent suicide? - [[User:MyUserName|MyUserName]] # [[/Sorry business/]] - What is sorry business and what role does it play in Indigenous communities in Australia? - [[User:MyUserName|MyUserName]] # [[/Stress control mindset/]] - What is a SCM, why does it matter, and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Suffering as emotion/]] - What is the emotional experience of suffering and how can people cope with suffering? - [[User:MyUserName|MyUserName]] # [[/Telemental health/]] - What are the pros and cons of TMH and what are the key ingredients for effective TMH practices? - [[User:MyUserName|MyUserName]] # [[/Topophilia/]] - What is topophilia, how does it develop, and what are the psychological impacts? - [[User:MyUserName|MyUserName]] # [[/Triumph/]] - What is triumph, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Unemployment and mental health/]]: What is the relationship between unemployment and mental health? - [[User:MyUserName|MyUserName]] # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? - [[User:MyUserName|MyUserName]] # [[/Wave metaphor for emotion/]] - In what respects is an ocean wave a helpful metaphor for understanding human emotions? - [[User:MyUserName|MyUserName]] # [[/Window of tolerance/]] - What is the window of tolerance and how this concept be used? - [[User:MyUserName|MyUserName]] # [[/Workplace mental health training/]] - What is WMHT, what techniques are used, and what are the impacts? - [[User:MyUserName|MyUserName]] # [[/Zoom fatigue/]] - What is Zoom fatigue, what causes it, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] ==Motivation and emotion== # [[/Financial investing, motivation, and emotion/]] - What role does motivation and emotion play in financial investing? - [[User:MyUserName|MyUserName]] # [[/Hostage negotiation, motivation, and emotion/]] - What role does motivation and emotion play in hostage negotiation? - [[User:U3213549|U3213549]] # [[/Money priming, motivation, and emotion/]] - What is the effect of money priming on motivation and emotion? - [[User:MyUserName|MyUserName]] # [[/Motivational dimensional model of affect/]] - What is the motivational dimensional model of affect and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Napping, motivation, and emotion/]] - What are the motivational and emotional effects of napping? - [[User:MyUserName|MyUserName]] # [[/Overchoice, emotion, and motivation/]] - What are the emotional and motivational effects of overchoice? - [[User:MyUserName|MyUserName]] # [[/Patience and impatience/]] - What are the psychological causes and consequences of patience and impatience? - [[User:MyUserName|MyUserName]] # [[/Reward system, motivation, and emotion/]] - What role does the reward system play in motivation and emotion? - [[User:MyUserName|MyUserName]] [[Category:Motivation and emotion/Book/2022]] 162was6rq6qpq04wt4tazid5qumjjjz 2410940 2410927 2022-08-02T07:52:01Z U3191574 (PHP) 2946555 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Academic help-seeking/]] - What are the barriers and enablers of AHS and how can AHS be fostered? - [[User:MyUserName|MyUserName]] # [[/Academic self-regulation/]] - What is academic self-regulation, why does it matter, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? - [[User:MyUserName|MyUserName]] # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it? - [[User:MyUserName|MyUserName]] # [[/Antidepressants and motivation/]] - What are the effects of popular antidepressants on motivation? - [[User:MyUserName|MyUserName]] # [[/Approach motivation/]] - What is approach motivation and how does it lead to behaviour? - [[User:U3189370|U3189370]] # [[/Behavioural economics and motivation/]] - What aspects of motivation theory are useful in behavioural economics? - [[User:U3141987|U3141987]] # [[/Behavioural model of health services/]] - What is the BMHS and how can it be used? - [[UserFV:MyUserName|MyUserName]] # [[/Beneficence as a psychological need/]] - What is beneficence and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention? - [[User:MyUserName|MyUserName]] # [[/Choice overload/]] - What is choice overload? What is the optimal amount of choice? - [[User:MyUserName|MyUserName]] # [[/Chunking and goal pursuit/]] - How does chunking affect goal pursuit? - [[User:MyUserName|MyUserName]] # [[/Cognitive entrenchment/]] - What is cognitive entrenchment and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[/Climate change helplessness/]] - How does learned helpless impact motivation to engage in behaviours to limit climate change? - [[User:MyUserName|MyUserName]] # [[/Closeness communication bias/]] - What is the CCB, why does it occur, and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirably outcomes? - [[User:MyUserName|MyUserName]] # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories? - [[User:KingMob221|KingMob221]] # [[/Construal level theory/]] - What is construal level theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Courage motivation/]] - What is courage, what motivates courage, and how can courage be enhanced? -[[User:U3213871|U3213871] # [[/Death drive/]] - What is the death drive and how can it be negotiated? - [[User:U3086459|U3086459]] # [[/Drugs-violence nexus and motivation/]] - What is the role of motivation in the drugs-violence nexus? - [[User:MyUserName|MyUserName]] # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD? - [[User:MyUserName|MyUserName]] # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? - [[User:MyUserName|MyUserName]] # [[/Equity theory/]] - What is equity theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/ERG theory/]] - What is Alderfer's ERG theory? - [[User:MyUserName|MyUserName]] # [[/Frame of reference and motivation/]] - How does frame of reference affect motivation? - [[User:MyUserName|MyUserName]] # [[/Freedom and motivation/]] - What is the effect of freedom on motivation? - [[User:MyUserName|MyUserName]] # [[/Fully functioning person/]] - What is a FFP and how can full functioning be developed? - [[User:U3217727|U3217727]] # [[/Functional fixedness/]] - What is functional fixedness and how can it be overcome? - [[User:U3214117]] # [[/Functional imagery training/]] - What is FIT and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Gamification and work motivation/]] - How can gamification enhance work motivation? - [[User:MyUserName|MyUserName]] # [[/Giving up goals/]] - When should we give up goals and when should we persist? - [[User:MyUserName|MyUserName]] # [[/Green prescription motivation/]] - What motivates green prescription compliance? - [[User:MyUserName|MyUserName]] # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour? - [[User:MyUserName|MyUserName]] # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis, what is the evidence, and how does it help to understand drug addiction? - [[U3218292|U3218292]] # [[/Honesty motivation/]] - What motivates honesty? - [[User:MyUserName|MyUserName]] # [[/Humour and work/]] - What is the role of humour in the workplace? - [[User:MyUserName|MyUserName]] # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? - [[User:MyUserName|MyUserName]] # [[/Kindness motivation/]] - What motivates kindness? - [[User:MyUserName|MyUserName]] # [[/Motivational music and exercise/]] - How can music be used to help motivate exercise? - [[User:MyUserName|MyUserName]] # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Nucleus accumbens and motivation/]] - What role does the nucleus accumbens play in motivation? - [[User:MyUserName|MyUserName]] # [[/Physiological needs/]] - How do human's physiological needs affect motivation? - [[U3203655]] # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19? - [[User:MyUserName|MyUserName]] # [[/Relative deprivation and motivation/]] - What is the effect of relative deprivation on motivation? - [[User:MyUserName|U3191574 (PHP)]] # [[/Retrospective regret/]] - What is the motivational role of retrospective regret? - [[User:MyUserName|MyUserName]] # [[/Revenge motivation/]] - What motivates revenge and how does it affect us? - [[User:MyUserName|MyUserName]] # [[/Self-efficacy and academic achievement/]] - What role does self-efficacy play in academic achievement? - [[User:U943292|U943292]] # [[/Self-efficacy and achievement/]] - What role does self-efficacy play in achievement outcomes? - [[User:MyUserName|MyUserName]] # [[/Sexual harassment at work motivation/]] - What motivates sexual harassment at work and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Signature strengths/]] - What are signature strengths and how can they be applied? - [[User:MyUserName|MyUserName]] # [[/Social cure/]] - What is the social cure and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Stretch goals/]] - What are stretch goals? Do they work? - [[User:MyUserName|MyUserName]] # [[/Sublimation/]] - What is sublimation and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Survival needs and motivation/]] - What are survival needs and how do they influence motivation? - [[User:MyUserName|MyUserName]] # [[/Task initiation/]] - What are the challenges with task initiation and how to get get started? - [[User:MyUserName|MyUserName]] # [[/Theoretical domains framework/]] - What is the TDF and how can be used to guide behaviour change? - [[User:MyUserName|MyUserName]] # [[/Time and motivation/]] - What is the effect of time on motivation? - [[User:MyUserName|MyUserName]] # [[/Time management/]] - How can one's time be managed effectively? - [[User:MyUserName|MyUserName]] # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - [[User:MyUserName|MyUserName]] # [[/Uncertainty avoidance/]] - What is uncertainty avoidance, why does it occur, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Urgency bias and productivity/]] - What is the impact of urgency bias on productivity and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Vocational identity/]] - What is vocational identity and how does it develop? - [[User:MyUserName|MyUserName]] # [[/Volunteer tourism motivation/]] - What motivates volunteer tourism? - [[User:Efost|Efost]] # [[/Wanting and liking/]] - What are the similarities and differences between wanting and liking, and what are the implications? - [[User:MyUserName|MyUserName]] # [[/Work breaks, well-being, and productivity/]] - How do work breaks affect well-being and productivity? - [[User:MyUserName|MyUserName]] # [[/Work and flow/]] - What characteristics of work can produce flow and how can flow at work be fostered? - [[User:MyUserName|MyUserName]] ==Emotion== # [[/Animal emotion/]] - What is the emotional experience of animals? - [[User:MyUserName|MyUserName]] # [[/Attributions and emotion/]] - How do attributions affect emotion? - [[User:MyUserName|MyUserName]] # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? - [[User:MyUserName|MyUserName]] # [[/Benzodiazepines and emotion/]] - What are the effects of benzodiazepines on emotion? - [[User:FulaAjeo22|FulaAjeo22]] # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? - [[User:MyUserName|MyUserName]] # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? - [[User:U3211603|U3211603]] # [[/Cognitive dissonance reduction/]] - What strategies do people use to reduce cognitive dissonance and how effective are they? - [[User:MyUserName|MyUserName]] # [[/Colonisation and emotion in Australia/]] - What are the emotional responses to colonisation in Australia? - [[User:MyUserName|MyUserName]] # [[/Compassion/]] - What is compassion, what are its pros and cons, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? - [[User:MyUserName|MyUserName]] # [[/Contempt/]] - What is contempt, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Core emotions/]] - What are the core emotions and what is their function? MyUserName # [[/Creative arts and trauma/]] - How can creative arts help in dealing with trauma? - [[User:MyUserName|MyUserName]] # [[/Cultural influences on shame, guilt, and pride/]] - How does culture influence shame, guilt, and pride? - [[User:MyUserName|MyUserName]] # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? - [[User:MyUserName|MyUserName]] # [[/Difficult conversations and emotion/]] - What communication and emotional skills are needed to successfully negotiate difficult conversations? - [[User:MyUserName|MyUserName]] # [[/Disappointment/]] - What is disappointment, what causes it, and how can it be managed? - [[User:U3216256|U3216256]] # [[/DMT and spirituality/]] - How can DMT facilitate spiritual experiences? - [[User:MyUserName|MyUserName]] # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? - [[User:MyUserName|MyUserName]] # [[/Ecological grief/]] - What is ecological grief and what can be done about it? - [[User:Brewerjr|Brewerjr]] # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress? - [[User:MyUserName|MyUserName]] # [[/Embarrassment/]] - What is embarrassment, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? - [[User:MyUserName|MyUserName]] # [[/Emotion knowledge/]] - What is emotion knowledge and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? - [[User:MyUserName|MyUserName]] # [[/Endocannabinoid system and emotion/]] - What is the role of the endocannabinoid system in emotion? - [[User:MyUserName|MyUserName]] # [[/Environmental grief/]] - What is eco-grief, its causes and consequences, and what can be done? - [[User:MyUserName|MyUserName]] # [[/Exercise and endocannabinoids/]] - What is the relationship between exercise and the endocannabinoid system? - [[User:MyUserName|MyUserName]] # [[/Expressive suppression and emotion regulation/]] - What is the role of expressive suppression in emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Fairness and emotion/]] - What is the relation between fairness and emotion? - [[User:MyUserName|MyUserName]] # [[/Fatigue and emotion/]] - What is the effect of fatigue on emotion and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Fear/]] - What is fear, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Fear of working out/]] - What is FOWO and how can it be overcome? - [[User:MyUserName|MyUserName]] # [[/Fundamental attribution error and emotion/]] - What is the relationship between the FAE and emotion? - [[User:MyUserName|MyUserName]] # [[/Gratitude and subjective wellbeing/]] - What is the relationship between gratitude and subjective wellbeing? - [[User:MyUserName|MyUserName]] # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Heart rate variability and emotion regulation/]] - What is the relationship between HRV and emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Humility/]] - What is humility, what causes it, and is it desirable? - [[User:MyUserName|MyUserName]] # [[/Hypomania and emotion/]] - What are the emotional characteristics of hypomania? - [[User:MyUserName|MyUserName]] # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[Indigenous Australian emotionality]] - In what ways is emotionality experienced by Indigenous Australian people? - [[User:MyUserName|MyUserName]] # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness? - [[User:MyUserName|MyUserName]] # [[/Inspiration/]] - What is inspiration, what causes it, what are its consequences, and how can it be fostered? - [[User:MyUserName|MyUserName]] # [[/Insular cortex and emotion/]] - What role does the insular cortex play in emotion? - [[User:MyUserName|MyUserName]] # [[/Interoception and emotion/]] - What is the relationship between interoception and emotion? - [[User:MyUserName|MyUserName]] # [[/Kama muta/]] - What is kama muta, what are its effects, and how can it be fostered? - [[User:U3183521|U3183521]] # [[/Linguistic relativism and emotion/]] - What is the role of linguistic relativism in emotion? - [[User:MyUserName|MyUserName]] # [[/Menstrual cycle mood disorders/]] - What causes menstrual cycle mood disorders and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and creativity/]] - How can mindfulness enhance creativity? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and driving/]] - How can mindfulness affect driving? - [[User:MyUserName|MyUserName]] # [[/Mindful self-care/]] - What is mindful self-care, why does it matter, and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Mixed emotions/]] - What are mixed emotions, what causes them, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Mudita/]] - What is mudita and how can it be developed? - [[User:MyUserName|MyUserName]] # [[/Natural disasters and emotion/]] - How do people respond emotionally to natural disasters and how can they be supported? - [[User:MyUserName|MyUserName]] # [[/Nature therapy/]] - What is nature therapy and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Narcissism and emotion/]] - What is the relationship between narcissism and emotion? - [[User:MyUserName|MyUserName]] # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy? - [[User:MyUserName|MyUserName]] # [[/Needle fear/]] - How does needle fear develop, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? - [[User:JorjaFive|JorjaFive]] # [[/Psychological distress/]] - What is PD, what are the main types, and how can they be managed? - [[User:MyUserName|MyUserName]] # [[/Psychological trauma/]] - What causes psychological trauma, what are the consequences, and how can people recover from psychological trauma? - [[User:U3210431|U3210431]] # [[/Psilocybin assisted psychotherapy/]] - How can psilocybin be used to assist psychotherapy? - [[User:U3083720|U3083720]] # [[/Rational compassion/]] - What is rational compassion and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? - [[User:MyUserName|MyUserName]] # [[/Religiosity and coping/]] - What is the relationship between religiosity and coping? - [[User:MyUserName|MyUserName]] # [[/Resentment/]] - What is resentment, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Risk-as-feelings/]] - What is the emotional experience of risk and how does it influence decision-making and behaviour? - [[User:MyUserName|MyUserName]] # [[/Self-esteem and culture/]] - What are the cultural influences on self-esteem? - [[User:MyUserName|MyUserName]] # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? - [[User:MyUserName|MyUserName]] # [[/Social media and suicide prevention/]] - How can social media be used to help prevent suicide? - [[User:MyUserName|MyUserName]] # [[/Sorry business/]] - What is sorry business and what role does it play in Indigenous communities in Australia? - [[User:MyUserName|MyUserName]] # [[/Stress control mindset/]] - What is a SCM, why does it matter, and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Suffering as emotion/]] - What is the emotional experience of suffering and how can people cope with suffering? - [[User:MyUserName|MyUserName]] # [[/Telemental health/]] - What are the pros and cons of TMH and what are the key ingredients for effective TMH practices? - [[User:MyUserName|MyUserName]] # [[/Topophilia/]] - What is topophilia, how does it develop, and what are the psychological impacts? - [[User:MyUserName|MyUserName]] # [[/Triumph/]] - What is triumph, what causes it, and how can it be managed? - [[User:MyUserName|MyUserName]] # [[/Unemployment and mental health/]]: What is the relationship between unemployment and mental health? - [[User:MyUserName|MyUserName]] # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? - [[User:MyUserName|MyUserName]] # [[/Wave metaphor for emotion/]] - In what respects is an ocean wave a helpful metaphor for understanding human emotions? - [[User:MyUserName|MyUserName]] # [[/Window of tolerance/]] - What is the window of tolerance and how this concept be used? - [[User:MyUserName|MyUserName]] # [[/Workplace mental health training/]] - What is WMHT, what techniques are used, and what are the impacts? - [[User:MyUserName|MyUserName]] # [[/Zoom fatigue/]] - What is Zoom fatigue, what causes it, what are its consequences, and what can be done about it? - [[User:MyUserName|MyUserName]] ==Motivation and emotion== # [[/Financial investing, motivation, and emotion/]] - What role does motivation and emotion play in financial investing? - [[User:MyUserName|MyUserName]] # [[/Hostage negotiation, motivation, and emotion/]] - What role does motivation and emotion play in hostage negotiation? - [[User:U3213549|U3213549]] # [[/Money priming, motivation, and emotion/]] - What is the effect of money priming on motivation and emotion? - [[User:MyUserName|MyUserName]] # [[/Motivational dimensional model of affect/]] - What is the motivational dimensional model of affect and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Napping, motivation, and emotion/]] - What are the motivational and emotional effects of napping? - [[User:MyUserName|MyUserName]] # [[/Overchoice, emotion, and motivation/]] - What are the emotional and motivational effects of overchoice? - [[User:MyUserName|MyUserName]] # [[/Patience and impatience/]] - What are the psychological causes and consequences of patience and impatience? - [[User:MyUserName|MyUserName]] # [[/Reward system, motivation, and emotion/]] - What role does the reward system play in motivation and emotion? - [[User:MyUserName|MyUserName]] [[Category:Motivation and emotion/Book/2022]] bqqsa5oz540p82hczoff5u3l7jkbpre 0 277667 2410813 2409724 2022-08-01T19:41:32Z Greg at Higher Math Help 2911873 Added "Get involved" section and links to example lesson plans. wikitext text/x-wiki {{Eventmath tabs}} __NOEDITSECTION__ <div style="float:left; width:60%;"> ==Math lesson plans based on current events== ===Welcome to Eventmath!=== Eventmath is a living repository pairing news articles and social media posts with math lesson plans. Anyone can * discover and edit [[Eventmath/Lesson plans|'''Lesson plans''']], * share ideas and get feedback by [[Eventmath/Contributing|'''Contributing''']] lesson plans, and * help make Eventmath a world-class resource by completing [[Eventmath/Tasks|'''Tasks''']]. This is a new project and is under development. If you're curious to learn more, you can read our [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath '''successful grant proposal''']. ===Get involved=== There are two great ways to get involved right now! You can * register for one of [[Eventmath/Meetups|'''our meetups''']] to help us add more lesson plans * use our '''example lesson plans''' in the classroom and add your feedback: ** [[Eventmath/Lesson_plans/Proportions_and_voting_power_under_the_Electoral_College|Proportions and voting power under the Electoral College]] ** [[Eventmath/Lesson_plans/Comparing_streaming_service_pay_rates_to_artists|Comparing streaming service pay rates to artists]] </div> <div style="float:right; width:30%">[[File:Eventmath_logo_bitmap_graphic.png|200px]]</div> <div style="float:left; width:99%"> ===Planned content=== * ''Project purpose'': brief explanation of the problem addressed by Eventmath and the solution it provides * ''Featured lesson plan:'' prominent recognition of high-quality lesson plans to encourage users to create the best possible content * ''Intro video:'' welcome message to introduce visitors to the site and the project goals * ''Eventmath news:'' ongoing list of news, updates, and other current info ([[Eventmath/Meetups|meetups]], media coverage, etc.) * ''Testimonials:'' featured quotes, links, and other info about Eventmath being used in classrooms </div> {{Eventmath end tab}} [[Category:Eventmath project pages]] rcoj83vyk34f477p8b117ud6bt2p0sj 2410821 2410813 2022-08-01T20:39:41Z Greg at Higher Math Help 2911873 Under Get involved, replaced link to general meetups page with a direct link to the upcoming workshops. There's still a link to the general meetups page under Planned content. wikitext text/x-wiki {{Eventmath tabs}} __NOEDITSECTION__ <div style="float:left; width:60%;"> ==Math lesson plans based on current events== ===Welcome to Eventmath!=== Eventmath is a living repository pairing news articles and social media posts with math lesson plans. Anyone can * discover and edit [[Eventmath/Lesson plans|'''Lesson plans''']], * share ideas and get feedback by [[Eventmath/Contributing|'''Contributing''']] lesson plans, and * help make Eventmath a world-class resource by completing [[Eventmath/Tasks|'''Tasks''']]. This is a new project and is under development. If you're curious to learn more, you can read our [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath '''successful grant proposal''']. ===Get involved=== There are two great ways to get involved right now! You can * register for one of [[Eventmath/Meetups/Summer_2022_workshops|'''our workshops''']] to help us add more lesson plans * use our '''example lesson plans''' in the classroom and add your feedback: ** [[Eventmath/Lesson_plans/Proportions_and_voting_power_under_the_Electoral_College|Proportions and voting power under the Electoral College]] ** [[Eventmath/Lesson_plans/Comparing_streaming_service_pay_rates_to_artists|Comparing streaming service pay rates to artists]] </div> <div style="float:right; width:30%">[[File:Eventmath_logo_bitmap_graphic.png|200px]]</div> <div style="float:left; width:99%"> ===Planned content=== * ''Project purpose'': brief explanation of the problem addressed by Eventmath and the solution it provides * ''Featured lesson plan:'' prominent recognition of high-quality lesson plans to encourage users to create the best possible content * ''Intro video:'' welcome message to introduce visitors to the site and the project goals * ''Eventmath news:'' ongoing list of news, updates, and other current info ([[Eventmath/Meetups|meetups]], media coverage, etc.) * ''Testimonials:'' featured quotes, links, and other info about Eventmath being used in classrooms </div> {{Eventmath end tab}} [[Category:Eventmath project pages]] dq6wf90kkzjzvb0vned1uiops9rt43j 2410864 2410821 2022-08-02T02:16:37Z Greg at Higher Math Help 2911873 Added an implicit mission statement, and clarified that the workshops are free and online. wikitext text/x-wiki {{Eventmath tabs}} __NOEDITSECTION__ <div style="float:left; width:60%;"> ==Math lesson plans based on current events== ===Welcome to Eventmath!=== Wouldn’t it be great if students could wield math as a tool for evaluating news and social media content? To help make this a reality, Eventmath pairs news articles and social media posts with math lesson plans. It's a living repository, where anyone can * discover and edit [[Eventmath/Lesson plans|'''Lesson plans''']], * share ideas and get feedback by [[Eventmath/Contributing|'''Contributing''']] lesson plans, and * help make Eventmath a world-class resource by completing [[Eventmath/Tasks|'''Tasks''']]. This is a new project and is under development. If you're curious to learn more, you can read our [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath '''successful grant proposal''']. ===Get involved=== There are two great ways to get involved right now! You can * register for one of [[Eventmath/Meetups/Summer_2022_workshops|'''our free, online workshops''']] to help us add more lesson plans * use our '''example lesson plans''' in the classroom and add your feedback: ** [[Eventmath/Lesson_plans/Proportions_and_voting_power_under_the_Electoral_College|Proportions and voting power under the Electoral College]] ** [[Eventmath/Lesson_plans/Comparing_streaming_service_pay_rates_to_artists|Comparing streaming service pay rates to artists]] </div> <div style="float:right; width:30%">[[File:Eventmath_logo_bitmap_graphic.png|200px]]</div> <div style="float:left; width:99%"> ===Planned content=== * ''Project purpose'': brief explanation of the problem addressed by Eventmath and the solution it provides * ''Featured lesson plan:'' prominent recognition of high-quality lesson plans to encourage users to create the best possible content * ''Intro video:'' welcome message to introduce visitors to the site and the project goals * ''Eventmath news:'' ongoing list of news, updates, and other current info ([[Eventmath/Meetups|meetups]], media coverage, etc.) * ''Testimonials:'' featured quotes, links, and other info about Eventmath being used in classrooms </div> {{Eventmath end tab}} [[Category:Eventmath project pages]] p2nvqcbsyqj18ucyoxguishh8bgo6cd 2410871 2410864 2022-08-02T02:33:13Z Greg at Higher Math Help 2911873 Wording. wikitext text/x-wiki {{Eventmath tabs}} __NOEDITSECTION__ <div style="float:left; width:60%;"> ==Math lesson plans based on current events== ===Welcome to Eventmath!=== Wouldn’t it be great if students could wield math as a tool for evaluating news and social media content? To make this a reality, Eventmath pairs news articles and social media posts with math lesson plans. It's a living repository, where anyone can * discover and edit [[Eventmath/Lesson plans|'''Lesson plans''']], * share ideas and get feedback by [[Eventmath/Contributing|'''Contributing''']] lesson plans, and * help make Eventmath a world-class resource by completing [[Eventmath/Tasks|'''Tasks''']]. This is a new project and is under development. If you're curious to learn more, you can read our [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath '''successful grant proposal''']. ===Get involved=== There are two great ways to get involved right now! You can * register for one of [[Eventmath/Meetups/Summer_2022_workshops|'''our free, online workshops''']] to help us add more lesson plans * use our '''example lesson plans''' in the classroom and add your feedback: ** [[Eventmath/Lesson_plans/Proportions_and_voting_power_under_the_Electoral_College|Proportions and voting power under the Electoral College]] ** [[Eventmath/Lesson_plans/Comparing_streaming_service_pay_rates_to_artists|Comparing streaming service pay rates to artists]] </div> <div style="float:right; width:30%">[[File:Eventmath_logo_bitmap_graphic.png|200px]]</div> <div style="float:left; width:99%"> ===Planned content=== * ''Project purpose'': brief explanation of the problem addressed by Eventmath and the solution it provides * ''Featured lesson plan:'' prominent recognition of high-quality lesson plans to encourage users to create the best possible content * ''Intro video:'' welcome message to introduce visitors to the site and the project goals * ''Eventmath news:'' ongoing list of news, updates, and other current info ([[Eventmath/Meetups|meetups]], media coverage, etc.) * ''Testimonials:'' featured quotes, links, and other info about Eventmath being used in classrooms </div> {{Eventmath end tab}} [[Category:Eventmath project pages]] cg2ppq30532eqtcjekdnlt13xa2sdsm 2410872 2410871 2022-08-02T02:41:47Z Greg at Higher Math Help 2911873 Revised wording. wikitext text/x-wiki {{Eventmath tabs}} __NOEDITSECTION__ <div style="float:left; width:60%;"> ==Math lesson plans based on current events== ===Welcome to Eventmath!=== Wouldn’t it be great if students could wield math as a tool for evaluating news and social media content? To make this a reality, Eventmath pairs news articles and social media posts with math lesson plans. It's a living repository, where anyone can * discover and edit [[Eventmath/Lesson plans|'''Lesson plans''']], * share ideas and get feedback by [[Eventmath/Contributing|'''Contributing''']] lesson plans, and * help make Eventmath a world-class resource by completing [[Eventmath/Tasks|'''Tasks''']]. This is a new project and is under development. If you're curious to learn more, you can read our [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath '''successful grant proposal''']. ===Get involved=== There are two great ways to get involved right now! You can * register for one of [[Eventmath/Meetups/Summer_2022_workshops|'''our free, online workshops''']] to help us add more lesson plans * share your experience with using our '''example lesson plans''' in the classroom: ** [[Eventmath/Lesson_plans/Proportions_and_voting_power_under_the_Electoral_College|Proportions and voting power under the Electoral College]] ** [[Eventmath/Lesson_plans/Comparing_streaming_service_pay_rates_to_artists|Comparing streaming service pay rates to artists]] </div> <div style="float:right; width:30%">[[File:Eventmath_logo_bitmap_graphic.png|200px]]</div> <div style="float:left; width:99%"> ===Planned content=== * ''Project purpose'': brief explanation of the problem addressed by Eventmath and the solution it provides * ''Featured lesson plan:'' prominent recognition of high-quality lesson plans to encourage users to create the best possible content * ''Intro video:'' welcome message to introduce visitors to the site and the project goals * ''Eventmath news:'' ongoing list of news, updates, and other current info ([[Eventmath/Meetups|meetups]], media coverage, etc.) * ''Testimonials:'' featured quotes, links, and other info about Eventmath being used in classrooms </div> {{Eventmath end tab}} [[Category:Eventmath project pages]] ial6wewtodc78ka5mq97zy9mo1tk0lq 2410873 2410872 2022-08-02T02:48:13Z Greg at Higher Math Help 2911873 Made a clearer call to action regarding the example lesson plans. wikitext text/x-wiki {{Eventmath tabs}} __NOEDITSECTION__ <div style="float:left; width:60%;"> ==Math lesson plans based on current events== ===Welcome to Eventmath!=== Wouldn’t it be great if students could wield math as a tool for evaluating news and social media content? To make this a reality, Eventmath pairs news articles and social media posts with math lesson plans. It's a living repository, where anyone can * discover and edit [[Eventmath/Lesson plans|'''Lesson plans''']], * share ideas and get feedback by [[Eventmath/Contributing|'''Contributing''']] lesson plans, and * help make Eventmath a world-class resource by completing [[Eventmath/Tasks|'''Tasks''']]. This is a new project and is under development. If you're curious to learn more, you can read our [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath '''successful grant proposal''']. ===Get involved=== There are two great ways to get involved right now! You can * register for one of [[Eventmath/Meetups/Summer_2022_workshops|'''our free, online workshops''']] to help us add more lesson plans * add an endorsement or feedback to our '''example lesson plans''': ** [[Eventmath/Lesson_plans/Proportions_and_voting_power_under_the_Electoral_College|Proportions and voting power under the Electoral College]] ** [[Eventmath/Lesson_plans/Comparing_streaming_service_pay_rates_to_artists|Comparing streaming service pay rates to artists]] </div> <div style="float:right; width:30%">[[File:Eventmath_logo_bitmap_graphic.png|200px]]</div> <div style="float:left; width:99%"> ===Planned content=== * ''Project purpose'': brief explanation of the problem addressed by Eventmath and the solution it provides * ''Featured lesson plan:'' prominent recognition of high-quality lesson plans to encourage users to create the best possible content * ''Intro video:'' welcome message to introduce visitors to the site and the project goals * ''Eventmath news:'' ongoing list of news, updates, and other current info ([[Eventmath/Meetups|meetups]], media coverage, etc.) * ''Testimonials:'' featured quotes, links, and other info about Eventmath being used in classrooms </div> {{Eventmath end tab}} [[Category:Eventmath project pages]] 353n4f0d95cg4qtqhtee7vzmjzkinw3 2 277737 2410892 2342791 2022-08-02T03:58:31Z Jtwsaddress42 234843 wikitext text/x-wiki {{User:Jtwsaddress42/People}} '''[[User:Jtwsaddress42/Bibliography_F|<big>F</big>]]''' {{User:Jtwsaddress42/People/Faraday,_Michael}} {{User:Jtwsaddress42/People/Fermi,_Enrico}} {{User:Jtwsaddress42/People/Feynman,_Richard Phillips}} {{User:Jtwsaddress42/People/Faraday, Michael}} {{User:Jtwsaddress42/Includes/Notes_&_Citations}} {{User:Jtwsaddress42/Navigation/Footer Navbar}} bsx8j7nz76p6dmonxk6q25fxgjfhyj4 2410893 2410892 2022-08-02T03:59:13Z Jtwsaddress42 234843 wikitext text/x-wiki {{User:Jtwsaddress42/People}} '''[[User:Jtwsaddress42/Bibliography_F|<big>F</big>]]''' {{User:Jtwsaddress42/People/Faraday,_Michael}} {{User:Jtwsaddress42/People/Fermi,_Enrico}} {{User:Jtwsaddress42/People/Feynman,_Richard Phillips}} {{User:Jtwsaddress42/People/Freud, Sigmund}} {{User:Jtwsaddress42/Includes/Notes_&_Citations}} {{User:Jtwsaddress42/Navigation/Footer Navbar}} hgolawkb2cup2bcscdb4i61t9d4fnvo 2 277884 2410882 2408345 2022-08-02T03:28:14Z Jtwsaddress42 234843 /* Cisek, Paul */ wikitext text/x-wiki {{RoundBoxTop|theme=3}} === [https://cisek.org/pavel/ Cisek, Paul] === <hr /> '''Notable Accomplishments''' * Neuroscientist * Phylogenetic refinement * Affordance competition hypothesis <br /><hr /> '''Publications''' {{User:Jtwsaddress42/Bibliography/Cisek,_Paul}} {{RoundBoxBottom}} <hr /> 39kcwre72lon4kmdzjqip216894he8i 2 277950 2410890 2408663 2022-08-02T03:50:04Z Jtwsaddress42 234843 wikitext text/x-wiki {{RoundBoxTop|theme=3}} === [[w:Michael_Faraday|Faraday, Michael (1791 - 1867)]] === <hr /> [[File:M Faraday Th Phillips oil 1842.jpg|thumb|Michael Faraday (1791 - 1867){{efn|portrait by Thomas Phillips (oil, 1842)}}]] '''Notable Accomplishments''' * Magnetic Field * Electromagnetic Induction * Electrolysis <br /><hr /> '''Publications''' {{User:Jtwsaddress42/Bibliography/Faraday, Michael}} <br /><hr /> {| align= center | width= 480px |{{User:Jtwsaddress42/Gallery/Michael Faraday}} <br /> |} {{RoundBoxBottom}} <hr /> 715o2dgg2ogahgd6jbg8t4865iyavvd 2 277951 2410887 2408356 2022-08-02T03:46:23Z Jtwsaddress42 234843 /* Fermi, Enrico (1901 - 1954) */ wikitext text/x-wiki {{RoundBoxTop|theme=3}} === [[w:Enrico_Fermi|Fermi, Enrico (1901 - 1954)]] === <hr /> [[File:Enrico Fermi 1943-49.jpg|thumb|Enrico Fermi (1901 - 1954)]] [[File:Beta-minus Decay.svg|thumb|Fermi and Pauli postulated that a neutrino (<math>\bar{\nu}_e</math>) would be emitted during Beta Decay.]] '''Notable Accomplishments''' * [https://www.nobelprize.org/prizes/physics/1938/fermi/facts/ The Nobel Prize in Physics 1938] - "for his demonstrations of the existence of new radioactive elements produced by neutron irradiation, and for his related discovery of nuclear reactions brought about by slow neutrons." <br /><hr /> '''Publications''' {{User:Jtwsaddress42/Bibliography/Fermi,_Enrico}} <br /><hr /> {| align= center |'''Stagg Field Reactor''' ''Chicago Pile-1, the first nuclear reactor to achieve a self-sustaining chain reaction''<br /> [[File:Stagg Field reactor.jpg|640px|Stagg Field reactor]] <br/> |} {{RoundBoxBottom}} <hr /> rz4l27djbcygyq2gvfvdiwqb5xe8xs5 2 278019 2410900 2410614 2022-08-02T04:51:48Z Jtwsaddress42 234843 /* Onsager, Lars (1903 – 1976) */ wikitext text/x-wiki {{RoundBoxTop|theme=3}} === [[w:Lars_Onsager|Onsager, Lars (1903 – 1976)]] === <hr /> [[File:Lars Onsager.png|thumb|Lars Onsager]] [[File:Thermodynamics.png|thumb|Pioneers of Thermodynamics]] '''Notable Accomplishments''' * [https://www.nobelprize.org/prizes/chemistry/1968/onsager/facts/ The Nobel Prize in Chemistry 1968] - “for the discovery of the reciprocal relations bearing his name, which are fundamental for the thermodynamics of irreversible processes.” <br /><hr /> {| align= center | width= 640px |{{User:Jtwsaddress42/Gallery/Lars Onsager}} <br /> |} <hr /> '''Publications''' {{User:Jtwsaddress42/Bibliography/Onsager, Lars}} {{RoundBoxBottom}} <hr /> 2qplupgc155do5qd2kxx0d7iau1fz5m 0 278752 2410797 2341665 2022-08-01T15:32:11Z Guy vandegrift 813252 wikitext text/x-wiki [[File:Colored_circle_permutations_of_3.svg|thumb|300px]] Abstract:Of the three kinds of questions, only one needs to be finished. Outline: 1. Reform: Two tasks for the reformer: first convince people the reform is needed, second lay out a plan. 2. Type 1-2 errors. 3. sss * [[Open Educational Resources|open educational resources]] wikiversity link *The advantages of assessing students ''before'' the course begins cannot be overstated. Mechanisms already exist that allow students to skip a required course,<ref>After getting my undergraduate degree in physics, I spent two [[wikipedia:Gap year|gap years]] studying graduate level physics, while working part time at an [[wikipedia:IHOP|IHOP]]. I was scheduled to take a three-semester course that covered [[wikipedia:Classical_Electrodynamics_(book)|Jackson's Electrodynamics]]. Upon my request to skip the course, the professor opened the book to a random page. I knew enough from the top of my head about [[wikipedia:Spherical harmonics|spherical harmonics]] to bypass the entire sequence.</ref> and the introduction of quasi-standardized<ref>I all them "quazi-standardized" for two reasons: (1) there will be no strictly enforced uniform standards</ref> undergraduate prelims will facilitate further evolution of such alternative assessments.<ref>For example, contribution to these undergraduate prelim exams could serve as an alternative assessment for those with test anxiety, dyslexia, or other neurological conditions</ref><u>I am confident that all the flaws associated with an initial effort to create an open source exam bank for undergraduate prelims can be overcome by providing students with alternative assessments.</u> ==Three types of test questions== {{center|''Current efforts should focus on #3 (scientific literacy)''}} Of the three kinds of test questions on a first effort to create our bank, current efforts should focus on the third: #Simple fact-based knowledge questions. #Traditional "problems"<ref>I place "problems" in quotation marks because the majority of such entities are pedagogical in nature and at best marginally related to real-life situations. No criticism of higher education is intended here; the complexity of the "real world" is such that our goal is not to </ref> that require a calculation. #'''Reading comprehension with fact-based follow-up questions.''' The first two types actually represent the ends of a continuous spectrum. To some extent, the student's solution to any question requiring a calculation is based on the student's ability to remember the solution to another problem. With the ever-increasing practice of [[Quizbank/Creating a bank so students won't ''break the bank''#Google-solving|Google-solving]], one can argue that the traditional exercise of using thought to solve a physics problem no longer exists. *[[w:Concept_inventory|Concept inventory]] questions. If the circumstances are favorable, these questions effectively measure a student's grasp of fundamental ideas.<ref>My experience is with [[w:Force Concept Inventory]]</ref> Unfortunately, the release of any [[Quizbank|quizbank]] information to students is essentially [[w:Teaching to the test|"teaching-to-the-test",]] and therefore diametrically opposite to any [[w:authentic assessment|authentic assessment.]] Ways to compensate for the absence of such questions shall be discussed below. *Essay questions and oral exams. These are by far the most effective assessments if the goal is to permit students to bypass a prerequisite course. I view such essays and oral exams as an opportunity to compensate for the obvious flaws associated attempting to design a collection of [[Quizbank/Cost-benefit analysis/Undergraduate prelims|undergraduate prelims]] for introductory college courses. ==Two short "essays" with sample questions== key points to mention here or elsewhere #True false questions #Disjoint nature of the two essays #Important to introduce all formulas in review #Reviewing material is permissible b/c it teaches students to skim. #Short essays and TF will permit students to contribute to the bank #Plan is to start "low-tech" and hope "high-tech" versions follow ===Review of binomial coefficients=== {{center|''One of the issues we face in designing an [[w:Open educational resources|OER]] involves the question of what a student is expected to know in order to [[w:Alternative_pathways_in_education#Credit_by_examination|challenge]] all or part of a college course. To some extent, this can be alleviated by carefully reviewing material material that most students are expected to know in a passage designed to assessing a student's skill at reading technical prose. For that reason, we begin with the binomial coefficient, with the understanding it will be skimmed by many students.''}} The binomial coefficient, <math>\tbinom{n}{k},</math> is often pronounced as "{{mvar|n}} choose {{mvar|k}}": <math>\binom{n}{k} = \frac{n!}{k! \; (n-k)!}</math><math> = \frac{(n_1+n_2)!}{n_1! \; n_2!},</math> where the [[w:factorial|factorial]] is expressed as <math>n! = </math><math>n \cdot (n-1) \cdot (n-2) \cdots 3 \cdot 2 \cdot 1 \,. </math> While this "definition" leads naturally to <math>1!=1</math>, it does little to justify the [[w:Empty product|empty product]] convention,<ref>Perhaps the most well-known argument that <math>0!</math> should equal [[w:1|unity]] involves the [[w:Gamma function|gamma function]].</ref> which is that <math>0!=1.</math> This convention also supports An examination of ''{{mvar|3}} choose {{mvar|k}}'' for ''{{mvar|k}}''<math>\in\{0,1,2,3\}</math> also supports the use of this convention<ref>In [[w:Set theory|set theory]] <math>k\in\{0,1,2,3\}</math> means that <math>k</math> is in the set <math>\{0,1,2,3\}.</math></ref> for both <math>k=0</math> and <math>k=3,</math> as shown to the left in the figure below: [[file:Powerset.png]] In this figure, the three objects are the colored circles, labeled <big>{{color|#E4717A|'''1'''}},{{color|#7BB661|'''2'''}},{{color|#7366BD|'''3'''}}</big>.<ref>One '''might''' be able to argue that the convention that there is one way to select zero objects is arbitrary by saying that it is meaningless to count the ways to do nothing. On the other hand, a strong case can be made there is '''one''' way to select all three items. Both formulas involve {{mvar|0!}}.</ref> On the right, we see that 3 choose 2 and 3 choose 1 are equal. The other pair of binomial coefficients shown in the figure above are obviously equal: When choosing from {{mvar|3}} items, selecting two items is equivalent to not selecting one item. Instead of "choosing" either one or two items, both coefficients represent splitting the items into two groups: One group contains {{mvar|1}} item, while the other contains {{mvar|2}} items. :<math>\binom 3 1=\binom 2 3 = \frac{3!}{2!\cdot 1!}=3</math> When introducing binomial coefficients to students for the first time, this example can be used to establish that the order in which objects are chosen has no significance. The above figure makes it abundantly clear that the choice <big>{{color|#E4717A|'''1'''}},{{color|#7BB661|'''2'''}}</big> is equivalent the choice <big>{{color|#7BB661|'''2'''}},{{color|#E4717A|'''1'''}}</big>. ===Number of words that can be created from a set of letters=== [[File:Multinomial theorem mississippi.svg|thumb|Multinomial coefficient as a product of binomial coefficients, counting the permutations of the letters of MISSISSIPPI.]] Here we use properties of the binomial coefficient to solve a more complex problem. Consider the set of positive integers less than or equal to eleven: ''{1,2,3,...11}''. While the binomial coefficients permit us to group the elements of this set into two distinct sets, we now seek to group the eleven members of this set into four groups. For reasons that will become clear, we label these four sets with the capital letters: I, M, P, S. The eleven integers shall be placed into these groups (or "bins") in the following manner: * 4 elements are placed in bin I * 1 element is placed in bin M * 2 elements are placed in bin P * 4 elements are placed in bin S For example, the number of distinct permutations of the letters of the word MISSISSIPPI, which has 1 M, 4 Is, 4 Ss, and 2 Ps, is :<math>{11 \choose 1, 4, 4, 2} = \frac{11!}{1!\, 4!\, 4!\, 2!} = 34650.</math> *[[wikt:toots]] *[[w:Binomial theorem]] *[[w:Multinomial theorem]] ==Educational "reform": easier said than done== {{center|''A goal and the path to that goal are different things.''}} private and public institutions competing interests and vaguely defined division of power perestroika driving on left to right I have hope for this ===Type-1 and type-2 assessment errors=== {{center|''Why it will be easy to construct a useful bank for undergraduate prelims.''}} The construction of a publicly available [[w:Open educational resources|OER]] bank questions [[Quizbank/Cost-benefit analysis/Undergraduate prelims|preliminary exams]] in [[w:Science, technology, engineering, and mathematics|STEM]] courses is greatly simplified by a surprising fact: '''There is little need for concern ???about the quality of the questions.''' This can be explained in the language of hypothesis testing, taking the null hypotheses to be that the student is prepared to skip a portion of the course.<ref>If the test exempts students from the first of multi-course series, most colleges would probably require that the student either take the next course, or successfully engage in a follow-up workshop or seminar on the subject. The simplest such followup would be to require attendance and some sort of performance in the last weeks of the course that the student wishes to test out of.</ref>. There is almost no reason to '''be concerned???''' about a [[w:Type I and type II errors|type-1 error]], where '''a qualified student fails the prelim an is told to take the course.''' Keep in mind that student has studied a topic before taking a course, ''at zero financial cost''. Perhaps the student will learn how ''not to learn'' a topic. Perhaps the student will uncover (or document) a test-taking disability. Perhaps the student has exposed a flaw in the test that the college needs to know about. All of these type-1 errors uncover issues that need to be resolved as soon as possible. Keep in mind that the alternative to learning about these issues before a student enrolls in a course is for them to be discovered after time and money has been wasted on an unsuccessful effort to complete a college course. A simple remedy for the type-1 error is to provide [[w:Alternative assessment|alternative assessments]] to students who fail the prelim but continue to believe they know the perquisite material. {{Red|MENTION THE OPTIONS}} ===A path through the wilderness=== {{center|''We can let competition between colleges for quality students drive the reform.''}} ==Temporary odds and ends to be included/discarded== [[w:Special:Permalink/991951112#Multinomial_Theorem_of_2ⁿᵈ_kind|This proof]] reminded me of the fact that I have been contemplating the problem of writing test questions specifically designed for an open source [[Quizbank|quizbank]]. [[How_things_work_college_course/Conceptual_physics_wikiquizzes/Velocity_and_acceleration]] [[Category:Quizbank/Essays]] nkvwr6lbd042u9dhknft4gfyrzv6q4q 2 279198 2410874 2408983 2022-08-02T02:49:26Z Jtwsaddress42 234843 /* Sources & References */ wikitext text/x-wiki == Sources & References == {{RoundBoxTop|theme=2}} {{refbegin}} <!-- Convert all 'cite journal' citationas to 'cite web' for citation links to work with reflist --> * {{cite book | last= Black | first= Ira B. | date= 1994 | title= Information in the Brain - A Molecular Perspective | publisher= MIT Press | isbn= 978-0-262-52188-8 | url= https://mitpress.mit.edu/books/information-brain}} * {{cite journal | last= Chomsky | first= Noam | year= 1967 | title= The Responsibility of Intellectuals | journal= The New York Review of Books | publication-date= February 23, 1967 | url= https://chomsky.info/19670223/ }} * {{cite web | last= Darwin | first= Charles | date= 1837 | title= Transmutation Diagram | publisher= Cambridge University Library | quote= Notebook B | website= University of Cambridge - Digital Library | access-date= October 18, 2021 | url= https://cudl.lib.cam.ac.uk/view/MS-DAR-00121/21 }} * {{cite book | last= Darwin | first= Charles | year= 1859 | title= On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life | publisher= D. Appleton and Company | location= New York | publication-date= 1869 | url= https://www.google.com/books/edition/On_the_Origin_of_Species_by_Means_of_Nat/ez9KAAAAYAAJ?hl=en&gbpv=1&dq=1869+On+the+Origin+of+Species+by+Means+of+Natural+Selection,+or+the+Preservation+of+Favoured+Races+in+the+Struggle+for+Life&printsec=frontcover }} ([https://www.youtube.com/watch?v=KX0EwgxiAg4&list=PLZ-bKJtH3G7Cnqo14_BiY09yett68y9Es&index=32 Audiobook - Part 1] - [https://www.youtube.com/watch?v=iBPVA3i8mKU&list=PLZ-bKJtH3G7Cnqo14_BiY09yett68y9Es&index=30 Part 2] - [https://www.youtube.com/watch?v=uURn7dnC66I&list=PLZ-bKJtH3G7Cnqo14_BiY09yett68y9Es&index=48 Part 3] - [https://www.youtube.com/watch?v=IM6fPaEHZgE&list=PLZ-bKJtH3G7Cnqo14_BiY09yett68y9Es&index=18 Part 4] [[File:High-contrast-camera-video.svg|24px|video]]) * {{cite book | last= Darwin | first= Charles | year= 1871 | title= The Descent of Man | publisher= Prometheus Books | publication-date= 1998 | isbn= 978-1-573-92176-3 | url= https://www.google.com/books/edition/_/Z0B9QgAACAAJ?hl=en&sa=X&ved=2ahUKEwjlkr_F-vH1AhWLIkQIHVl_C6QQ8fIDegUIAxCeAg }} ([https://www.youtube.com/watch?v=So3GkSrFtnU&list=PLZ-bKJtH3G7BYYscYohAFZAq65RwCf-K-&index=153 Audiobook - Part 1] - [https://www.youtube.com/watch?v=6gqLfFPblZ0&list=PLZ-bKJtH3G7BYYscYohAFZAq65RwCf-K-&index=164 Part 2] [[File:High-contrast-camera-video.svg|24px|video]]) * {{cite book | last= Darwin | first= Charles | year= 1872 | title= The Expression of the Emotions in Man and Animals - with a preface by Konrad Lorentz | publisher= The University of Chicago Press | location= Chicago & London | publication-date= 1965 | isbn= 0-226-13656-6 | url= https://www.google.com/books/edition/The_Expression_of_the_Emotions_in_Man_an/p_ZJCgAAQBAJ?hl=en&gbpv=1&dq=The+Expression+of+the+Emotions+in+Man+and+Animals&printsec=frontcover }} ([https://www.youtube.com/watch?v=AopL_EE9SIQ&list=PLZ-bKJtH3G7BYYscYohAFZAq65RwCf-K-&index=90 Audiobook - Part 1] - [https://www.youtube.com/watch?v=_auuHHXm3ZI&list=PLZ-bKJtH3G7BYYscYohAFZAq65RwCf-K-&index=101 Part 2] [[File:High-contrast-camera-video.svg|24px|video]]) * {{cite book | last= Darwin | first= Charles | year= 1887 | title= The Autobiography of Charles Darwin 1809–1882." With the Original Omissions Restored. Edited and with Appendix and Notes by his Granddaughter Nora Barlow | publisher= W.W. Norton & Company | publication-date= 1969 | isbn= 0-393-00487-2 }} <!-- * {{cite journal | last= Deuchar | first= Elizabeth M. | year= 1970 | title= Diffusion in Embryogenesis | journal= Nature | volume= 225 | number= 5233 | pages= 671 | publication-date= February 14, 1970 | pmid= 16056680 | doi= 10.1038/225671b0 | url= https://www.nature.com/articles/225671b0 }} --> {{User:Jtwsaddress42/Bibliography/Dobzhansky, Theodosius}} * {{cite book | last= Edelman | first= Gerald M. | date= 1987a | title= Neural Darwinism - The Theory of Neuronal Group Selection | publisher= Basic Books | location= New York | isbn= 0-465-04934-6 | url= https://books.google.com/books?id=hOdqAAAAMAAJ }} * {{cite journal | last1= Edelman | first1= Gerald M. | last2= Gally | first2= Joseph A. | date= 2001 | title= Degeneracy and Complexity in Biological Systems | journal= Proceedings of the National Academy of Sciences, USA | volume= 98 | number= 24 | pages= 13763–13768 | doi= 10.1073/pnas.231499798 | pmid= 11698650 | pmc= 61115 | bibcode= 2001PNAS...9813763E | url= https://www.pnas.org/content/98/24/13763 }} * {{cite AV media | last= Eisenhower | first= Dwight D. | year= 1953 | title= Cross of Iron - Address to the American Society of Newspaper Editors | publisher= American Rhetoric Online Speech Bank | medium= .mp3 | publication-date= April 19, 1953 | url= https://www.americanrhetoric.com/mp3clips/politicalspeeches/dwighteisenhowerpeace.mp3 }} [https://www.americanrhetoric.com/speeches/dwighteisenhowercrossofiron.htm Delivered at the Statler Hotel, Washington, D.C.] (transcript) (0:25:46) * {{cite journal | last= Hall | first= Brian K. | year= 2000a | title= The Neural Crest as a Fourth Germ Layer and Vertebrates as Quadroblastic Not Triploblastic | journal= Evolution &amp; Development | volume= 2 | number= 1 | pages= 3-5 | publication-date= January-February 2000 | pmid= 11256415 | doi= 10.1046/j.1525-142x.2000.00032.x | url= https://onlinelibrary.wiley.com/doi/abs/10.1046/j.1525-142x.2000.00032.x?sid=nlm%3Apubmed }} * {{cite AV media | last= International Society for Evolutionary Protistology (ISEP/ISOP) | year= 2021 | title= “Tree of life” symposium in memory of Tom Cavalier-Smith | publisher= The International Society for Evolutionary Protistology | publication-date= July 23, 2021 | url= https://www.youtube.com/watch?v=SfeRKaDZyz0 }} [[File:High-contrast-camera-video.svg|24px|video]] (3:31:32) * {{cite AV media | last= King Jr. | first= Martin Luther | year= 1967 | title= Beyond Vietnam - A Time to Break Silence | publisher= American Rhetoric Online Speech Bank | medium= .mp3 | publication-date= April 4, 1967 | url= https://archive.org/download/BeyondVietnamATimeToBreakSilence4467/Beyond%20Vietnam-A%20Time%20to%20Break%20Silence%204-4-67.mp3 }} [https://www.americanrhetoric.com/speeches/mlkatimetobreaksilence.htm Delivered at Riverside Church, New York City] (transcript) (0:56:20) * {{cite journal | last1= Romer | first1= Alfred Sherwood | year= 1972 | chapter= The Vertebrate as a Dual Animal — Somatic and Visceral." | editor= Dobzhansky T., Hecht MK, and Steere WC | journal= Evolutionary Biology | publisher= Springer | location= New York, NY | pages= 121–156 | doi= 10.1007/978-1-4684-9063-3_5 | isbn= 978-1-4684-9065-7 | url= https://link.springer.com/chapter/10.1007%2F978-1-4684-9063-3_5 }} * {{cite AV media | author = United States House of Representatives | year= 2022d | series= January 6 Hearings | volume= 4 | title= Fourth Hearing on Investigation of Capitol Attack | publisher= CSpan | medium= Recorded June 21, 2022 1:00 PM ET | publication-date= June 21, 2022 | url= https://www.c-span.org/video/?521075-1/fourth-hearing-investigation-capitol-attack }} [https://january6th.house.gov Select Committee to Investigate the January 6th Attack on the United States Capitol] [[File:High-contrast-camera-video.svg|24px|video]] (2:43:17) <!-- * {{cite book | last= Stanier | first= Roger Y. | date= 1970 | chapter= Some aspects of the biology of cells and their possible evolutionary significance | title= Organization and Control in Prokaryotic and Eukaryotic Cells | series= Twentieth Symposium of the Society for General Microbiology | volume= 20 | pages= 1-38 | editor= Charles HP, Knight BCJG | publisher= Cambridge University Press | url= https://www.google.com/books/edition/_/hD3dxwEACAAJ?hl=en }} --> {{refend}} {{RoundBoxBottom}} <hr /> 72tk395vnlichnm97jnupsv94l6xi2q 0 279994 2410834 2410423 2022-08-01T22:50:41Z Dave Braunschweig 426084 Dave Braunschweig moved page [[User:Jacsaw/sandbox]] to [[Advocacy in Technology and Society]] without leaving a redirect: Moving to main space wikitext text/x-wiki = Advocacy in Technology and Society = ==== ''a Living Syllabus Project'' ==== === Introduction === 2020 brought the  inequality across society into sharper focus. COVID-19 spread across the world highlighting the inequities of disproportionate impact, response, and community preparedness. Starting in the United States, and echoed across the world, protests for racial justice and continued action for Black lives was newly galvanized across sectors of society. Economic strains and widespread unemployment raised awareness of widening wealth inequality in the US, adding new dimension to old debate about social protection and safety nets. Additionally, with most of the workforce working remotely from their homes, dependency on technology infrastructure emerged almost overnight, illuminating access disparity and the [[wikipedia:Digital_divide|"digital divide]]". This past year called for reflection and action at the intersections of social life, with technology playing a critical role in how information is consumed, produced, distributed, and controlled in 2021. This course is designed to build a critical foundation for understanding the role technology plays in social welfare and social change and the important role of advocates in this space. Beginning with a foundational history, this course will set the stage for the rapid technology development in the 21st century and prepares learners to actively interrogate, question, critique, and evaluate the emerging technologies. This course considers the technology and data-based systems in a social welfare context in the following ways: # How can social workers and advocates use technology and a data-based practice to further their advocacy efforts? # How are current and emergent technologies challenging the work, and how can social workers be more engaged advocates in the pursuit of better, fairer, and more just technology? This course will provide an overview of the role that technology plays in civic engagement and social movements in the digital age, in the United States, with some international examples. Students will develop the ability to critically analyze technology tools in a social welfare context and make recommendations for fairer tech. Students will contribute content and help build and shape an open knowledge platform through ‘a living syllabus’ === Learning Objectives === ==== Through this course, students will be able to : ==== * Identify technologies omnipresent in contemporary society * Describe social and community advocacy in online spaces & digital context * Identify the different stakeholders who affect and are affected by emerging technology * Articulate the role of social workers in technology development and technology in social welfare context * Identify current technology tools (apps, platforms, and movements) * Analyze the social effects of emerging technologies (e.g.  discriminatory algorithms, digital marginalization, access issues) * Critique a current technology of their choosing (ie. an app/platform/movement)  from a social work framework, identifying the technology used, the challenges and social impact from the perspective of the different stakeholder groups, identify how the technology might oppress or empower individuals or groups * Contribute to open access, common knowledge platforms, build a ‘social workers in tech’ context === Weekly Topics === # Defining the Digital Age : from analogue to digital # Brief History of the Internet and Open Access # Understanding the Landscape: Identifying the Stakeholders, the role of Social Workers # Emerging Technologies - Artificial Intelligence and Algorithmic Decision Making #[https://www.ruhabenjamin.com/race-after-technology Race After Technology], [https://design-justice.pubpub.org/ Design Justice] , and [https://data-feminism.mitpress.mit.edu/ Data Feminism] # Community Data during COVID-19 # Civic and GovTech # Environmental Justice & Tech #Critiquing and Evaluating Tech {| class="wikitable" |+ !Module !Topic !Description |- |1 |[[/Defining the Digital Age/|Defining the Digital Age : from analog to digital]] | |- |2 |[[/Digital Landscape + Open Access/|Digital landscape and Open Access]] | |- |3 |[[/Who's Online? Building the next 30 years of the Internet/|Who's Online? Imagining the next 30 years of the Internet]] |This module gives an overview of internet usage in the present day and discusses the immediate future of the internet and internet access. |- |4 |[[/Social Work + Technology/]] |This module explores the advocacy social workers can do with technology in mind; creating new technology tools, advocating for vulnerable populations affected by technology, and how to create a more fair and just use of technology in the world. |- |5 |[[/Emerging Tech - Identifying the Stakeholders/]] |This module prompts social workers to gain a deeper understanding of the extent to which emerging technologies are impacting individuals on a micro, mezzo, and macro level. This module explores how the process of identifying stakeholders can lead to a greater understanding of the impact of emerging technologies on the well-being of communities while highlighting the areas in which social workers can intervene. |- |6 |[[/Community Data and COVID-19/]] |This module highlights community data initiatives that were used to illustrate the devastating impacts of COVID-19 on different communities, while challenging the lack of equity and intersectionality in how data is captured. This module also explores disinformation concerns during the pandemic, and how we can minimize the collection of personal data while managing public health. |- |7 |[[/Building Informed Critique/]] |This module explores the risk of clicking through without consideration of how information is shared, stored, and sold. |- |8 |[[/Social Work Futures/]] |This module focuses on the intersections of social work and technology in an innovative manner. |- |9 |[[/Design Justice + Creative Media/]] |This module seeks to question society's norms, values, assumptions and how they are encoded in and reproduced through the design of sociotechnical data-driven systems The module will explore innovative community-based organizations that seek through technology to dismantle a matrix of domination (white supremacist heteropatriarchy, ableism, capitalism, and settler colonialism) and bring marginalized communities to the forefront in technology. |- |10 |[[/Civic and GovTech, Regulation in the time of Big Tech/]] |This module gives an analytic review of the relationship between people and government through the use of different techniques. This includes communication, service delivery, decision-making, and general politics. It also delves into how technology can be controlled in the time of big techs. |- |11 |[[/Social Media, VR + the metaverse/]] | |- |12 |[[/Tech and Data-based advocacy outside of the US context/|Tech and Data-based advocacy outside of the US context]] |This module solicits input from advocates who work outside of the United States context. Centered around two central questions, contributors provide insight from their experience on the following: # What are the biggest opportunities you see for using technology to further your advocacy efforts? # What are the biggest challenges and/or barriers you face in using technology to further your advocacy efforts? |} === '''Course Content Development''' === The course content was developed by Jaclyn Sawyer and was taught synchronously at Columbia University, School of Social Work (Spring 2022). === '''Contributors''' === The students in this class contributed to each module by summarizing, synthesizing, critiquing, and adding their perspective to the material and building a "living syllabus". md4jhu2wy6w726c6mar64f6e2yshztx 0 280002 2410838 2384895 2022-08-01T22:50:42Z Dave Braunschweig 426084 Dave Braunschweig moved page [[User:Jacsaw/sandbox/Defining the Digital Age]] to [[Advocacy in Technology and Society/Defining the Digital Age]] without leaving a redirect: Moving to main space wikitext text/x-wiki == Module Description == g0vugc4eryrnw8dve8tipb8ey0zy21o 0 280069 2410845 2384896 2022-08-01T22:50:44Z Dave Braunschweig 426084 Dave Braunschweig moved page [[User:Jacsaw/sandbox/Tech and Data-based advocacy outside of the US context]] to [[Advocacy in Technology and Society/Tech and Data-based advocacy outside of the US context]] without leaving a redirect: Moving to main space wikitext text/x-wiki == Advocating for better tech == # What are the biggest opportunities you see for using technology to further your advocacy efforts? # What are biggest challenges and/or barriers you face using technology to further your advocacy efforts? {| class="wikitable" |+ ! !Contributor !Profession !Country where you work |'''What are the biggest opportunities you see for using technology to further your advocacy efforts?''' !What are biggest challenges and/or barriers you face using technology to further your advocacy efforts? !Resources ''(insert links)'' |- |1 | | | | | | |- |2 | | | | | | |- |3 | | | | | | |- |4 | | | | | | |- |5 | | | | | | |- |6 | | | | | | |- |7 | | | | | | |- |8 | | | | | | |- |9 | | | | | | |- |10 | | | | | | |} 4cqh0n6a0u4mxxj0afizbkcjgwee2pu 0 281095 2410907 2410758 2022-08-02T05:33:21Z Hamish84 1362807 Making spaces wikitext text/x-wiki = <ref>'''William Shakespeare''' (bapt. 26 April 1564 – 23 April 1616</ref>A message. = What humanity needs is not any individual approach but a governance powerful body of excellence that has modern technology, knowledge, and freedom they can use to disseminate clear information. Clear information about a new language structure of absolutes beneficial to an International forum, and eventually to reach a Universal status.Its benefits reach toward conceptual language on a planet that speaks more than 7,000 languages. No matter the language spoken the concepts of: Air - Food - Water are recognised. The overall development of conceptual language can only be beneficial and will be as appropriate to Absolutes definitions. Reaching for the Stars might show us the way! Universal Language of Absolutes (A very grand title but it took many years to explain its value) = Our history. = Born in Scotland in 1927 left school at 14 years of age. Married at 21 years of age and we had two children. We emigrated to New Zealand in 1953 and lived there for approximately thirty years. During our stay there I did a Liberal Studies Course at Canterbury University Christchurch and graduated. We have since had ten books published through Google books on the subject matter at hand and my wife Jean Caldwell McMillan is the co-author to most of the work presented here. My wife Jean was an avid reader of many works on philosophy and psychology. She was influenced by the works of Erich Fromm. Jean died 9th of January 2011. To refresh the original purpose of  our  earlier writings my wife and I  went on an odyssey  looking for any data, ancient or otherwise, on human consciousness, specifically related to Alzheimer’s disease. Now at 95 years of age (well past my used by date) it may well be that I am a candidate with a focus on my own pending dementia. If so, then the theory and the method I now write about is holding it at bay. To address the health of my mind in this way could be the catalyst that retains its own functional activity. The creation of a  semantic template is well documented below. No semantic definition of absolutes or principles can be ill-defined. They are always interconnected, interdependent and infinite. Each configuration constructed by anyone has meaning particular to them, although its value is universal. That  is why it is never personal property! [[File:JimNJean.png|center|thumb|228x228px|Jim and Jean]] = The Beginning. = "The road's half traveled when you know the way" [[File:Tree lined path - geograph.org.uk - 2269906.jpg|center|thumb]] Oxford dictionary definitions: Principle: "A fundamental truth used as a basis of reasoning". Absolute: " Complete - Entire - Perfect - Pure. These particular dictionary given definitions offers us guidelines to ‘existing conditions’ necessary for complementary understanding, and experience. We can only examine that which is real, basic evidence, that is fundamentally true, and we must ‘use’ it, to establish that which is reasonable. The general consensus is that there are no Absolutes. The following material is set out to show the very reverse is true and that everything that is is Absolute. Establish that there are no dichotomies that will leave the primary terms alone to create a semantic template. There are no dichotomies. Mythical dichotomies distort Reality. Everything is: The computer you use today has always existed, it is the arrangement of particles that have materialized it. The subject matter "Universal Language of Absolutes' is promoted to provide a new understanding of spoken language. This understanding was initially constructed by the cognitive experiences of both my wife and myself many years ago. Just like the principle of a jigsaw puzzle, meaning lies significantly in the fact that all pieces of the puzzle are interdependent and interconnected. When completed they provide a picture of the whole. We have endeavoured to produce a picture of the evolutionary process of language in human history because the evolution of language prefixes all modes of thought in human culture. The material directs the reader towards a new view that all that evolves is in a vertical direction, not the linear direction commonly understood. Human consciousness is of itself the phenomenon of evolution and to recognize its existence is part of the process. Shakespeare expressed this succinctly through the voice of Juliet who proclaimed, “a rose by any other name- would smell as sweet.” [[File:Comestible rose in the Laquenexy orchard garden, Moselle, France (01).jpg|thumb|center|237x237px]] === Conceptual language. === My wife and I recognized how profound the extension of this observation would mean conceptually. Of all the languages spoken on this planet, it would be fair to say that all of them would contain the properties of, air, food, and water conceptually, etc. This is a form of consciousness equality that is available to us all. It points to the reality of our constant relation to each other and our existence. We can never exist in a world of individuality, but only in relation to the consciousness of one another. Consequently, that exceptional experience can only be shared superficially. We cannot ‘know’ any other life experiences other than our own introspection. <gallery> File:Einstein 1921 by F Schmutzer - restoration.jpg </gallery>" ''Albert Einstein 1921. We experience our thoughts and feelings as something separate from the rest. A kind of optical delusion of consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us."'' ''Albert Einstein, in One Home, One Family, One Future,p.99'' Einstein came very close. In reality, every human being has a backpack from the day they were conceived. In the backpack every experience in their mother’s womb is experienced. At birth and throughout their lives, everything that happens to them in life is registered and creates their personality. That life with all its experiences can never be known to anyone else, consequently, we can never “know” another person. It creates equality of consciousness that we must understand. We can know details about a person, but that is all. That life is sacrosanct. Who we are really goes beyond normal human experience and into the realm of the Absolute. Werner Karl Heisenberg (physicist). The Heisenberg Uncertainty Principle. As a young layman with no knowledge of Heisenberg but interested in principles it seemed to me that the Uncertainty Principle was just a contradiction in terms. In later years I found that Heraclitus describes life as being in a state of flux a replica of the Uncertainty Principle which in fact can be defined as an absolute state. Within the context of knowing who we are and the backpack we carry our life in, we can never know each individual life as that life experience is singularly their own and sacrosanct. It now seems that the Heisenberg Uncertainty Principle can fall into the category of being an Absolute. Evolution proceeds in advance of our need to evolve. In our pure active state, we are.There is no static end (an abomination) - only beginning. As we cannot know what tomorrow will bring, living with expectations is rather futile. Nature has its own agenda. Zen Koan recorded 1228: 'An instant realization sees endless time. Endless time is as one moment. When one comprehends the endless moment He, or she, realizes the person who is seeing it.' We do not own Space. We do not own Time We do not own Energy We do not own Matter "What we call the beginning is often the end And to make an end is to make a beginning. The end is where we start from We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know the place for the first time" T.S Eliot = Everything is in scale. = The present moment is the point in which Eternity has placed us – we all live in that moment, and whether we like it or not, we exist in it, experience it, have knowledge of it, and we all share it, measure by measure. There are no dichotomies. Illusion is a measure of Reality, as Stupidity is a measure of Intelligence If one keeps measuring illusion it is an attempted downward spiral to nothingness.There is no opposite to Reality – that illusion is a measure of Reality. There is no such thing as ‘nothing’ in the elemental construction of Homo sapiens. All the innate ‘something’s’ are the fundamentals of our being human and all our experiences. The correct use, and understanding of who we are, is an extension therefrom. Does it require any interpretation on anyones part to say ‘we are? Any attempted denial of that statement would be perverse use of the language, and delusional. === Try saying ‘we are not” === ‘We are’ is the foundation of all affirmation, and within that spectrum, we can know, and be. ‘We are’ is self-evident Truth. We can neither know, nor experience what isn’t. Eternity is the here and now, that is why it is possible to explain the experience of Eternity. Nothing is ever lost in Time. We are located in expansion. ‘twas a moment’s pause,- All that took place within me came and went As in a moment; yet with Time it dwells, And grateful memory, as a thing divine. Wordsworth Prelude, Book V111 We already know – the ''basic'' condition that must exist for us to re-cognise. It is at that moment of pure affirmation, when all that is, is manifest. Unless there was mutual identity we could not know anything. It is why we are urged to evacuate the Platonic cave. Sadly most prefer the shadows rather than confront who they are. We already know – the basic condition that must exist for us to recognise. It is at that moment of pure affirmation, when all that is, is manifest. == Lost Shadows == The shadows move Lost in confusion Lost in despair Imagination shrouds the real Looking back Looking forward Is this the Centre? James Brines. = Basic Principles. = We are all in the business of living and attempting to understand the principles involved in that human process up to the end of life. The implicit principles necessary for life eventually disappear and all measurable criteria pronounce a body to be devoid of life. Throughout historical agreement we know what that means, and we act accordingly on *common knowledge*. We know that dying is a necessary factor of life. It is a Natural law that if we live - we also die. Natural law is Universal, for us to *know* that a body lives; we also *know* that a body’s life will end. Albeit that reports tell us that today millions of people die of disease, starvation, wars, we of necessity accept that as the ongoing reality because again we are universally connected and know the results of such carnage. Because it is in our realm of common knowledge we have graveyards, crematoriums, undertakers, doctors who pronounce bodies to have died. We understand the consequences of leaving such bodies unburied, the diseases that would prevail. Again, please explore the definition of principle (Universal principle) and try to go to the limitations of the definition without using mythical dichotomies. Principle: A fundamental truth used as a basis of reasoning. All of these questions are based on singularity (the Cartesian dogma)… If Descartes had only introduced inclusion into his musings (they were taken as conservative singularity) he may have realized the difficulty of addressing thought as reality. He then may have quite easily concluded that universally - *We are!!* That pronouncement is inclusive, and conclusive in every way, *we are - and we know!!* Because my knowledge is not a private, personal piece of property concerning principles, but Universal (Archimedes et al), then that innate knowledge has completeness we can share. Whatever identical resonance we may be able to share (concerning completeness), that can only be accomplished by understanding the principles involved and their constant relationship to each of us. Clearly the plethora of present and past discoveries establishes the existence of that which is fundamentally true, and the foundation for law. The principles are established, and always have been, we are in the business of making them transparent and complete. However tenuous the link we are all connected through communication, the air we breathe, the ground we walk on, the universe we live in, the common principles we live by. We all must have sustenance to survive, or we will not live. (See above) <nowiki>*</nowiki>There is nothing else to experience.* When we actively explore the reality of anything, all principles involved in that exploration are complementary, and honest, and we understand the wasteful divisive mythology that people attempt to attach to them. We cannot */partially know/* the truth, it must be complete. Dichotomies attempt to deny the existence of truth, and are misleading. How do we more reasonably completely know anything? === Naturaly. === How do we completely know? The complexity of language systems with contemporary usage requires new and creative structures to provide clear information. Internal and external reasoning capacities can only develop in concert with Man’s recognition of the principles that essentially form our lives. The inevitability of human consciousness rising beyond its historical beginnings posits a future outside our normal perceptions, and  a factual reality that points to the existence of new perceptions that are infinite.   It is natural to know when we are no longer trapped in any mythical ideologies that gives credence to dichotomies that stifle the recognition of simple principles. There was no cause for knowledge to be established - it has always existed.* Ask yourself - ‘how do you know to ask any questions at all?’ There is no hidden dimension or mystical world. The only philosophical reality is ‘that which is’. To access that we need a new structure to explore ‘knowledge’, a new transformational language. Real knowledge is not amassed information, nor is it the establishment of dogmas, isms, or mythical belief systems. Basic principles are the source and foundation of all /*knowledge*/e. Until that is recognized, extension from mythical sources only leads to a denial of one’s own senses. Trying to conceive of a contemporary world without principles is to posit a world without reason, or intelligence. The principle of pure knowledge could be said to exist in another dimension given the present state of human understanding. To progress that mistaken belief there would be strong support in the need for a comparative reality. Curiously it seems that philosophy (the seeking of knowledge) constantly discounts any knowledge that does not come within the sphere of established philosophy, and the comparative reality dictum. That consequence profoundly distils the purity of any experience and alienates the observer in their confrontation of that which is real. The measure of that ‘comparative reality’ knowledge bounded by dichotomies is so restrictive that it lies in a mythical dimension where denial of its very purpose is the order of the day. Evidently it will not allow doubt to undermine its own denials. A basic principle of Nature is /*knowledge*/ and it constantly communicates innately in every living structure. How to grow, develop, and disseminate. Knowledge is reciprocal truth that depends on our relationship and the recognition of principles operating. Knowledge (unlike information) is not stored in an individual box; it is ubiquitously manifest in everything we do. Knowledge is the experience of a positive reality, and its true construction is a dependable source of secure information (not to be misused). We daily have the opportunity of witnessing ‘knowledge’ in action as expressed by the ‘the principles of knowledge’, namely the principles themselves. The questions lie in a continuing mythical belief in a mystical unknown (the Cartesian stance) which because of its non-existence can never be known. It is a belief that is detrimental (because it attempts to deny all existing factual knowledge) to dealing with Nature and Reality and the fundamental necessity of our relationship with them. Making that relationship transparent is our basic obligation and the ongoing evolutionary activity. Although there is an obvious avoidance to address the definition of principle itself, it is a factor that must be paid attention to, to realize that ‘common knowledge’ is the only reality that exists. Amazingly although there is avoidance of principles - truth - reason etc, there is acceptance of the Cartesian dogma,. Paradoxically this determined acceptance of Descartes supports the reality of innate knowledge existing (I think - therefore I am) which establishes for him innate knowledge; however mistaken he is concerning the interpretation of his experience. = Leverage. = Long before I read of Archimedes and his various principle discoveries I was using the principle of leverage in a variety of ways, prying lids off boxes, moving articles with a lever well beyond my physical strength to do so without said lever, and I knew how to do it. Transferring that knowledge to a student or apprentice is relatively easy because innately they also /*know*/ how to do it. Every aspect of human industry uses the principle in a myriad of ways because it is our obligation to constantly progress the principle and confirm the constant utilization of knowledge. We wholeheartedly adopted Archimedes principles (et al) because we recognized their fundamental utility. It is preposterous to question the widespread /*factual knowledge*/ of all principles, more especially so when we cannot escape their ubiquitous daily existence in all our lives - Nature and Reality do not lie. One may abstain from admitting their existence. To do so is simply attempting the impossible, and is devoid of all reason. Real knowledge has been put into the realm of the mystical unknowable even to the point that knowledge practitioners go to the outer extremes and deny the gift of their natural senses. With their adherence to what they consider is knowledge they become captive automatons to any prescription for life that is expressed in that ’knowledge medium’, which then becomes the authority. When ’knowledge’ is addressed as having a collective source in Universal principles then we have the potential to experience its complete reality (microcosm - macrocosm) without any imaginary, or divisive comparative content. There is then a re-orientation process toward our true being and recognition of our own reality in relation to the natural processes we share. Real knowledge is elementary and Natural. We know, because that which is knowable is constantly expressed by the principles involved. We all share those principles and can correctly infer the most simple and obvious truths. All social life functions by our adherence to the implicit laws operating within them. Seeking experimental contradiction to a fact of life offers us nothing but proof which is the establishing of ‘complete knowledge’ however ‘more reasonably’ one wishes to extend the exercise. The construction of new philosophies must seek a mandate to fully explore the relationship between experience and innate knowledge as the foundation for pure knowledge to emerge. To repeat, knowledge or truth do not reside in any individual domain, nor are they the private possession of any human being. We know, because ’knowledge’ is an innate natural possession that we constantly share - otherwise civilization could not exist. == Knowledge: Evident facts about mutual standards that provide us with security. == We do not become human beings because we can ‘think’; we become more human because we learn to conform to the implicit principles in Nature and Reality. Denying them - denies our being. It is natural to know. There was no cause for knowledge to be established - it has always existed. Knowledge is an evolutionary process. Human beings developed from primitive innate instinctual knowledge to contemporary cultures. Some know more than others through experience, and make that knowledge transparent. Insistence on how we can ’completely’ know is an ephemeral philosophical question that attempts to deny that we can have ’knowledge’ at all, as you understand it. Knowing that we ’know’ the inherent completeness of everything through the existence of principles, is the natural catalyst to make ’that which is’ transparent. Heraclitus: "No man steps in the same river twice " He believed in the "Unity of Opposites (Absolutes). He cried for the needless unconsciousness of mankind.. “Exploration of a mythical dichotomy below for the purpose of establishing principles. Principles that are not a 'mind' construct, but the very essence of our being. Independence, is the curious and dangerous malady where humanity has lionized negative mythology in denial of its own reality. The human fundamental reality has at its base the simple natural law that we are dependent beings. From conception, the human embryo is entirely dependent on the health and well-being of its mother to provide it adequate healthy sustenance to enable its entry into the world. That form of innate dependability the human species carries with it throughout its spectrum of life. Every aspect of human activity is premised on the availability of air, food, and water without which the organism cannot survive (this would be an incontrovertible 'more reasonable' truism or an Absolute). In a perverse way, that which is our natural state has become the target for what appears to be open defiance of the laws that govern our behavior. When a basic premise is either used mistakenly, or deliberately, its consequences can be socially far reaching, for any deviation however far it is extended is a distortion of the truth, and a denial of who we are. The erroneous conclusion through exercising responsibility that we can confer independence to our actions has gained a distorting and ubiquitous influence which paradoxically undermines the very responsibility practiced. Within the context of being dependent we can correctly be responsible for our own actions but with the surety of knowledge that that responsibility is contingent on the measure of life giving forces available that we are dependent on. The mythical dichotomy 'independence' connotes with the myth of separateness, division, alienation, and the force of these particular myths is expressed in wars, genocide, criminality. Alarmingly the mythological term has become a residual in our lexicon and is used more widely with acclaim than its true counterpart. To uphold delusional 'independence' as a value to strive for erodes our human heritage by diverting useless energy toward a dubious goal, and consequences that leave us questioning our means of arrival. Sadly it is a loss of being with the paradoxical view that the energy expended will deliver up a personal reality. The cult of independent individuality with its mythical ideology based on personal intimacy is now taken for granted, which then passes into the acceptance of the spurious dichotomy as a tangible reality. This in turn disposes the adherents to discredit the very essence of their being, and in the process dehumanizes many cultures. The presumption of independent individuality leading hopefully to a superior future is in fact counter-productive to the purpose, and leads eventually to corrupt power, and subsequently the invention of immoral policies that continue the negative spiral, which in the end has no ethical base to extend from. The alienation from our substantial being creates inevitable tension anxiety, and the need to somehow relieve that anxiety with any artificial means available. = Responsibility. = Being responsible for any social activity would best be enacted with regard to the effect it will have (directly or indirectly) on the lives of those who are dependent on a beneficial outcome. To recognize with some significance the basic structure of our being in turn significantly increases the measure of our understanding of human relationships. Being dependent is not a mental construct choice - it is a state of being, and there can be no being-ness without at least one other being, there is then the possibility that the principle of true egalitarianism becomes the manifest reality. Being-ness can only be identified and expressed in relationship. This is why the cult of 'independence' is eventually so socially destructive, as it creates that alienation which attempts to deny each real human need, and leads to a depravation of honesty. Human relationships between children and adults where independence is the accepted norm is severely undermined when the educational process predominantly teaches an unnatural form of living (either intentionally or unintentionally). The educational process is then reduced to the adoption of a fiction, which in turn puts at risk any educational program. The effect of interpreting mythical dichotomies as described here is symptomatically ascribed to the existence of all other principles that govern life. The construction of any ethological debate should not be premised on comparative perspectives, based on human thought, but rather on the issues that we can recognize as being universally compatible, therein lies the common denominator point of extension. The focus of attention on comparative perspectives denies justice to 'what is'. To contemplate the 'right or wrong' of any circumstance is a deviation from the truth. The correctness of any debate (however minute the finding), is the justifiable extension, and the only true trajectory we are morally obliged to travel. Truth is not defined, nor experienced by comparison, but by 'what we are'. An orange is to an orange, what an orange is to an orange. To define correctly there should be careful and disciplined action toward establishing 'true factors' that we constantly use to promote reasonable standards. = Time. = In that moment of time, we have the potential to merge on an equal basis with the reality that exists, and to know what true interaction is. That is the point of 'direct experience'. It is then that we know the truth about ourselves and the beauty of this Universe which also reveals to us the folly of our present conditioning. In that experience, it becomes very clear that all so called cerebral activity has nothing to do with reality. The fundamental repository of our knowledge and relationship with life is our being-ness, which is not located inside a mental box to be analyzed, accepted, or discarded at will, but the very privileged natural gift of being. What some scholars deal with is the appearance of life prescribed by the illusion of comparative perspective which functions on the basis of dichotomous ideology. It is in effect a denial of our humanity to conclude that all things that exist [from our perspective] exist only in the mind, that is, they are purely notional. It compounds the denial of 'what is', and an extraordinary refusal to observe transparent life. It should be noted that there is ample contemporary exposure to the Cartesian doctrine, and in this regard, I would refer you to the works of Professor Gilbert Ryle, notably his publication The Concept of Mind. The heuristic principle applies throughout when establishing our connectedness with reality. It is only through our contact with reality that we can discover, and equate with the mutual structure of the principles that govern all existence. Have already noted that it is also a peculiar form of arrogance that presumes that life is only a notional existence beyond the boundaries of the 'mind in a box' assertion. It would be foolish of anyone to assert that ice cream has a cold smooth satisfying texture and taste on a warm summers day unless they had actually experienced it, preferably on more than one occasion. For anyone who has never enjoyed that experience, it would be foolish of them to discount the very numerous accounts of such an experience that is available just because they had not been party to that event. From a logical point of view, given the avowed experiences of ice cream eaters, we could reasonably ascertain the validity of each experience by documenting their separate opinions. Each participant would have 'direct experience' in the consumption of ice cream, which at that point in time has the potential for that participant to experience the full measure of that factual reality. We have the natural capacity to experience coldness, smoothness, which equates with the reality that exists, and the potential for those realities to unify. It is not a question of how to get outside of our minds (mind in a box position); we are constantly outside our so-called minds performing acts of transparency throughout our entire existence. The belief that our constant engagement with reality can never be based on a rational acceptance of 'what is', is at the least, very sad. The Platonic Cave shadows are a metaphor for the (mind in a box) syndrome. The need to reach simple, and obvious conclusions and accept them for the reality they are provides the opportunity to engage the complete reality of the moment. It is indeed going too far beyond the reality of the moment searching for philosophical profoundness which does not exist, that fails to establish the constant principles that always operate. Pure principles are not amorphous shadowy ideals; they are represented in everything that exists. The only way we can equate our inner knowledge of reality is through direct experience of its truth. Within that context then, life cannot take on a notional existence but is an existence that is very real, and that we continually share through our innate knowledge. That our so-called minds are defined by comparison - incompleteness - dualism would have extreme difficulty in pursuing the proposition that we are defined by our direct relationship with reality which is expressed in our innate ability to directly interact with 'what is'. The reality of interconnection, and interaction, are not idealistic concepts of a notional nature, but actual and consistent transparent realities. We do not live in a shadowy world that is hidden from our direct experience, but we are constantly engaged in the process of life, and we do not have the right, nor the choice, to deny it. The man whose book is filled with quotations has been said to creep along the shore of authors as if he were afraid to trust himself to the free compass of reasoning. I would rather defend such authors by a different allusion and ask whether honey is the worse for being gathered from many flowers. Anonymous, quoted in Tryon Edwards (1853) The World’s Laconics: Or, The Best Thoughts of the Best Authors. p. 232 == Create your own semantic template. == That will consist of an alphabetical list of Absolutes that are all interdependent and interconnected. Their unifying construction creates a ‘new’ consciousness meaning. That ‘meaning’ is yours specifically. The greatest knowledge you can ever have is your own! That meaning also creates its own moral construction that cannot be misused . The semantic template is available to everyone, and its dissemination is our responsibility. “Consider that the language structure, concepts, and definitions now in use no longer always deliver, accurate, reasonable, and responsible information. Indeed at times, they can be quite ambiguous. The statement ‘mutual agreement’, and its physical manifestation in whatever form, is its own dialectic, and will carry within it all other principles necessary for the activity to proceed. Given the Socratic assertion that if something is true then it cannot lead to false consequences no matter how circular any argument may be. Then extrapolating the statement into extended definitions must only lead to a better understanding of the inherent truths available. This can promote recognition of the underlying essence of all things, which can become more real than our conventional understanding of Reality. There is a contemporary need to find new definitions, new paradigms to explore the concepts that govern our existence. Where a circular argument is based on an untruth, then it cannot lead to a truth. The reverse of that is that when the truth is established, it cannot be denied. Establishing ‘mutual agreement’ as a center from which we can reach out for extended knowledge in its ever-evolving radius, is not a limitation, or a stop, it is only a beginning! When any concept is truly established the superficial exemplification ceases to dominate, and we can truly experience the apparent essence of ‘what is’. Paradoxically to resource innate knowledge, we must recognize our profound ignorance of Nature, and Reality. Completeness does not lie in individuality. This is an extreme form of monastic expectation. There can be no individuality (or completeness) unless there is at least one other individual. This is the true foundation of completeness. Whenever we are privileged to experience that instantaneous essence of one other, then we know in that moment that we experience ourselves. It is complete complementation with the knowledge paradoxically that it is an endless process. There are many paradoxes we live within that strain our conventional views of what is ‘more reasonable’. Any true relationship experience is not based on a causality premise, but on an experience that is necessarily complementation. Individuality in terms of completeness is a fundamental circular argument back to one, which in its form of denial excludes any form of reasonable argument to the contrary. It is a non sequitur, which denies the pressure of facts that are in abundance, despite the evidence of their reality. To observe ‘mutual agreement’ is looking at things as they are. True observation of ‘mutual agreement’ in action is observing essence transparency – it is knowing ‘who we are’. That form of recognition is essence duplication. The proposition that we can observe the Truth may well be the highest attainment of Realities properties, for Truth is knowledge. Consider the hypothesis of a human entity (an individual) being born in a black space with no other form of life in that environment. How could there be Agreement? How could there be Intelligence? How could there be Understanding? How could there be Recognition? How could there be Love? How could there be Law? How could there be Reason? All of the above principles are the transparent manifestation of Nature and Realities properties that are constantly evolving. They are ethical imperatives, and we have developed the positive properties of language to establish them for our use. We can only be defined through relationship principles for they offer us the best hope to recognize the factors that lead to complementation. There is a fundamental need to grasp simple common-sense essentials. The Here and Now is not a temporary transitional time phase that we move in and out of. It is a constant certainty that is essential to recognize, so that our focus of attention has a foundation. Centrism can imply a fixation, which also implies vulnerability, which can be perfectly true if it does not lead to extension. To understand who we are, it is essential that we recognize and become aware of the very principles that we operate from. They encapsulate all the measure of any human societies ethics, morals, and laws, which is a continuous evolutionary educational process within which the realization of its total essence is always available. To use the doctrine that reason is a reliable tool to discover Truth – therefore ‘mutual agreement’ in the context ‘correct information’ translates to the Truth to reason! Evolution is a constant dynamic process. The human phenomena of ‘who we are’ is only understood in our union with each other, and ‘what is’. The paradox again is that there never is any separation. Separation is a mythical non-existent. The principles that are our necessities have continuous expansion properties that as humans we are privileged to assist their propagation. The human constellation in its evolutionary march must use these fundamental principles to ensure continuity. To maintain coherence and consistency our source is centered in the principles and factors that we have interpreted from our association with Nature, and Reality. Whatever we write that is of any consequence, or at any other time, is written with the hope that stronger interpreters than us overtake what we present. To ‘see’ Reality as we have seen, and be intoxicated by it, as we have been, will ensure its progression.” == Discovery == The consciousness whole is the sum of all its parts and experiences. As we are all on an evolutionary path, our life and knowledge hopefully develop in the right direction. The exploration took us through a plethora of data and opinions about reality from authorities on science, religion, philosophy and metaphysics. Nowhere could we find a definitive conclusive argument, or agreement, that met our needs. For us, the question came down to “Is there anything at all that provides some form of construction, and certainty?” Something that has its own inherent ethical standards. The alternative proposition to that is a nihilistic “nothingness”. A pathological proposition that makes no sense. Heraclitus’s “unity of opposites” seemed the most promising. Our understanding of that now made dichotomies a semantic illusion. If achieved in a mindful way it is the act of uniting them, providing a conscious correct experience of ‘what is’. We live our lives with secondary knowledge that everything that is – is always interconnected and interdependent. Yet our illusory experience belies that form of knowledge. It is here that we understood Heraclitus and his “unity of opposites”. Mentally uniting opposites replaces the existing illusion of their existence – there are no dichotomies! Once the illusion is gone a new solution manifests that is peculiar to the mindfulness operator, and belongs to a higher form of consciousness. Heraclitus was known as “The Weeping Philosopher”. He wept for the needless unconsciousness of mankind when the ‘unity of opposites” was always available. He was also known as Heraclitus The Obscure. A title we suspect that came about because the successful conclusion to uniting opposites and replacing the illusion, opened a door to a new dimension. Semantic description at this time may not have been available. This brings us to the ancient Yin and Yang symbol of the ‘unity of opposites’. As separate entities in Chinese philosophy, they are complementary, and in fusion they represent the whole. So as dichotomies they don’t exist. The whole is the elemental answer to any fusion of opposites, whatever that may be. Symbolize a line as being infinite in the sense that any line can be categorised as being infinite. Apply an infinite number of points in any line. Intersect any line through any point by another line then we have a specific identifiable point at the intersection, which at that point in time has an infinite quality, yet constant and complete. Any such point has Matter, Energy, Space, and Time, the epitome of the microcosm. We may locate a Reality point that establishes the Truth. Conventional mechanical ‘thought processes’ deal with dichotomies that are based on a comparative perspective ideology, and consequently, skew any real experience of that which is real. We must use correct ‘measuring sticks’ to secure proper standards, but from the point of view that there is a belief in dichotomies, it will always be a compelling argument that aspects of reality can be contradictory. The element of denial within human historical memory accumulates to establishing dichotomies as being real. We are defined not by how ‘different’ we are, but by our commonality of existence. When we locate that Reality point we will then know that the definition in itself has a whole, and complete explanation of ‘reason” in all possible senses. All the reality we can deal with is here, and now. There is no possibility that ‘infinite regress’ (an imagined reality) is any part of our immediate experience. Infinite regress through thought processes, deals with questionable imponderables. It is a descending spiral, which further removes one from reality, which only produces illusion, and correct meanings are always deferred. It is making a holy virtue out of complexity. The epitome of completeness is the active realization of the operational principle. Conclusion: A brick – a house. Each complete in themselves. A house is not composed of one single brick, but each brick in its composition is complete, and whole in itself in that it has matter, energy, space, and time. In that context, it is a microscopic whole which has implicit within it the macrocosmic whole, a house. We cannot define that which isn’t. We constantly use negative dichotomous terms in language, which are in essence factually indefinable, and therefore non-existent, but they are used as though we can support a view as to their existence. At this time we constantly use mythical concepts as though they had real substance. That erroneous belief in turn diminishes that which is real and compounds the problem of recognition of Reality. The flat earth society no doubt had to be persuaded of the mythical nature of their beliefs. This dictates that we must research ‘that which is” to achieve an understanding that supports that reality. Separation is the mythical measure we use in an attempt to justify the real identity of either ‘relationship’ or ‘completion’, but it has no substance in fact.That we are connected, that we are related, that we are communicating, that we agree that ‘mutual agreement’ exists, all of these factors fall into the category of ‘that which is. There is nothing that is real that is not immediately available to us, there is ‘mutual agreement’. Attempting to view true relationships as having a necessary separation link, or dichotomy is a clear misunderstanding of the nature and completeness of all that we are related to. == Connectedness. == A relationship is defined as we are by the measure of contact (especially homo sapiens) that is apparent. It would be true to say that I have a measure of relationship with everyone who reads this material. No matter how tenuous the link we have a measure of relationship with all life – we are related! Depending on the strength of that relationship defines ‘who we are’. ‘Who we are’ is not defined by any spurious separation from life, quite simply because we cannot be separated from it, we are engaged in it at every moment in time. Any attempt to establish ‘separation’ as a reality is an attempt to deny ‘who we are’, and another exercise in futility! Again artificial interioralisation of concepts or principles leads only to a denial of their external reality. We are all connected by the very simple fact that we all exist on this planet. It is a very simple axiom that all life on this planet is supported by the conditional properties this planet provides. It is also a very simple, and more reasonable axiom to conclude that no matter how tenuous the link that all life in this regard has very concrete and definitive forms of relationship. We all must breathe, we all must eat, and we all must drink, and if you need any further certainty of ‘completion’ relationships, we certainly, all must die! [[File:Wikimedia|thumb|center|]Arcimedes] To set in qualifications from the premise that there is a ‘separate mind’ (a kind of Platonic cave) to get outside of. This premise precludes either in part or in whole the evidence and experience of Nature, and Reality, within which our beingness is located. It would all be beyond our grasp if indeed our conventional concepts of consciousness was adhered to, which in effect attempts to deny us that direct ‘relationship’ to ‘what is’, and the completeness of that experience. Knowing or being, despite solipsistic theories to the contrary, does not exist in penetrating one other mind, but in the democratic recognition that we know and have our being in relationship, and the mutual, and natural convergence of everything there is. Homo sapiens (race, color, or creed aside) necessarily conduct themselves in ways that extend recognition, and understanding at every level, without the constant need of ‘completion’ recognition that is inherent in all our interrelated actions. The notion of completion may be beyond what you call your ‘conscious grasp’, and therein I suspect lays the difficulty in recognition. The flat earth society eventually moved on to a realization that their visionary scope was shrunken, and severely limited. They were deprived of a planetary (never mind a universal) relationship that one can only imagine severely curtailed the very expansion of consciousness necessary for humanity to progress. We have evolved some little way because of our understanding of the natural relationship. The centered in the mind condition - which connotes with the separation ideology - screams to be released from that mythology, and engage itself directly in real relationship with everything that is. = Relationship. = Separation is the mythical tool we attempt to use to maintain a false continuity of an imaginary individuality that does not exist. The taking for granted conditional mythologies (the flat earth society) engage the victims in what can best be described as serious problems in recognizing the very limitations that restrict their development. We must learn to view wholes, which equates with viewing ‘completeness’. The whole is greater than the sum of its parts, but the parts are not necessarily separate conceptual parts. We can ‘see’ the whole when we are able to identify the factors that constitute their existence as a complete reality. That which is complete in Law = Agreements that produce secure and dependent outcomes. We know in essence the concept of ‘completeness’, and we demonstrate the evidence in myriad ways. Each act is a microcosm of the whole – view from the other end of the telescope! In the traffic analogy the driver, and all other drivers, conform to the law by driving off when the light turns green. There is an implicit agreement about the value of traffic laws, and traffic lights that control the flow of traffic. At that moment there is a complete relationship understanding of those values. The ‘complete’ or ‘wholesome’ activity of motorists waiting at traffic lights for the green signal to go, and they then move off, validates all the factors implicit within the properties of ‘mutual agreement’. Throw a ball from one side of the room to the other. The ‘whole’ or complete traverse of the ball is the instant it leaves your hand until it arrives at the other side of the room. You have already given credence to the concept of ‘mutual agreement’ as a reality. When there is a correct definition made in terms of ‘necessary factors’, then it has implicit within it the concepts of ‘completeness’ or ‘wholeness’ within the measure in which it is used. Whatever we communicate for the benefit of future generations should not be based on mythical assumptions, but should be based on necessary factors. It is ordinary life that portrays all the dignity, honor, and the complementary wonder of the human species. We are here – we are present – we are communicating. We have an obligation (which we necessarily fulfill) to make transparent the basic principles that govern our existence. That, which is factual, provides us with a correct motive for behavior, and we do a disservice to Reality when we attempt to deny it. We exist and live in a world where acts of ‘completeness’ expressed in one form as ‘mutual agreement’, are continually enacted. It is the form of expression, and continuance of processes that we constantly use to arrive at reasonable solutions, and we employ factors that are necessary to provide us with a clear, and unambiguous understanding. They motivate reasonable behavior toward activity that we can accept as being a logical process. == Natural Experience: == No form of life can exist in and of itself, it is brought into existence through a relationship with its own environment, or its species. The obvious egotistical monistic nature of oneness (if there was such an entity) could not leave any room for the realization of anything that might disturb its comfort zone. There is no real knowledge where any concentration is on the “I am” syndrome. "No man is an island, entire of itself; Every man is a piece of the continent, a part of the main." - John Donne, 'No Man Is An Island, Meditation XVII - Devotion Upon Emergent Occasion. “We are’ is an inclusive affirmation that deals with “what is’. There is no constructive dialogue, no real understanding, without a relationship. Based on personal experience, we are a distinct, and unique species born of Nature and Reality that has combined to provide us with the innate ability to recognize the very properties that created us, and utilize them through an evolutionary process toward ever-increasing transparency. That transparency can only become available through a matching process between innate knowledge, and the reality we share, a reality that is our heritage. We like all other forms of life are the product of Nature, and subject to its laws, and principles. Necessity created a language that evaluated Reality, and provided us with guidelines to emulate its constant properties. The measure, and quality of knowledge is dependent on the realization of ‘what is’. The crux of correct knowledge is to know the base that we function from. The principles that are implicit within Nature, and Reality we have now translated through the evolved language systems with symbols and definitions that we now use to share the experience. When principles are fundamentally true and recognized for what they are, misguided belief systems will evaporate. To ask what is the source of the principles we present is ipso facto to ask what is the source of Nature, and Reality, and we repeat, that is an exercise in futility, but that does not mean that we cannot recognize that which is natural to us, and express it, as best we can through language. The experience of ‘who we are’ is the Ultimate transparency that transcends all doubt, or denial. We can know with an understanding that is pure and indisputable, that is the motivational drive for evolutionary continuance. To understand who we are we must address Nature, and Reality, and ask ‘what is’ Here, and Now, with an understanding of the dichotomies that exist in language. The ‘Eureka’ moments, epiphanies, enlightenments, etc, are all evolutionary evidence of who we are, and when we can translate them into principles, and concepts, then the assertions of an Archimedes (and many others) are recognized, and properly used. Through Nature, and Reality we can establish what Truth is! Is it true to say that most people conform to the rules of the road? It would be more reasonable to assert that the answer is yes! Consequently, we can say that we have ‘Mutual agreement’, and ‘Co-operative Understanding’ as observable realities. There is no conceptual source through Time, or history where there is an end. There is only ‘beginning’. Here and Now is always the ‘beginning’, and a more reasonable platform to explore than any exploration into the past concerning our true identity. Contemporary terms like Absolute, Complete, End, we use to match our conditional understanding of ‘what is’. When we use contemporary conditional language to address concepts like Truth, Knowledge, Understanding, they are limited by the measure of our progression, but we use them all as stepping-stones. Language is a constantly evolving process. When we agree that there is ‘some certainty’ and ‘limited knowledge’ you have agreed to the concepts of certainty and knowledge as factors that are part of our natural reality. All of us function within the framework of certainty, and knowledge, to some degree. Given that we agree to their existence, these are the factors that can lead us to the experience of ‘who we are’. They are a part of us that can lead us to recognition of ‘what is’, and make a transparent reality of the very things we do on a daily basis. We do not need absolute and certain knowledge to perform everyday tasks, but those performances are structured contemporarily because of our advanced understanding of the things we do, based on our own innate reality. To honestly perceive the consistency of ‘what is’ (to be interdependent, and mutually connected) in interaction, can and does promote the visible reality of ‘who we are’. That visible evidence translates into the knowledge of our complete presence. We know with certainty that our beingness is of pure essence, and from that experience, we are obliged to formulate, as best we can, the structures that are responsible for making that transparent (witness the explosion of human progression, without the necessity in Time of experiencing fully ‘who we are’). To take a more reasonable stand please observe the multiplicity of human action where interdependence, and connection naturally proliferate. To realize that we are ‘interdependent, and mutually connected’ is the realization of a consistent fundamental truth – ‘what is’. Through identifying ‘what is’ as an internal reality we can make transparent the factors that are our natural construct. It is only through sharing this reality that we recognize it!! These factors are not based on any ideology, belief systems, or opinions; they are composed of the Reality that is available to us all. We are unique in that we have the opportunity to be able to use their value in the manner that is implicit within their structure. That use is evidence of our understanding of Reality. What could have happened without the assertion that traffic lights are a safe way to control crossroads, or the assertion that the rules of the road are beneficial for our safety? Any true experience, epiphany, enlightenment, etc, of ‘who we are’ provides fundamental, and indisputable knowledge of that Reality. Applying the recognized principles provided by Nature, and Reality consistently advances the evolutionary process, hence we have Science, Philosophy, Religion, Education, Art, and Law. When there is a Pure Realization of who ‘we are’ through relationship recognition, it is unquestionably the recognition of the encapsulated, and innate principles we all share, and there is no place for the dissolving of another Real identity through that recognition. Indeed it is a privileged insight into the epitome of purity. Nature and Reality can give us direction and guidance to our human existence, and we repeat, it is an exercise in futility to seek any cause to their beginnings. We have proffered the concept of two as a basis from which human reality can be ‘experienced’. To recognize through, and equate with the true substance, and essence of one other is to automatically experience the totality of ‘who we are’ in full measure. This does not mean that the terms ‘totality’ and ‘full measure’ convey an ‘end’ to ‘what is’. We have consistently offered recognizable facts (not assertions) that are part of our natural human activity, and give correct direction and meaning to our basic essence. We do function within the structure of ‘mutual agreement’, and we do communicate and ‘make known’ – basic obligations. These are evident simple examples of innate knowledge, and our understanding of ‘what is’ made transparent. To repeat we could not recognize anything without innate knowledge. All knowledge is a continual matching process 1 + 1 = 2. toward the realization of ‘who we are’ the development of language structures that correctly establish basic reality as it is, provides continuous knowledge that makes transparent the very nature of that reality. It is vitally important to recognize that we have active communal agreements concerning the existence of basic principles and concepts that form the very foundation of our lives. Constant change and movement in Space-Time - Energy - Matter are applicable absolutes to be recognised, which equates to evolution. There never will be a static property involved in the evolution advance. Evolution encompasses its own absolute properties to provide cognitive connection confirmation. The evolution of conceptual language exists to provide natural equality and to promote cognition between language states. No matter the languages spoken the concepts of air - food- water is the same and can provide a gateway to explore the future dynamics of human relationships. === Knowing. === Knowing what all the truth is is not some miraculous state of perception. Nor is it a high academic achievement of amassed information. It is simply an objective common-sense view of ‘what is” and in reality what must be. It is what must be for life to function within the principles that exist that are its natural foundation. We are always of necessity the living expression of a reality that must be experienced in the whole. Our recognition of the same principles operating universally is also our recognition of who we are. That proliferate ‘sameness’ is an evident easily recognizable identity. When a child is afraid of an imaginary monster in the dark, we generally do not accuse them of being absurd, or that they are lying. Appeasement comes with an explanation of reality at that level which is truthful. An explanation, which the child can grow up with, and find comfort. It is simply introducing a child to a level of truth that is more real to them. In every instance, the only reality that ever exists is truth. However, distorted it may be expressed. One of the major distortions as the result of thought processes is to consider that we can manufacture something other than the reality that exists. The ‘fact’ that you recognize contradictory or absurd statements is that behind them there is a measure of truth. The habitual liar lives in a world in which he or she imagines that truth is something they can manufacture. Where human ‘thought’ constructs its reality in terms of dichotomies it can never deal with the truth because it continually makes those comparative perception judgments. Those judgments are always in question because again they cannot deal with reality as it is. No matter how absurd or contradictory any statement is, that is the measure of truth expressed. Ergo whatever it is that is expressed, or made manifest, is the truth to some degree. Ergo everything that is, is the truth. It is our responsibility to recognize it for what it is. It may be appropriate to review previous observations on dichotomies and gradient scale. Consistently we have contended that there are no dichotomies, which then properly puts each principle into the category of an absolute. To identify ‘truth’ as an absolute in that category then everything that is must have a measure of truth. It is a very simple and sensible approach to establish ‘what is’. It is the means of identifying a reality that must have truth as a base – whatever it is, and however nonsensical it may appear. All principles have an elementary gradient scale that we must use to identify knowledge that is honest. That gradation scale knows no dichotomies. Dichotomies are always the imaginary properties of pseudo subjective reasoning. Necessary factors establish that gradient scale where only objective realities exist to furnish a healthy subjective reason with truth, and so we learn to apply the conjunction to address reality for what it is. Truth comes in an abundant variety of ways in its commonality – and therein lays its overall ‘complete’ power, despite any denial to the contrary. Embedded knowledge as we see it is neither experience nor knowledge without principle content. A person may be well educated in all aspects of the geography of a beautiful South Sea island, but have no practical experience at all of its beauty. Being clever about a subject does not necessarily equate to an understanding of the subject. Nor should it lend itself to posing as adjudicators on a proposition preset we imagine by the same adjudicators, or essentially the same school of embedded thought processes. Long before human evolution, the principle of leverage has always existed in all Nature (as have all other principles, wherever there is space – time – energy – matter). Our adaptation to the existence of principles has added to the sum total of ‘knowledge’ as we know it, including the concept of knowledge itself. That form of ‘knowledge’, and our ‘knowing’ is natural and not any personal or esoteric acquisition. Just as a fish knows what its natural habitat is, or a bird to fly in the air, the human species uses all available principles it recognizes to add to its knowledge (already said). Any valid theory of knowledge must have as its base constructive definitive principles to support it, and it is evident that our accumulated common knowledge equates to our common experience. No matter how erudite or convoluted any argument may be, if in the end it is reduced to inane observations that have no factual basis in principle, then it is time to abandon them. Do try to consider the sort of ‘mind’ processes that offers us up a world that knows nothing but separation. How can we possibly evaluate what ‘wholesomeness’ really is? How can we possibly evaluate ‘who we are’? How can there be any theory of knowledge without addressing Nature or our innate and biological relationship with it? Any attempt to debate ‘who we are’ and the completeness of that concept must have some sense of reality on the real meaning of ‘completeness’, and some understanding of the principles that are the nucleus of human society. To wrap any argument around a non-existent concept that can never be realized is apropos to attaching oneself to a system of belief in things that do not exist. One can make ponderous and convoluted statements about those beliefs but in reality, they are morally and ethically misguided. The ability to correlate correct definitions to the reality of life offers up that direct link to the truths that are common to us all. It corrects the presumptive notion that there can be ‘different’ perspectives on the same reality. There can be ‘differences’ but there can not be ‘different’ measuring sticks for the same reality. No matter the multiplicity of perspectives, they can never alter the core principle of ‘what is’. Historically evolutionary progress can best be measured by the adoption of recognized principles. Reality at whatever level we find it can only be understood by addressing ‘what is’. Nothing can be understood by attempting to relegate it to a non-entity through questionable theories of ‘knowledge’, which in essence negate the very content of knowledge itself. The perpetuation of any theory of knowledge, which cannot recognize the principles that are its foundation, can only be a shadow of its own reality. Construct the ‘necessary factors’ around the skeleton and a body will take form. If any particular theory of knowledge cannot identify simple truths, how can we possibly question how anyone ‘knows’? A dichotomy is the human attempt to deny the existence of a whole reality of a principle. We have the principle of leverage and its necessary gradient scale. Mutual agreements of a consistent reality, at a communal level, are a passive form of the Eureka moment, which recognizes fundamental principles that relates to truth. In every social structure, there are varying degrees of recognition, which determine social use. The mosaics of differences, which make up the rich pattern of life, are a testament to human creativity. Principles offer up a form of predictability in which our brain forms knowledge through the process of interaction. The experience accumulated through each moment, forms exponentially in use, or becomes transparent immediately in a Eureka moment, in which we know. Real knowledge is through the constant interaction with natural principles, much more than the transference of divided embedded information. The problem we face is that ‘knowing’ or ‘how we know” is never a personal possession. Any theory of knowledge no matter how in that respect, is true interaction. Peeling a potato and ‘knowing’ it, is rewarding enough! All principles are the repository of pure erudite knowledge. We recognize Universal principles in play at all times in Nature and its by-product – human societies. The correlation between determined interdependence to objective reality requires our intellect to ‘honestly observe ‘what is’, and assimilate that subjectively. Then the equality of the external and the internal becomes a reality and we ‘know’. Knowledge is the process of natural action, reaction, and interaction. It is nonsensical to ask how do we ‘know’.Every moment in time is complete because it must contain all the principles that form its nucleus. It can only be like that to facilitate the immediate experience of Eternity, or the wholesomeness of any of its principles. On the gradient scale of experience, we all exist somewhere on that scale. It is called life. === Relativism: === Relativism can be consistent with interconnectedness and a gradient scale of knowledge. What it cannot do is confuse the relationship that correct gradient scales of principles have in reality. Hot and cold would be on a temperature scale – no dichotomy! Leverage could only be measured by its own scale (say a child’s sea-saw to a high-rise industrial crane) – no dichotomy! The human being is a human being whether it is a child or an adult – no dichotomy! The domestic cat is the same animal species as a wild lion – no dichotomy! The domestic dog is the same animal species as a wolf – no dichotomy! How can we manage to classify these as dichotomies? Gradient scales are the natural human mechanisms used to recognize constant principles. They ensure the human perspective is aligned correctly to identify ‘what is’. The distorted human perspective is the result of human thought processes unable to establish constants that must exist in each moment of time. The problem with embedded information is that it becomes stultified and it can stifle healthy reaction. The injection of recognizable principles invigorates and brings new life that offers countless avenues for human energy to be released. More importantly, those energies are used to enhance the evolutionary process. We are collectively gifted with the potential to elevate life itself. We can correctly use such information by transforming its content so that its inherent truth is made recognizable. It would be impossible for life to function if it was composed of ‘different’ opposite realities. All theories of knowledge are in essence interconnected and can only contain validity when the principles that are the coalescent mechanisms are recognized. The unification of the truth that must exist in any theory needs to be harvested and used to offer up a body of ‘knowledge’ that has commonality of meaning. The identification of principles, truth, knowledge, and their subsequent establishment can only be achieved through direct interaction with Nature and life. Gifted with life we have an obligation to demonstrate its capacity to use every resource to sustain and nourish its own environment. We all know through the constant natural process of action – re-action – interaction. Depending on the quality of that process, knowledge will take its appropriate place on the gradient scale. That we ‘know’ is natural. It is not some extraordinary esoteric attainment, posited by a body of theories that, by their very nature, look for difficulties where none exists. Universal belief systems based on mythologies can have an entrenched view of good principles being established because of their beliefs. Indeed the perpetuation of the beliefs throughout history offers a dynamic that is counterproductive to the ‘realization’ of principles that are necessarily true. Principles used in this approximate way, paradoxically hold no real meaning, and in fact, impose unhealthy dysfunctionality. When there is a critical change toward establishing correct principles, it is axiomatic that the diffusion of mythologies becomes an automatic process. True interaction lies in the knowledge that correct action is its own reward Any other interpretation is less than tangible. The accuracy in interpreting basic principles, and the alternate knowledge implicit in the interpretation, will always establish the primary principle sought. Archimedes et al. Truth can be found in the oddest places. (Archimedes bath image here please) Archimedes cognition on how to weigh metals in water through displacement. === Truth. === Language is the construct of human action and the word “truth” seems to hold pride of place by the power of its usage and the meanings it evokes. It is preferable if we could turn our attention to the unity of principles (including truth) that are the construct of every language we use. By uniting the principal terms we can elevate the meanings we desire. Reasonable constructs and the correct duality of established principles always lead toward meaning. It is the only form of meaning that leads to its own extension eg. how to weigh metals – how else could it be? All principles have reciprocal value one to the other. No foundation principle can stand alone. They can only exist in a union, one with the other, the source of reciprocity. All absolutes are universal. There is no hierarchy beyond the meanings they evoke in their joint construction. The binary connotations, however, one may express them, provide a constant reality beyond conventional consciousness. That experience is the immediate reward through disciplined application of their use. That discipline takes the form in all human action (such as the bathing scene above) disposed toward the correct functionality of basic principles. The daily connections we make always include the distinct possibility of their recognition, when we make those connections in a mindful state. From any common sense, or ‘more reasonable’ position, it would be more productive to view reality as possessing at every level the same innate values or principles consistent with our ability to measure, or recognize them. To view reality as having ‘different’, or antagonistic properties, is simply a misguided view of ‘what is’. That form of perspective is counter productive when it attempts to establish mythical dichotomies as realities in their own right. When the realities of principles are made transparent, we can then ‘more reasonably’ make use of them to further their basic existence. Here we use reason to exemplify their necessary function, and once established it becomes (if necessary), ‘more reasonable’ to locate them in all things. ‘More reasonable’ seems to have the particular philosophical motivation, not toward simple, sensible, and reasonable evidence, but more likely toward that ‘immaterialism’ ideology, and continually seeking for an elusive protracted answer is hardly ‘more reasonable'. Since we are apparently confined to a human perspective, we must settle with the latter position: the apparent state of representation of the world. The de-materializing of any object through the practice of ‘perceptual illusion’ is an attempt to deny the reality that exists. Where perceptual illusions are concerned, innate direct communion with that which is, suspends the effect of such illusions. All the properties in a chair are recognized as the reality that exists. That is materialism. A chair does have the principles of form, design, structure, colour, substance, etc. However it is analysed – it is a quantifiable reality. === Truth and Reason. === The reason could easily be defined, and validated, as the correct application of common sense. More expressions of common sense can only endorse the completeness of any concept. A true experience of reality does not require endless explanations as to its ‘wholeness’. It just is. Truth is in reality a network of implicit principles in which it is the predominant energy in each of them. They are identifiable by their interdependent nature (see network below) not the least of which is common sense. Dictionary given definitions of ‘truth’ place it in a very common sense acceptable category. One of which is ‘accuracy of representation’. Note how the two definitions in this paragraph coalesce. The human drive toward recognizing and understanding the place of principles (constants) correlates to the energy we expend on questioning ‘who we are". The constant principles of action, reaction, interaction, are the automated natural impulse toward ‘establishing’ a human reality, and human identity. The process of evolving within that process has an egalitarian dynamic that powers it. In essence, it is a natural gift that we must accept. Each life and its identity contains all its personal experiences which can never be known to anyone else. In a sense, we can never “know” another person. Their life is sacrosanct. We can know a lot about them, and there it ends. Truth is at the top of the gradient scale that measures the veracity of all things that are complete and related and paradoxically all reality is the truth. It gets back squarely to ‘who we are” and where we exist on that scale. To view gradient scales as having no truth to their structures is denying truth itself. For the entire interconnected, interdependent network of principles, each of them has a gradient scale whereby each measure expresses truth in its own manner. All forms of leverage, from the minuscule to the lever that will move the world, are in of themselves, true and exact at that point. It is the only way we can recognize their existence, and use that complete truth at that time, to move up the scale. Time is the relative measuring stick that determines the amount of knowledge we can absorb. Consider the advanced extensions to the Archimedes principle of leverage throughout time. Network scale example. Truth Knowledge Common sense Responsibility Reason Understanding Intelligence All interdependent, and interconnected with all other principles and absolutes. No ‘thought processes’ or ‘mind’ constructs can create reality. All we can ever do to gain knowledge is ‘act’ react’ and ‘interact’ within the confines of our immediate reality. The quality of that action is determined by the nature of available information. When there is freedom from embedded thought processes, there is a natural human ability to relate to the existence of truth as it is expressed in reality, and our brain records it accordingly. Thus, the principles of civilized societies evolve. Where there are predominant belief systems, the implicit energy will naturally direct itself toward human standards that blend all ethics together. That implicit energy will find its true home in the principles it seeks. The connected strength of those principles offers sanctity of experience that demands no sacrifice. Everything that is, must of necessity, have a true comparative value (not a distorted dichotomy value) for honest recognition to be realized. which is to ‘know’. All things are relative but only within their own true scale. It is the process by which we can identify reality, as it is. Principles cannot operate on any scale practicing negative discrimination. Thus a healthy individual can be at the top of the scale and someone with various health issues can be near the bottom of the scale. But that is how Healthy they are. There can be no relativity when ‘mind’ or ‘thought processes’ believe in mythical dichotomies. When human perception is distorted by such beliefs, they create a false reality and deny access to the true state. Where there is a network of connected basic standards that are universal, then it is possible to use them and be nourished accordingly. The scale of natural human progression provides recognizable evidence that we are constantly developing. Reality is the direct and conclusive evidence of possibilities realized. Therefore, the reality is always the source of all possibilities where truth exists. When the truth is used as the universal measure of ‘what is" there can be no discord as to its accuracy. It can only measure the principles that are implicit in everything there is, its natural milieu! Truth can only deal with ‘something’, it cannot measure an imaginative negative counterpart. Truth is the constant implicit property in every universal scale of principles. Thus, reality becomes transparent. Have writ large on the value of distorted comparative perception judgments. The standard of correct knowledge always carries with it, its own appraisal. Where principles are concerned there is an obvious scale of identification (e.g., leverage and the numerous references) that is all-inclusive and provides us with evidence of its existence. We could say with some truth, that the industrial crane has more leverage than a child’s see-saw, but we cannot deny the truth at the lower level or the reason applied. Where principles are concerned, truth is not a possibility, it is a constant reality (e.g., leverage). When mythological dichotomies are recognized and established for what they are, the process of ‘ironing’ them out and experiencing their constant reality will translate into the reality, which they are, and used accordingly. To evoke that new sense of reality, the mechanisms of ‘selective immaterialism’ need to be dismantled. Where human experience is presented with something it does not understand and is unable to appreciate the principles involved, the reaction can invoke a sense of fear. That condition can be a primary breeding ground to establish a language of dichotomies and put a selective name to something it does not deserve. The diffusion of a false singular dichotomy into the natural healthy state of the common good puts responsibility into its proper place. Within the process of diffusion, there is the natural and equal absorption of our true reality. The transition between separation and inclusion will be a seamless process because it is our natural state. Objectivity and Commonsense: Explore the plethora of principles - truths - constants that are the mark and phenomena of Homo sapiens. We cannot have any doubt about our existence in this present reality. The truth of reality is and can be experienced wholly and completely by anyone at any point in time. All human progress is the result of such experiences, manifest in principles throughout time and their subsequent ‘use’ evolves exponentially. The overwhelming evidence is our reality, now. A simple analogy of objectivity and commonsense. Somewhere in our early development, someone put the ingredients of a loaf of bread together, somehow baked it, and hey presto, the first experience of a loaf of bread. It is now a form of sustenance, which feeds billions of people. We no longer need to experience that ‘truth’ that ‘knowledge’. It is unnecessary because it is subjective assimilation and the act of external and internal activity. Because ‘our’ brain functions in a manner that can identify the natural elements it exists in –space, time, energy, matter, we learn to ‘know’ and recognize ‘completely’ a child’s see-saw. Knowing is a natural evolutionary function. The quality of ‘knowing’ has its own natural determinants which of necessity contain the measure of principles required for universal recognition. An Archimedes insight (or anyone else’s) could not become a universal reality unless those determinants were in play. I know the very same way we all know – by experience via action, reaction, and interaction. Truth persists and what Archimedes experienced was true and complete. Any experience of any truth, principle, or constant can be as ‘complete’ within a grain of sand, or knowledge of a pyramid. All experience of that nature is an experience in ‘time’, and when it is the truth, we use it accordingly. There is no mechanical translation, or opinion of ‘necessary factors’ as they are constant universals. That, which is ‘complete’, is transparent universal knowledge e.g., the principle of leverage. The precision of terms must include ‘necessary factors’. Necessary factors translate into a common universal language so there is no loss of meaning. All truth – principles-constants – absolutes, that stands the test of time we use accordingly. Thus, human societies evolve, and we evolve without the necessity of having to re-experience any of the principles we recognize and establish. That simply would not be a ‘natural’ commonsense proposition and an entire waste of unnecessary energy. The truth of any principle at any point in time, and at that point in time, can be experienced completely by anyone. Whatever it may be if the principle is established – from then on, it will evolve. There are some misguided notions that ‘truth’ ‘knowledge’ and ‘experience’ have some secret value that is unattainable by ordinary experience. It is a ‘natural’ evolutionary reality that what we ‘know’ becomes useful. We have a mutual responsibility to recognize, use and honor the principles, which are the common property we share. = Human consciousness. = [[File:Lane past Coombshead Farm - geograph.org.uk - 589598.jpg|center|thumb|369x369px|"The roads half travelled when you know the way"]] [[File:Sombrero Galaxy in infrared light (Hubble Space Telescope and Spitzer Space Telescope).jpg|center|300x300px]] === The Universe === ''<big>We do not own Space</big>'' ''<big>We do not own Time</big>'' ''<big>We do not own Energy</big>'' ''<big>We do not own Matter</big>'' The human capacity to understand the question of sovereignty or ownership of Space - Time - Energy - or Matter can only be accepted when any basis of dispute includes two dispositions - human and spiritual indigenous ties throughtout history. With the evolutionary appearance of indigenous peoples throughout this planet, their way of life should make it paramount that their existence be recognised as a natural law that has providence! Their culture and way of life has its own identity in which Space - Time - Energy - Matter is expressed as they experience it. That proof also lies in the existence of caves thousands of years old, and the existence their art and culture. So the constructs defined below are a new approach to understanding the concept of ‘knowledge’ and its proper place in an evolutionary expanding universe. Knowledge acquisition requires appropriate recognition through action, reaction, interaction, in which proper perception and comsciousnessvalues are applied. That form of construction requires the dismantling of previously embedded information. This requires a new direction to formulate a sound basis from which to extend. Construction of an analytical methodology to establish a form of ‘knowledge’ that is best suited to distinguish in a contemporary reality. A reality that adequately conforms to common notions of that which is true, and can only exist without any false relation to that which does not exist. All science needs the certainty that established absolutes provide. A pragmatic construction of real knowledge would propose that all reality expresses a form of evidence or proof and that the observer and the observed contain innate properties necessary to establish a foundational agreement. That form of agreement would necessarily function on the basis that everything that is – is truth. Any other interpretation would be a disconnect from reality, and the interdependent correspondence that must exist for true recognition of any absolute. Controversy will always emerge when the discourse on bifurcation and the introduction of non-existent dichotomies are used as arguments to be explored. Pragmatism would say that the human species would need basic properties to formulate any form of reliable epistemological analysis to explain and simplify the reality that forms their existence. That reality could well be recognized initially as the absolutes of space-time - energy-matter, through innate perceptual data that corresponds to an outward structure that is constantly evolving. That which is absolute is the determinative factor in establishing the existence of truth. When a chair, is a chair, is a chair, its recognition is established when we ‘commonly’ apply subjective and objective measures to that which is truly external to that which is truly internal. That which does exist becomes ‘common’ knowledge, and accepted as being true. Thus, the distinction between a priori knowledge and a posteriori knowledge no longer exists. Where there is consensus, everything is. The formulation of absolute criteria that offers ‘simplicity’ as a tool to measure all and everything, dispenses with the confusion of ‘difficulties’ historical philosophies engage in. It also offers an observable synthesis that clarifies the confusion. Within the combined properties of those absolutes then everything potentially would be. Everything that is by that definition is original, ipso facto, everything that has no false relationship, and therefore true (no dichotomies). Given contemporary human development, it would be ‘common’ pragmatism to accept the proposition that we exist within the absolutes of space-time – energy-matter. Within those absolutes and their innate properties, it would also be pragmatic to assert that ‘everything then is’ (whatever everything may be). Pragmatism would also dictate that ‘everything that is, is its own form of truth, and must contain available constructs of meaning. Therefore knowledge and understanding would be obtainable to that common experience, and at whatever level that experience is activated it is in interdependent unison with the source. The continuing establishment of basic absolute principles (their generalities and their specifics) forms our reason. 1+1= 2 is a ‘simple’ but true universal constant generality. That form of generality is used because we recognize the specific principles of a balanced equation that adds up and makes sense. Simple generalities with their inclusive specifics form the foundation of human reason and its constant evolution. Simplicity is the bane of a ‘mind’ that must have difficulties. = Availability. = The unifying feature that makes ‘knowledge’ available to us all, are the innate universal principles in all things. Archimedes established the principle of leverage. To paraphrase - ‘give me a lever, and I will move the world’. Once the law is established it can then be put to good use. The principle of leverage is manifest in countless ways, and put to good use! The principle of leverage is a constant available to us all, and always has been. Through his application Archimedes conveyed his insight in practical terms, and made aware of the principle and the laws governing it. We now use those laws. We no longer need to philosophise on its existence as a truth. Similarly whenever 2+2 = 4.  Whatever we use to make the equation – apples, oranges, bricks, the mathematical equation is a constant, and the principle of correctness applies. Here 2+2 =4 is empirical proof that the principle of correctness and agreement exist. Principle, or law: ‘a fundamental truth used as a basis of reasoning’.           If it were otherwise we could not measure anything. That which appears abstract then, requires no implausible argument as to its non-existence. Such is the nature of all universal principles, they exist whether the notion of a ‘mind’ can perceive their existence, or not. Because ‘thought’ does not create the reality of principles – universals- truth, it cannot from its mythical standpoint understand the simplicity of a Reality where ‘everything is’, nor the simple and factual conclusion – how could it be otherwise! Where ‘everything is’ evidently encompasses the whole evolutionary dimension. It is not an ideological enclosed static that stultifies expansion of an unfolding Reality. Any pure knowledge experience that ‘everything is’ ,(quite apart from the common-sense truth of the statement) is to experience the Absolute in any immediate part of anything that exists, which establishes its own truth forever. To examine a road code of law with that knowledge, and view the actions of drivers at traffic lights, it is more than reasonable to conclude with some conviction that there is  to some degree, Agreement – Knowledge – Understanding, and Conformity to that code of law. It becomes a ‘more reasonable’ proposition within Reality to understand that that code of law is multiplied exponentially, and the principles practised,  wherever drivers, motorcars, and traffic lights exist. It is the nature of the type of knowledge we are measuring that determines the measure of reason that can be applied to any given form of Reality. We can conclude that 2+2 = 4 is a reasonable mathematical calculation that contains the principles of Agreement – Understanding – Conformity. Given the accepted knowledge of these innate principles we can with more reason apply such a calculation Universally. Knowing is agreement with ‘what is’. Knowledge is not the attempted denial of any existing reality. That is a contradiction in terms. Mechanistic observation is akin to viewing from the outside, a straw in a glass of water. The straw always looks bent, but when removed from the glass we realise it is straight. To claim an experience of that which is Absolute, is not a claim of an experience from a higher domain, or an  isolated incident – it is common-place, numbered by just how many we are. Attempting to denigrate such experience is denying the everyday actions that contain the innate principles of a constant Reality. All life functions within the constraints of the laws that are the constructs of Nature and Reality. All life is an expression of the Absolute. It is when that expression is realised, not only in an instinctive sense, but in a real sense, that we penetrate reality beyond a comparative framework of mythology. To claim that you ‘know’ intrinsically what the principle of leverage is, or the principle behind the mathematical equation 2+2=4 is to claim experience of the Absolute. The Absolute is not some abstract esoteric truth – it is that which is immediate. Whether in awareness or not, we constantly comply, to some degree, with the laws of a constant reality. Therein lies the difficulty for a comparative framework mythology – the Absolute is everything! = Basic Equation. = However much the simplicity of the equation is, it contains the properties of correctness – balance – equality – mutual identity – meaning, which in its ‘simplicity’ presages all future mathematical equations. In that universal meaning, there is particular knowledge of consistent truth. That form of ‘simple’ consistency, creates its own natural equilibrium, and its ‘usefulness’ evolves exponentially up the reality scale. Here is where we need to give proper credence to ‘that which is’. All generalities have profound and specific principles as their ‘common’ identity, which are absolute. Unless those components are recognized, both objectively and subjectively, they are reduced to a comparative value spectrum (using dichotomies) as a misguided ‘simplistic’ factor. Deductive reasoning is then deprived of all value, and leads to the inevitable spurious question ‘how do we know? Given the above criteria to establish a correct basis for knowledge that is recognizable, and of a kind that can be used universally, ‘simplicity’ can be recognized as a tool that promotes its own established formula. That which we constantly use. Everything is the truth with regard to the methodology. How that truth or generality is expressed denotes the measure of the principle that is at its core, and forms that measure of reasoning we enjoy. Fortunately, although the truth is an innate property, it is not a ‘personal’ property per se, nor is the ‘experience’ of its reality. Its natural evolution is progressive. In that progression we are in common, the beneficent recipients that ‘evidently’ conform to its constant existence. Philosophical dissertations have become a monopolistic form of opinions that always seem to presume the ‘rightness’ of difficulties in establishing the source of our being, and are unable to put in ‘simple’ terms the question of ‘who we are’. There comes with that the denial of evidence that permeates human history, which establishes the principles of our ‘common’ reality. Those opinions carry with them a colossal library of questionable erudition that becomes embedded, using questionable values to support their argument. The most distinguished opponent of such arguments (Ludwig Wittgenstein 1889-1951<ref>{{Cite book|url=http://worldcat.org/oclc/1203018418|title=Tractatus logico-philosophicus|last=author.|first=Wittgenstein, Ludwig, 1889-1951,|isbn=978-1-78527-656-9|oclc=1203018418}}</ref>) proposed that language logic was a necessary tool to dismantle the convoluted ‘mind’ propositions that have permeated the philosophical hierarchy. Those ‘mind’ propositions only served to construct meaningless concepts as to ‘who we are’. Although he gained prominence in philosophical circles, his work was directed more toward academia. In his Tractatus Logico – Philosophies he quotes: “The limits of my language mean the limits of my world - What cannot be shown cannot be said”, and “There can be no representation of the logic of facts”. Having a belief in mystical truths that were inexpressible, his statements above are indicative of the embedded language of dichotomies. His form of ‘knowledge’ ‘philosophy’ gave (without question) the concept of ‘mind’ credence to formulate its own logic to clarify its own form of reasoning. Wittgenstein, by not recognizing within the language the distortion that dichotomies create, was unable to approach the evident constructs of true meaning that lie within the interdependent relationship of absolutes, and their constant existence. Evidential reality is all there ever is. The ‘meaning’ or ‘knowledge’ that becomes evident in reality is ‘commonly’ accepted and used accordingly. To repeat, the evidential reality is all there ever is. In that regard, the exponential drive toward ‘difficulties’ amassed a historical discourse of misinformation, which is used to address the very ‘difficulties’ created by spurious value systems. In effect, dealing with ‘nothingness’. We have managed to turn ‘truth’ into a problem, into a difficulty, when the truth is simplicity itself. An oxymoron of gargantuan proportions. This contemporary malady solidifies a diversion that discounts the reality of continuous progression. Progression in which ‘common sense’ is a motivating principle that promotes human evolution. The consistency of specific principles allows us to achieve correspondence. Real knowledge is not a fabrication of convoluted prescriptions. It is the ‘coalescence’ of ‘what is’ to internal reality. That established, evolutionary progress is assured, and reality factors are recognized for what they are. True meaning can best be attained by the interaction and interdependence of natural principles and so recognized as such. That meaning which contains all the specific components of reality is experienced as ‘true knowledge’, measure by measure by anyone. That form of ‘experience’ is not a ‘mind’ process, but a very natural state of realization consistent with our level of action, reaction, and interaction. A correct and pragmatically form of knowledge-seeking foundational answers to perennial questions would seek a direct passage to our ‘commonality’, the beacon that offers guidance. Consider the quantity and quality of knowledge we all pursue that has meaning and usefulness. To posit the notion that there are no dichotomies is a cataclysmic proposition that seems nonsensical to established embedded constructs of knowledge. Constructs of knowledge that offer only confusion, and continually pose impossible questions, whose absence would provide clarity. That absence of confusion would dispel and dismantle a reality of ‘mind’, which functions on its own selection of problems. To address reality as having only absolute constructs dispels the confusion of duality and its inability to ‘use’ relativism in its proper fashion. All general absolutes contain specific principles representing facts; the essential properties that confirm reality. This reference directs the observer to observe, and go beyond the restrictions of a ‘mind’ governed by dualism, monism, or any other spurious form of philosophy that distorts the very reality it exists in. Where there are no dichotomies, all we can deal with is ‘what is’, and the logic of ‘necessary factors’ thus destroying the possible inclusion of anything described as a “paradox”. = Pure Experience. = To design a chair our brain requires to exercise the qualities and properties necessary for its manifestation e.g., strength, balance , design, functionality etc, etc. A chair, is a chair, is a chair, the product of innate knowledge.   If we did address any tentative agreement that ‘I’  is a ‘fiction’, could not our ‘conscious experience’ of that ‘fiction’ be just as fictitious. It would follow that whatever perception of Reality we experience must also be fiction. Our contention, as always, has been that ’I think - therefore I am’ by Descartes is the greater fiction for reasons already explained. To consider to whatever degree that we can function on the basis of a fictional ’I’ precludes any attempt to honestly address ’who we are’. Saying that ‘conscious perspectives’ are limited and inconsistent with apparent reality are quite correct. It then brings into question the validity of ‘conscious perspectives’ to guide us toward ‘what is’. The entangled fictional relationship between ‘mind’ ‘I’ ‘thought’ ‘consciousness’ impose formidable barriers to that which is evident. Base observations on the construct and interpretation of what ‘knowledge’ is. Human experience is limited by its mechanical interpretation of Reality, especially ‘cogito ergo sum’. If everything that is, is its own measure of Reality (the differences) then everything must be measured, at whatever level, as being that part of the whole with all principles intact, making that measure available to be experienced as the Absolute. With absolutes there is no antagonism. We cannot exist or experience anything without a Universal complementary source of identification. It is notable that within the structure of Cartesian dualism, Descartes' personal address to innate knowledge he attributed to ‘thought’ which he identified as being distinct from his body. How different Western philosophy may have been if his attribution had been toward his brain and the existence and evidence of other physical entities that functioned every bit as efficiently as he did. The premise that Descartes operated from ‘never to accept anything as true’, was simply a wrong ended approach which brought him into conflict with his passing acceptance of innate knowledge, that the idea of God was innate to his being. To view the proposition that ‘everything is true’ allows reason to seek and identify that measure of truth. No quest can be productively based on cynicism or denial, nor adherence to belief systems that separate experience, knowledge, and Reality. We have the obligation to question whatever reality has placed before us , but if we constantly deny its existence and attempt to ‘disappear’ it from our experience, then we are in danger of never experiencing that reality. Not experiencing Reality as it is, is equivalent to not experiencing ‘who we are’, and is indeed the only human source and validity of truth, although Descartes held the erroneous belief that such knowledge was independent of any experience. That belief we suspect was some form of impetus toward his ’cogito ergo sum’.                   Knowledge and experience are co-existing ‘necessary factors’ So long as anyone believes that human experience is based solely on indirect conscious interpretation (mechanical disposition), therefore any ‘knowledge’ derived from experience will be incomplete. Clearly it is the quality of ‘knowledge’ that one experiences (e.g., Archimedes) that leads to a common certainty of evidence realised through direct experience. That quality of knowledge can be available when we observe directly the activity of drivers at traffic lights with the knowledge that it is a very common activity recognized internationally. In every case we can logically pronounce the premises to be true, therefore we have a conclusion that is also true - whether that conclusion is defined as Mutual Agreement, or Common Acceptance, it does not matter - they are mutual principles. It is this form of logical knowledge of innate principles that is the precursor to knowledge of the Absolute logically defined within all reason for ‘what it is’ . Where the basic premise is true that there is ‘Mutual Agreement’ between a multiplicity of drivers at traffic lights then we can with certainty conclude that the same principles exist Universally. We can also draw concrete conclusions, and establish knowledge, that it is not ‘absolutely necessary’ to experience by observation the multiplicity of drivers conforming to their particular road code. We have already established that knowledge. Knowledge and experience are not separate philosophical theories. One cannot be without the other.. Everyone has the potential to experience the Absolute paradoxically, in part or in whole. Everything that is, must contain the properties of the Absolute, otherwise nothing could be. To ask questions about human experience based solely and inevitably on our interpretation of ‘knowledge’, and co-existing with that, its particular meaning in human existence. So long as we can only deal with our conscious interpretation as representing Reality then we derive functionally less meaning than we are entitled to. When we see other humans consistently using levers to open crates then we can recognize a ‘social intelligence’ operating which equates to understanding that is not based on opinion, but is a clear expression of human activity that has correspondence. All of the principles involved in that experience can coalesce to provide that form of Reality that requires no interpretation. It becomes recognizable knowledge. How we understand that knowledge is through the realisation and identification of the principles involved, which become immediately transparent. The Absolute could be categorised as a knowledge experience that encompasses all and everything. Whatever is manifest is that measure (complete in itself) of the Whole with all its principles intact. Where there are at least two actions that are identical we can reach a common-sense conclusion that a definitive principle is operating. When that corresponds with innate knowledge then we have the complete cycle. The definition itself is language opening the door to an experience of Reality. No one can know in isolation. An imaginary ‘I’ restricts any experience of who “we are”,  and is not a necessary part of human experience.   Explaining experience beyond imaginary thought processes requires a definitive language that deals with the principles of Reality itself. Pure experience. The world-wide disposition that has no grounding in Nature and Reality becomes captive to any mythical fear that  offers a target to give some form of direction or stability. There is nothing more simple than to make Reality transparent - its evidence abounds. We can pronounce the principle properties that provide guidelines to its existence whereby the reality is made apparent. = Expansion. = To address concerns on ‘negativity’. Negativity is in essence the inability to establish a measure of Reality. Mechanistic processes of denial are the attempt to understand and make transparent that which is apparently unexplainable, and resolve a condition whose energy is driven toward finding that core of affirmation. The evolutionary principle from all available evidence is that human beings as a species progress. This seems a paradoxical contradiction to the embedded proposition that we can never know the ‘truth’. The consequence of such a traditional premise is that denial and negativity both hold paramount positions. We are conditioned to accept the premise that there is in fact no premise that will enable us to go beyond presently accepted norms of experience. We are conditioned to accept that the ‘truth’ is inexpressible. The evolution of the human species is constantly subject to contemporary ingrained social habits, which give some kind of credence to that particular point of existence. Indirect conscious interpretation classifies itself as a solid perspective to govern and justify human activity, which in many historical ways has proved disastrous.                   Our continued intention is to expose detrimental barriers to the realisation of ‘who we are’, and in that process establish a smoother, more realistic approach to ‘who we are’. The Archimedes legacy. When we establish knowledge of something that exists through a multiplicity of experience and evidence, then from every reasonable standard we can establish that it is true, ergo that which is true is Absolute. The principle of leverage is well grounded in social intelligence, and our natural knowledge of that does not need erudite explanations of its presence, nor any ‘conscious interpretation’ to realise its existence, or its practice. Evolution eventually removes restrictive passages to direct experience, the very purpose of evolution. The principle of leverage is not a matter of opinion, it is the realisation of actuality and our continued ‘more reasonable’ response each time the principle is applied. Children learn to speak their language primarily through experience without any direct, or indirect conscious interpretation, and so, universally we ‘know’ the most powerful means to communicate. Were we to move 50 miles in any compass direction from the town we live in, there is a certain predictability that we will meet others who speak the same English language that we do. If in that experience we find that these premises we have drawn about our travels were true, then the conclusion we would come to in particular, is that when we communicate we make known what we know. …………………………………………………………. '''Please note the date:''' '''Oct 2005.''' I am offering up this older material below to provide insight as to the progression of this work. There may well be some duplication to date. During this period my wife and I worked in collaboration to ensure an equality of experience. = Stepping Stones 1. = There is nothing other than what is – there is no hidden Reality that we need to seek, it embraces us at every turn. The Archimedes experience is the pure experience of Ultimate Reality, which provides indisputable knowledge. Reality is the source of complete knowledge, it is the constant source that has provided us with all human development, from the writings of William Shakespeare, to the computer development of Bill Gates. What they have produced is now an evident part of our reality that we can engage in. We can experience ‘mutual agreement’ through epiphanies, insights, enlightenment , understanding, Eureka moments etc, they are all one and the same. Implicit within the macrocosm is the microcosm – it cannot be otherwise. The more we conform within the microcosm the more we begin to appreciate that Reality contains everything, and that we can realise through experience its manifestation. Each Eureka moment is that personal point of experience that connects us with the Truth. The principle of leverage was always available, it took an Archimedes to explain it to us. Each Eureka moment necessarily engages with the reality of complete knowledge, and utilises its share at that time. When we have complete knowledge of who we are in that personal moment, then we understand that these, egalitarian properties, are rightfully shared by everyone and that we have experienced that which is infinite. It does not mean that the process of evolution is over – it has only just begun. It does mean that we can no longer continue coasting through this existence in a near comatose state. If there is a hypnotic fixation in holding the principles of Reality as being separate, and different, then the potential realisation of their immediate unity, and communion, becomes problematic. Knowledge, and experience are one and the same – they are not different!! Experience = Immediate knowledge of basic reality that is factually correct, and that we can reasonably use. Knowledge = Immediate experience of secure, and accurate information that is constantly stable, and sustains principles. Reality = Complete Knowledge. As the microcosmic part of the total macrocosm we are immersed in reality. The real question should be, ‘how can one not know Reality, or ‘who we are’. Mutual agreement is evident when we know we can go to the bank, and deal with money transactions. Mutual agreement is evident when we know we can go to the supermarket and exchange money for goods. Mutual agreement is evident when we know we can send our children to school to enhance their education. Evident proof is validation of what is – it is not a matter of anyone’s opinion, nor is it an assumption of ours. Neither do we assume, or offer any opinion, on the Universal Reality that there is ‘mutual agreement’ that we need air, food, and water to sustain us. Evident proof is also the basis for the mechanics toward realisation of ‘complete knowledge ‘ of who we are. Reality can be realised through concentration on its basic principles. We use language to express our understanding of who we are.It is relatively easy, it is reasonable, and it is responsible. We convey through language our measure of intelligence, and to the best of our ability conform to the basic rule of communication – ‘we make known’ Implicit within that exercise is ‘mutual agreement’. We may differ in some specifics, but we meet the basic obligation of communication – ‘we make known’, and always we progress to some degree. Simultaneity is one of the constant principles that we all share and they come from Here, Now, the Present, where they have always been. Everything is. Our being is always engaged in the present, and we each have an obligation to understand our relationship to what is. The present is the only point of contact we can ever have with Reality. To some degree or another, each one of us is directly connected to Reality (we do not have any choice in the matter), and we can potentially evaluate ‘what is’ through the utilisation, and examination of factual reality. We are the microcosmic part of that Universal Macrocosm, and because we already have that innate information it is a matching process when we have a Eureka moment, an epiphany, an understanding beyond question. Nothing enters our minds - we already know! Everyone has innate knowledge of the principle of leverage. It requires correct examination of ‘what is’ for realisation to occur. It is then a relief to have ‘mutual agreement’ on the things we would wish to make transparent to others. To use a traffic analogy, it is evident that there is ‘en masse’ mutual agreement when we know to drive off when the traffic light turns green. Mutual agreement is translated into people obeying traffic rules (otherwise chaos). Two cars, two drivers, sitting directly alongside each other at traffic lights, discuss their understanding of their Road Code in this particular position, and what they should do. When the light turns green there are a myriad of principles that apply when they drive off simultaneously. They have both demonstrated their ‘complete knowledge’ of the significance of the green light from this perspective. There is Mutual Agreement. There is Predictable Conformity. There is Common Ground. Each one complements the other. They are both right. One more remove: From an outsider’s point of view – they both know! The green light could be categorised as a Eureka moment, it sets in play all the above principles, whether the drivers are aware of it or not. From the perspective of two outside objective observers who know the traffic rules, if asked, did the two drivers at the lights obey the rules – the answer would be yes, there would be mutual agreement. They have complete knowledge of this particular circumstance concerning drivers, and green lights. Could it be that certain schools of thought are curtailed by a questionable refusal to recognize what is, and have a preference for creating a difficulty where none exists! No one can examine what isn’t! There is no such thing as ‘nothing’. Something is – what is it? A Scottish engineer functions on the same principles as an Italian Pope. Because Archimedes was prominent as a mathematician, his realisation of the principle of leverage, and his understanding of the difference in water displacement between silver and gold was widely reported. This does not mean that realisation of ‘what is’ is an exclusive experience. As said previously Archimedes did not realise something new – it has always existed, and all forms of life would have utilised the leverage principle to some degree or another (watch a bird build a nest). At that time there were probably many thousands of people who had some understanding of the principle, but Archimedes was the one who made statements about it. As in any Eureka moment, we can experience infinity and who we are. It is mutual agreement (an understanding) between the part, and the whole. It is when the principles are in unison Eureka! Reality is there to be examined, and experienced, it is not separate from us, nor should we try to make it so. = Stepping stones 2. = Knowledge is not conditional by the activities of what may be called ‘thought’ or ‘consciousness processes’. Real knowledge is that which is available to all, and to be shared by all. It cannot be contained by the ‘experiencer’ and then not ‘known’ by the accident of experience. It is the actual innate experience itself which conclusively establishes the truth. It can only deal in the truth which is its modus operandi of dissemination. For me to say that ‘everything is’, is a statement of fact which cannot be denied, and an intellectual dishonesty to attempt to deny the evidence by philosophical machinations. Hostility toward the truth leads inevitably toward attempted negation - looking for nothingness! ‘Being here’ demands its own recognition - attempting to deny it is simply perverse. Knowledge is the realisation of ‘what is’.. Rene Descartes ‘I think - therefore I am ‘did no service to human evolution, or education. It established in Western societies especially, the culture of individualism, with the precursor that so-called ‘thought’ was the inward evidence for existence, and for the following unfortunate claim that we have a ‘mind’, or to use the euphemism, a soul! Experience is true knowledge. When that experience marries up with its innate counterpart then recognition is realised (cognition). In simple terms, a light goes on in the brain. There can be no real knowledge without truth. All thought qualifies experience and attempts to reduce truth to near nothingness which is a widespread conditional activity. We cannot manufacture knowledge, or the principles which are its properties. No matter the amount of correct information anyone can ingest, it does not become knowledge until there is tripartite coalescence between inherent knowledge - ingested correct information - and ‘what is’. Then we truly recognize that which is Absolute. Within Nature we have the distinct privilege of evolving in a Universe that can only recognize the attributes of social cohesion. Knowledge is not anyone’s personal possession. Whatever measure of experience we may have of it, it is only available as a Universal sharing experience to be beneficially used. Human activity whereby we witness people using tools for leverage, or drivers at traffic lights obeying the rules of the road, are observable markers that contain the properties for understanding our own reality. Unless seen for what they are, they are only mechanical platitudes with an equally mechanical response. We could rightly claim that that at least is some response, but of no real value. The natural process that operates when we see that which is innate, overrides any erudite explanation from an academic base however intellectual its original source. ‘Thinking’ for oneself cannot make judgments about a ‘natural’ experience. When we see human duplicate functions in operation then we are in communion, and at another level we recognize who we are. When we actively see the activities of the human brain in action we are not dealing with any internal ‘will - o’ - the wisp’ that no one can ever experience. We exercise that prerogative (human activity) at every moment in time, but quite apparently without that focus of attention that denotes realistic recognition. To seek identity in sectarian, or secular belief systems to overcome the contemporary feeling of loss of identity leads to the acceptance of anything that offers some form of stability. That is then used to strengthen that which is euphemistically addressed as the ’self’. To retain that security the acceptance of information transmitted throughout generations, is absorbed into the culture, and defended to the death against those who would question that belief system. The greatest knowledge we can ever have is our own and it has the potential to transcend all else and provide insight into infinity. The most tragic human condition is the lack of experience of identity in a multiplicity of identities in which we all share. The real problem is not one of ‘identity’, but a lack of ’communion’. Whether we like it or not, whether we are aware of it or not, the principle of ’communion’ must always exist to some degree for evolution to proceed. It is within the experience of that principle that we understand the fallaciousness of that much heralded ‘self’ which draws down so much energy in an attempt to establish itself as a reality. Within positive language structure possibilities (no dichotomies), there should be the disposition toward the realisation that our relationships to cognize into ‘communion’ must be addressed as specifically dependent. Social attempts to be ‘independent’ are the very remove from reality and signify reduction attempts toward nothingness. Adherence to, and the cultivation of faith and belief systems give little elbow room for any factual occurrence to be anything other than a comparison to the myths that are held. The cultural and educational socialisation of generations of children must carry with it, its historical belief systems that overwhelm the natural instincts. Observe an animal out of its natural habitat and locked in a cage for its entire life. It would be a salutary exercise if we could dispense with the term ‘mind’ from our vocabulary and magnify the use of the word brain to promote a realistic discussion on ‘who we are’. My action of levering open a wooden crate and knowledge of it is one and the same. Our remarkable brain functions like that, the purpose of a brain, the natural repository of innate knowledge. The assertion of principles is critical to avoid all activity being submerged by questioning their very existence, and being unable to see directly. It would be a rarity today, if anyone using a lever to pry open a wooden crate would have the same enormity of experience that Archimedes had, nor the need to make pronouncements about it. It has all been done prior to our awareness of its value with the accompanying data attached. Our brain knows the value of a lever and activates our body accordingly when needed. It could be categorised as evolutionary transmission. The observance of someone prying open a crate with a lever, or drivers conforming to the road code at traffic lights, is a function of the brain in action, not a mythical entity in a singular locality that denies its own senses. When the brain is not burdened by distorted belief systems it then has the potential to experience ’that which is’, which is always constant. When we understand the function of a lever, or the presence of traffic lights, then we can activate the principles involved because we already know how! The negative impact in the use of dichotomies in language lies in their distraction from the truth, as our brain processes the words we use in relation to Reality. The tendency to attempt to separate inherent truths through the words we use disrupts that natural correspondence necessary for identification. A chair, is a chair, is a chair. = Stepping stones 3. = Where principles are concerned the constituent linkages in language are identity markers to that which is real - reference points. Without dichotomies there is no separation, or ambiguity between what we experience, and ‘what is’. Philosophy in its attempt to address something through denial is an elementary confusion. To say that that is a chair, and then attempt to deny it invoking philosophical theorems concerning the human ability to experience it, is a severe contradiction on the existence of the object , and the observer. When this form of contradiction is then taken as a constant, it then precludes any common-sense and definitive answer to the existence of a chair. '''For philosophers, George Orwell’s ’to see what is in front of one’s nose needs a constant struggle’ would be apt.''' Real concepts cannot exist in any mythology, therefore all that we experience is inevitably the truth that is there to beproperly categorised for what it is. The proper use of language in this context will identify whatever it is to correspond with present reality. Misuse of language (dichotomies and mythologies) leads only to the acceptance of a fractured state where nothing is whole and represents confusion. The dissipation of the supposed problem is never realised. Fiction has been elevated to the status of an accepted reality. Very early evolutionary physical dangers allowed the development of fictions that offered some form of imaginary protection beyond limited physical ability. That contemporary humanity endorses the mythology of ‘I’ is testament to the psychological fear that still exists and requires its proper recognition. Emphasis must be placed in the relationship between language and reality for understanding to proceed. The persistence of dichotomies has their own persistent confusion which then promotes a false reality through misleading information. Microcosm and macrocosm are one and the same in a Universe where ‘everything is’. Isolated viewpoints are exactly that, and are unable to view the expanse in which we are encompassed. We must learn to view reality through both ends of the same telescope. When we understand the extensive scope of ‘truth’, then we know that its values and properties do not change - which relates to ‘completeness’. Philosophical, ideological, and intellectual endeavour , try to shape the structure of ‘what is’ based on pre-dispositional knowledge, which can only ask the same questions, and look for the same answers. Not to experience that which is absolute or whole is the normal result of the confusion of language which has no correspondence to that which is real. To discuss with a philosopher the possibility that ’mind’ per se does not exist, and to dissolve it as a concept would place them in a realistic position, would indeed be a difficult proposition. The strength of that difficulty lies in another imaginary concept, that that ‘mind’ represents ’I’, and it is anathema to that fiction to consider its own demise! = Stepping stones 4. = There are no dichotomies. Everything is, and everything that is, is complete, everything is an Absolute complete Reality. You are experiencing your measure of that reality. It cannot be otherwise that you are experiencing that measure of completeness. When we come to terms with it we have the innate capacity to see the Absolute in a grain of sand. That is knowledge. Belief in dichotomies is the mythical barrier to that particular experience - which is only denial, supported by erudite protestations that human construct dichotomies exist. At a mechanical level Intelligence and Stupidity appear to be separate identifiable conditions, and they appear to be antagonistic. Stupidity is in Reality a measure of the Intelligence which is always constant. If someone was in a state of mythical utter and complete stupidity we would not attempt any form of emancipation from that condition. We know that that is misguided and proceed with techniques to advance intelligence. Consider the proposition that there are no dichotomies, and within that possibility all questions become irrelevant. Presuming that there are no dichotomies allows the process of establishing ‘necessary factors’ to proceed, and allows each measure of wholesomeness to be realised. Experience is the criteria for knowledge. Some Reality experiences were simply transposed into particular belief systems and elevated into a pseudo spiritual dimension, or a philosophical conundrum. Where there is a belief in a divisive fiction (dichotomies) there is automatic mechanistic restriction to that which is Real. There is a capacity beyond ego and intellect which can commune with ‘what is’, and recognize its properties. Reality is constant. Within the accepted comparative framework there is the view of principles as having different divisive categories e.g., as above, Intelligence and Stupidity, and classify them within ‘thought’ structure as dichotomies and give credence to them as being an antagonistic reality. The consequence of that, is, that one is always a remove from recognizing the structural properties of immediate existence. Any construct of knowledge necessary to evaluate ’what is’ will address the properties (principles) that are the constituent constant markers available in that which is the microcosm and the macrocosm. That identity (the Absolute) is found in any sphere of Reality.Everything is - and everything that is, must be experienced for what it is, and not for what anyone denies it to be. There is no mythical human construction that can deny ’what is”. Everything is - without dichotomies. To repeat, we do not have the ability to create ‘nothingness’ - ‘that which is’ has no imaginary comparative human construct. To attempt to deal with such constructs, and give credence to them is always the denial of ‘what is’, and adherence to ‘thought’ processes whose only purpose is to cement that activity. Indeed realising that the concepts of dichotomies are human mythical constructs, denying true perspective, is the beginning of insight. The dissipation of such processes through addressing the principles of Reality allows us the potential to experience directly ‘what is’, in simple terms -the truth! Intelligence is a ‘necessary factor’. Addressing stupidity is a denial of reality at whatever level we find it. Intelligence and Stupidity are not antagonistic, they are one and the same principle with measurable degrees of existence. Only from a comparative framework standpoint is credence given to any mythical form. The above observation is not negating the process, it is questioning the markers which evolve into imaginary separation (trapped in a comparative framework mythology). That particular process can and does create a false mythical reality that appears divisive. We cannot exist within a divisive reality! Reality must be complete for us to recognize its existence. Where there are no dichotomies within the premise that ‘everything is’, there exists no antagonistic position. The distinction between human constructs of positive and negative are matters of mythical perspective wherein no experience of the Absolute is available. It is because the human ’mind’ per se places its own construction on its immediate experience, and must have its particular interpretation based on what it considers ’knowledge’. There is a difference between ’mechanical knowledge’, and ’pure knowledge’. From the mechanical knowledge standpoint which can only deal ‘in indirect conscious interpretation’, it is quite correct to say that that form of knowledge is incomplete, and it always will be. Pure knowledge experienced via our brain knows no separation, nor antagonism, and is responsible for our ability to recognize the actions of others who may pry open wooden crates with a lever, or drive off uniformly at traffic lights. Within that cohesive activity it precludes ’a matter of opinion’ and by themselves can become subjects of a pure knowledge experience. To repeat, it is a form of ’communion’ with ’what is’, and available to all. Where drivers at traffic lights universally conform to their particular road code, and where universally there is a language which identifies their activity as Mutual Agreement, or any other logical definition, we can concur with the common-sense conclusion that we have universally established that within language and common activity, there is indeed a truth formed. The coalescence between universal language and universal activity are the logical constructs that create civilizations. There is a vast social network of common activity that solidifies the logic into an honest and persuasive conclusion that confirms innate common principles –knowledge. = Stepping stones 5. = The Art of making sense of everything. How to understand principles. # Principle. A fundamental truth or proposition that serves as the foundation for a system of belief or behaviour or for a chain of reasoning. # All principles are interdependent, interconnected, and infinite. # Each one is dependent on the other two. Examples of a principles template and how to define them without dichotomies. Communication. Truth. Standard. Proof. Express. Contribute. Mutual. Direction. Advance. Comfort. Organize. Certain. Immediate. Interest. Improve. Present. Constructive. Gain. Trust. Progress. Source. Knowledge. Basic. Original Reality. Awareness.Freedom. Purpose. Connect. Understand. Support. Peace. Cause. Unity. Ability. Rights. Honest. Discover. Positive. Energy. Balance. Good. Courage. Willing.  Control. Use. Association.  Observe. Reason. Easy. Wealth. Simple. Law. Increase. Order. Flow.Co-operation. Exact. Quality. Accuracy. Strength. Responsible. Operating. Creative. Measure. Recognition. Accept. Constant. Obligation. Include. Dependence. Relationship. Value. Success. Principle. Equality. Stable. Share. Love. Sustenance. Action. Identity. Intelligence. Education. Secure. Facts. Agreement. Information. For. Rules.Clear. Yield. Example: Success = Securing facts                = Responsible co-operation               = Constructive knowledge So success by definition is : Securing facts through constructive knowledge and cooperating  responsibly. All definitions of success from your template are infinite. You will find your own suitable definition. ==== There are no dichotomies! ==== Any principle is correctly defined by any two other principles. You create a new language of Absolutes. Using conjunctions you can write your own book. The man whose book is filled with quotations has been said to creep along the shore of authors as if he were afraid to trust himself to the free compass of reasoning. I would rather defend such authors by a different allusion and ask whether honey is the worse for being gathered from many flowers. Anonymous, quoted in Tryon Edwards (1853) The World’s Laconics: Or, The Best Thoughts of the Best Authors. p. 232 Amen to that! “One is not born, but rather becomes a woman” Simone de Beauvoir. “Time does not change us. It just unfolds us” Max Frisch. We experience ourselves our thoughts and feelings as something separate from the rest. A kind of optical delusion of consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Albert Einstein, in One Home, One Family, One Future. = Who we are. =   It is notable that within the structure of Cartesian dualism, Descartes' personal address to innate knowledge he attributed to ‘thought’ which he identified as being distinct from his body. How different Western philosophy may have been if his attribution had been toward his brain and the existence and evidence of other physical entities that functioned every bit as efficiently as he did. The premise that Descartes operated from ‘never to accept anything as true’, was simply a wrong ended approach which brought him into conflict with his passing acceptance of innate knowledge, that the idea of God was innate to his being. To view the proposition that ‘everything is true’ allows reason to seek and identify that measure of truth. No quest can be productively based on cynicism or denial, nor adherence to belief systems that separate experience, knowledge, and Reality. We have the obligation to question whatever reality has placed before us , but if we constantly deny its existence and attempt to ‘disappear’ it from our experience, then we are in danger of never experiencing that reality for what it is..   For anyone to say that ‘everything is’ is a simple linguistic absolute that no amount of ‘more reasonable’ requests (above) can deny. Those requests only appear to be governed by the difficulties of ‘mental complexities’, and embedded ideologies. To accept that ‘everything is’ as an absolute, is a realistic basis to establish any reality, and comprises the basis for reason to be activated. If there is ‘nothing’, nothing can be achieved. Within that which is Absolute there are no dichotomies. Therefore there are no antagonistic positions available. Everything that is, is a measure of the Absolute.  We are always in the present, everyone and everything. Instant elementary ‘knowledge’ which we all share, and must admit to. In being alive, we do not have the ability to not be here, and we do not have the ability to not know! To have a problem in addressing what Truth and Knowledge are, to the point of denying their existence, then that problem exists well below the scale of Reality. To repeat knowledge is not the proprietary right of any individual, it is enshrined in the principle of agreement that we mutually exercise to establish its own reality. The reality of experience is not, nor ever will be, a personal possession that we can have and hold. Its reality becomes more alive when we see the same activity being practised by others. Then we know we are sharing that reality, and that experience. We cannot "have" the principles that exist, but when we undrestand them then we are obliged to use them with integrity. That form of integrity in any language, is an added foundation stone to any belief system Mechanistic ‘I’ has no concept or understanding of ‘pure knowledge’. Only when we break free of the myth that some clarity becomes apparent, and we have the opportunity to engage with what is real. Philosophy it appears to me is constrained by individual ‘thought’ processes, which (without experience) cannot escape from that individuality. Those ‘thought’ processes conjure up a human history of inflexible, and impossible propositions which only serve to protect that individuality. Descartes ‘cogito ergo sum’ has compounded the difficulties by strengthening the incorrect premise of a false individuality. ‘I’ is a phantom consciousness much like a phantom pain experienced after a limb is amputated. The brain registers the pain signifying that something should be there. Likewise our brain has that same relationship with Nature and Reality. It is analogous to our brain dealing with a ‘phantom reality’ knowing that something is missing but is continuing to evolve to establish the whole. There exists a ‘phantom chasm’ between our brain and Reality and an understanding of its properties. We are robbed of real meaning. Evidence, recognition, and the truth are the principles it uses to reform. Within their structure is the meaning of reforms. Nature does not impose any morality on us, the principles implied in morality are there for us to understand and use. Our brain has the capacity, once reality is correctly examined, to recognize ‘that which is’. Once realised it becomes embedded. That ‘phantom consciousness’ is an experience removed from its proper environment. It takes its proper place when we experience reality for ‘what it is’, which provides the totality of meaning. True experience allows us entry to the quality of knowledge that is a continuous reality. So long as anyone believes that human experience is based solely on indirect conscious interpretation (mechanical disposition), any ‘knowledge’ derived from that experience will be incomplete. That form of philosophical negative conclusion can come down to not believing that Reality exists (a chair is not a chair, is not a chair etc,), or that our experience of ourselves and others is real, and discount any other form of knowledge that threatens that belief. There is an intellectual dishonesty in denying the existence of principles. Hostility towards the truth leads inevitably to negativity. Being here demands, not denial, but the right to be recognized. We are the recipients of a ubiquitous communication system – making known. We can only understand that which we know. We make known all the time. Knowledge of Reality – Truth – the Absolute is a collective inclusive experience of the principles we share, and never the property of any individual. To ‘know’ ‘who we are’ is an inclusive experience of the principles involved. Never ‘cogito ergo sum’.   Philosophers in investigating the nature of knowledge and the Universe, firmly established for themselves that the source of reason and logic was located in a mythical concept ‘the mind’. From the wrong basis evolved elaborate and metaphysical constructions which removed the investigations further, and further, from the truth. To comprehend the material world, and give it credibility, the recognition of implicit principles is paramount. We need to construct a language that provides that form of recognition. Any philosophical theory of ‘mind’ that will deny the evident structure of solid objects is misguided by the injection of a mythical entity (mind) that determines that seeing solid objects is a ‘perceptual illusion’. That form of determination is singularly narcissistic, empowered by the self-induced threat that venturing into a ‘materialistic’ world is a loss of that illusory self, and all the belief systems it has constructed to protect it. That erroneous established view that not addressing ‘materialism’ as a profound Reality, and as only a ‘perceptual illusion, is compounded by the belief that that form of illusion is implicit in every human view available. We cannot manufacture knowledge that leads to a mechanistic understanding of ‘what is’ , nor the principles which are its properties – however much dogma is practised. We can only aspire to relate to ‘necessary factors’ that are the implicit fundamentals of existence. For me to use a lever to open a crate is a form of communion with Archimedes through the principle he enunciated. It is now not ‘necessary’ for me to go through the same experience as Archimedes to establish that ‘necessary factor’ or ‘principle’. It is now common-place, and common-sense to utilise the principle. The extract below provides some explanation of the brain processes in action Universally, and coincides with any reasoning on the observance of the leverage principle, and the actions of motorists conforming to the Road Code wherever traffic lights exist. ''The right-to-left shift of mental control looked increasingly like a universal phenomenon, capturing the essence of every learning process on every time scale, from hours to years. An individual faced with a truly novel situation or problem tackles it mostly with the right hemisphere. But once the situation becomes familiar and is mastered, the dominant role of the left hemisphere becomes evident. It looked like the empowering patterns capturing the essence of the situations (or rather the whole class of similar situations) were, once formed, stored in the left hemisphere. (The Wisdom Paradox. Professor Elkhonon Goldberg. P202)<ref>{{Cite journal|last=Vandermeulen|first=Jo|date=2008-08|title=Verstand komt met de jaren|url=http://dx.doi.org/10.1007/bf03077135|journal=Neuropraxis|volume=12|issue=4|pages=137–139|doi=10.1007/bf03077135|issn=1387-5817}}</ref>'' = Limitations. = However limited our view of connectedness is, or however tenuous the reality our experience is, ‘everything is’, and everything is connected. Innate knowledge and the fundamental nature of Man is the prior source of knowledge that seeks and identifies that connectedness. Attempting to address what we don’t know is that mythical infinite regress toward that parallel mythical ‘nothingness’. To always address what we know establishes Reality. To establish knowledge of principles, start from ‘we are here’. The natural principles within the diversity of human culture and activity when recognized as mutual ‘necessary factors’ will have the effect of enhancing and directing vital energy toward the very process of  the communion we seek, and the gradual penetration of a reality that always exists. Knowledge of Reality is not ‘different’ in other locales. The fundamental principles are the same. Addressing ‘what is’ instead of denying ‘what is’ is the basic construction of real knowledge. Within the structure of the Absolute we are all the same with a magnificent differential in our expression of the principles of necessity. That expression is our ongoing effort toward its own experience which gives it life and meaning. That experience in turn exposes us to an immediate Reality that is in communion with the fundamental structure of our being. All that we can contribute toward that is 50%, the other half is in our momentary relationship with Reality – then we know! That form of knowledge is always available through that form of experience, and it always comes in the form of confirmation which reforms. Until that experience our prior condition appears mechanistic, without direction, or understanding. Reality, life, is not mechanistic. We are the recipients of innate principles with the constant potential to experience those principles in action (Archimedes et al). Dogmas, ideologies, are the restrictive practices used to blur the recognition of principles operating to a level that understanding of that common and constant activity is virtually denied. Our natural capital (principles) is degraded to the point that their factual evidence is reduced even to the point that they are categorised as a ‘perceptual illusion”. We can trust facts 2+2=4. Simplicity has its own majesty. Anything circumscribed by reason requires control of our emotions. To comprehend the material world, bring it alive, and give it credibility, the recognition of its implicit principles is paramount. We need to construct the language that provides evidence of that Reality. Any philosophical theory of ‘mind’ that will deny the evident structure of solid objects, is misinformed by the injection of that mythical entity (mind) that determines that seeing solid objects is a ‘perceptual illusion’. That form of determination is singularly narcissistic, empowered by the self-induced threat that venturing into a ‘materialistic’ world is a loss of that illusory self, and all the belief systems it has constructed to protect it. The erroneous established view that not addressing ‘materialism’ as a profound reality, and as only a ‘perceptual illusion’, is paradoxically compounded by the belief that that form of illusion is implicit in every other human view available, thus it then makes its own sense, form and justification to the illusion! The evident question we must ask, ‘how does a ‘mind’ conclude that ‘immaterialism’ exists universally? Surely it is a simple but massive contradiction in terms. If there is nothing there but ‘perceptual illusion’, how can you attribute it to other ‘minds’. The oxymoronic effect of narcissism is that it is the very denial of ‘who we are’. Man is not composed of an overwhelming self-love. That mythical embedded belief cannot consider the possibility of underlying principles that are the real life force of Man in his relationship with Reality. The truth of that, is that humankind (in spite of itself), evolves towards its own Reality. The only human values that exist, lie in Man’s recognition of the principles involved that provide human direction. Our ‘material brain’ is a product of Nature's evolutionary process, and has innate within it the same principles that exist in all matter. That ‘which is’, is the truth, and our brain evolves to process that at every level, and we constantly manifest that in every action we take – whether we like it or not. The fundamental similarities between human beings is that we are not only evidently human, but that we also function and construct societies that we recognize as beneficial to our immediate well-being. All social function is determined by our brain capacity and its ability to postulate the relationship it has with Universal principles. = Illusion. = Considering that we can contradict things is an illusion. We can never contradict the truth. We do not have the ability to create proprietary constructs of reality. That ‘which is’, can only make its basic properties transparent to us through direct experience. Imaginary concepts must in the end conform to a measurable construct that we can identify. Within the structure of any philosophical theory of ‘knowledge’ it must contain the basic elements of truth at all times, or there is nothing!! To say that ‘everything is’ is motivated by pure reason experience as an objective, and subjective reality and as an axiomatic grammatical premise that no amount of mental acrobatics can deny. We can only deal with ‘something’, whatever it may be. There is no metaphysical construct that can provide evidence that ‘nothing’ exists, outside a mythical mind. Explaining experience beyond ‘thought’ processes requires a definitive language that deals with the reality itself. We all Know. It is innate. The ‘difference’ between us is only the measure of the knowledge that is made manifest, and that knowledge continually proliferates. The ‘individual perspective’, and the illusory ‘I’ which dominates, is the barrier to any relation to ‘what is’, and the malady of never experiencing the truth directly!   Truth, knowledge, agreement are the abundant and embedded Absolutes that form the structure of human evolution. That we constantly utilise and improve on their use is evidence of their reality, and the material transparency within every social structure. The survival and proliferation of such realities should be the evidence to establish that ‘that which is’ is Absolute. When we focus our ‘perspective’, opinion, or a hypothetical consideration of a space, time, or identity to question a Universally accepted fact, it is hardly a categorical argument to dismiss that which is true as nonsensical. Any denial that 2+2=4 is a fundamental truth hardly takes into account that the reality of such basics are vital to the success of higher mathematics. Unless the basics are continually correct, and evidently so, then no correct solutions could evolve. We know that within any basic structural ‘use’ that the calculation is correct. We commonly accept its correctness as an embedded reality. All forms of lower or higher mathematics would have the axiomatic principle of ‘correctness’ as their basis to extend from. Also, they would have as an axiom that the reverse is true. The 2+2=4 is, in its reality, the epitome of balance and construction. The 2+2 reality forms its correct conclusion when the principles of mathematics are propounded and they conform to transparent truth and arrive at 4. Only when it ‘adds up’, does it become a truth that we all recognize. Our greatest ignorance is taking for granted the proliferation of such truths through an ideological blinkered perspective. Because truth takes a commonplace form it is no less fundamental. Unless there is correct knowledge as to the existence of fundamental truth, that ‘which is’, goes unrecognised. That form of truth must be applicable to all. Truth exists in everything – it is an evidential reality. Searching for an esoteric truth is chasing shadows. Every truth is a ‘necessary factor’, and fundamental to our existence. Because of the imposed limited perspectives (via education, ideology, beliefs) that which is evidently true, and transparent, is delegated to a position of simple practicality with conditions placed on it which further deletes its substance, and we have the awful predilection of conforming to the attempted destruction of that which is true. Do we have a problem with seeing something, which is correct, as also being true? All truths are fundamental. They are not subject to attempted denial because of any diminished realisation at any point in time. Where there is reasonable evidence of balance, equity, and agreement we can conclude that a truth exists. Once innate information of that truth becomes transparent, it becomes an embedded useful human utility that must have some measure of fundamental truth as their starting point. From any common-sense, or ‘more reasonable’ position, it would be more productive to view reality as possessing at every level the same innate values or principles consistent with our ability to measure, or recognize them. To view reality as having ‘different’, or antagonistic properties, is simply a misguided view of ‘what is’.  That form of perspective is counter productive when it attempts to establish mythical dichotomies as realities in their own right. When the reality of principles are made transparent, we can then ‘more reasonably’ make use of them to further their basic existence. Here we use reason to exemplify their necessary function, and once established it becomes (if necessary), ‘more reasonable’ to locate them in all things. The dematerialization of any object through the practice of ‘perceptual illusion’ is an attempt to deny the reality that exists. Where perceptual illusions are concerned, innate direct communion with that which is, suspends the effect of such illusions. All the properties in a chair are recognized as the reality that exists. That is materialism. A chair does have the principles of form, design, structure, colour, substance etc.  However it is analysed – it is a quantifiable reality.   = Human representation. When we understand the validity and existence of principles in all things, it is easy to understand that ideological dogmas are never the foundation for real knowledge, or that direct experience of ‘what is’. Our real perspective is not some individualistic experience that confines us, it is that expanse in which we exist that offers us the view of that expanse. Everyone has the potential to go beyond their ‘apparent’ human perspective limitations. Shifting our sense of perception toward that which is basic, paradoxically extends the experience of that which is true. Let general knowledge be directed toward the performance that identifies the measure of principles that are enacted. Therein lies the production of knowledge that offers a sustainable growth of that vital universal aspect of knowledge, where, reason and truth, can prevail. Any correct definition is language itself, opening the door to that reality experience which is critical. Only when we know and experience that the same reality (with all its principles intact) exists for all of us can we then recognize the mythical distinctions that are taken as being real. The majestic experience of that reality goes well beyond historical beliefs. Exploring simple ‘necessities’ is not based on any sacred text, but the privilege of recognizing a sensible evolutionary path through life. Whatever may be in the future, is implicit in the material world now, and it has always been so. gbqbjw8itaj7gx0nvteljfzbiwtmwcn 2410908 2410907 2022-08-02T05:41:51Z Hamish84 1362807 Added media wikitext text/x-wiki = <ref>'''William Shakespeare''' (bapt. 26 April 1564 – 23 April 1616</ref>A message. = What humanity needs is not any individual approach but a governance powerful body of excellence that has modern technology, knowledge, and freedom they can use to disseminate clear information. Clear information about a new language structure of absolutes beneficial to an International forum, and eventually to reach a Universal status.Its benefits reach toward conceptual language on a planet that speaks more than 7,000 languages. No matter the language spoken the concepts of: Air - Food - Water are recognised. The overall development of conceptual language can only be beneficial and will be as appropriate to Absolutes definitions. Reaching for the Stars might show us the way! Universal Language of Absolutes (A very grand title but it took many years to explain its value) = Our history. = Born in Scotland in 1927 left school at 14 years of age. Married at 21 years of age and we had two children. We emigrated to New Zealand in 1953 and lived there for approximately thirty years. During our stay there I did a Liberal Studies Course at Canterbury University Christchurch and graduated. We have since had ten books published through Google books on the subject matter at hand and my wife Jean Caldwell McMillan is the co-author to most of the work presented here. My wife Jean was an avid reader of many works on philosophy and psychology. She was influenced by the works of Erich Fromm. Jean died 9th of January 2011. To refresh the original purpose of  our  earlier writings my wife and I  went on an odyssey  looking for any data, ancient or otherwise, on human consciousness, specifically related to Alzheimer’s disease. Now at 95 years of age (well past my used by date) it may well be that I am a candidate with a focus on my own pending dementia. If so, then the theory and the method I now write about is holding it at bay. To address the health of my mind in this way could be the catalyst that retains its own functional activity. The creation of a  semantic template is well documented below. No semantic definition of absolutes or principles can be ill-defined. They are always interconnected, interdependent and infinite. Each configuration constructed by anyone has meaning particular to them, although its value is universal. That  is why it is never personal property! [[File:JimNJean.png|center|thumb|228x228px|Jim and Jean]] = The Beginning. = "The road's half traveled when you know the way" [[File:Tree lined path - geograph.org.uk - 2269906.jpg|center|thumb]] Oxford dictionary definitions: Principle: "A fundamental truth used as a basis of reasoning". Absolute: " Complete - Entire - Perfect - Pure. These particular dictionary given definitions offers us guidelines to ‘existing conditions’ necessary for complementary understanding, and experience. We can only examine that which is real, basic evidence, that is fundamentally true, and we must ‘use’ it, to establish that which is reasonable. The general consensus is that there are no Absolutes. The following material is set out to show the very reverse is true and that everything that is is Absolute. Establish that there are no dichotomies that will leave the primary terms alone to create a semantic template. There are no dichotomies. Mythical dichotomies distort Reality. Everything is: The computer you use today has always existed, it is the arrangement of particles that have materialized it. The subject matter "Universal Language of Absolutes' is promoted to provide a new understanding of spoken language. This understanding was initially constructed by the cognitive experiences of both my wife and myself many years ago. Just like the principle of a jigsaw puzzle, meaning lies significantly in the fact that all pieces of the puzzle are interdependent and interconnected. When completed they provide a picture of the whole. We have endeavoured to produce a picture of the evolutionary process of language in human history because the evolution of language prefixes all modes of thought in human culture. The material directs the reader towards a new view that all that evolves is in a vertical direction, not the linear direction commonly understood. Human consciousness is of itself the phenomenon of evolution and to recognize its existence is part of the process. Shakespeare expressed this succinctly through the voice of Juliet who proclaimed, “a rose by any other name- would smell as sweet.” [[File:Comestible rose in the Laquenexy orchard garden, Moselle, France (01).jpg|thumb|center|237x237px]] === Conceptual language. === My wife and I recognized how profound the extension of this observation would mean conceptually. Of all the languages spoken on this planet, it would be fair to say that all of them would contain the properties of, air, food, and water conceptually, etc. This is a form of consciousness equality that is available to us all. It points to the reality of our constant relation to each other and our existence. We can never exist in a world of individuality, but only in relation to the consciousness of one another. Consequently, that exceptional experience can only be shared superficially. We cannot ‘know’ any other life experiences other than our own introspection. <gallery> File:Einstein 1921 by F Schmutzer - restoration.jpg </gallery>" ''Albert Einstein 1921. We experience our thoughts and feelings as something separate from the rest. A kind of optical delusion of consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us."'' ''Albert Einstein, in One Home, One Family, One Future,p.99'' Einstein came very close. In reality, every human being has a backpack from the day they were conceived. In the backpack every experience in their mother’s womb is experienced. At birth and throughout their lives, everything that happens to them in life is registered and creates their personality. That life with all its experiences can never be known to anyone else, consequently, we can never “know” another person. It creates equality of consciousness that we must understand. We can know details about a person, but that is all. That life is sacrosanct. Who we are really goes beyond normal human experience and into the realm of the Absolute. Werner Karl Heisenberg (physicist). The Heisenberg Uncertainty Principle. As a young layman with no knowledge of Heisenberg but interested in principles it seemed to me that the Uncertainty Principle was just a contradiction in terms. In later years I found that Heraclitus describes life as being in a state of flux a replica of the Uncertainty Principle which in fact can be defined as an absolute state. Within the context of knowing who we are and the backpack we carry our life in, we can never know each individual life as that life experience is singularly their own and sacrosanct. It now seems that the Heisenberg Uncertainty Principle can fall into the category of being an Absolute. Evolution proceeds in advance of our need to evolve. In our pure active state, we are.There is no static end (an abomination) - only beginning. As we cannot know what tomorrow will bring, living with expectations is rather futile. Nature has its own agenda. Zen Koan recorded 1228: 'An instant realization sees endless time. Endless time is as one moment. When one comprehends the endless moment He, or she, realizes the person who is seeing it.' We do not own Space. We do not own Time We do not own Energy We do not own Matter [[File:Universum.jpg|center|thumb]] "What we call the beginning is often the end And to make an end is to make a beginning. The end is where we start from We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know the place for the first time" T.S Eliot = Everything is in scale. = The present moment is the point in which Eternity has placed us – we all live in that moment, and whether we like it or not, we exist in it, experience it, have knowledge of it, and we all share it, measure by measure. There are no dichotomies. Illusion is a measure of Reality, as Stupidity is a measure of Intelligence If one keeps measuring illusion it is an attempted downward spiral to nothingness.There is no opposite to Reality – that illusion is a measure of Reality. There is no such thing as ‘nothing’ in the elemental construction of Homo sapiens. All the innate ‘something’s’ are the fundamentals of our being human and all our experiences. The correct use, and understanding of who we are, is an extension therefrom. Does it require any interpretation on anyones part to say ‘we are? Any attempted denial of that statement would be perverse use of the language, and delusional. === Try saying ‘we are not” === ‘We are’ is the foundation of all affirmation, and within that spectrum, we can know, and be. ‘We are’ is self-evident Truth. We can neither know, nor experience what isn’t. Eternity is the here and now, that is why it is possible to explain the experience of Eternity. Nothing is ever lost in Time. We are located in expansion. ‘twas a moment’s pause,- All that took place within me came and went As in a moment; yet with Time it dwells, And grateful memory, as a thing divine. Wordsworth Prelude, Book V111 We already know – the ''basic'' condition that must exist for us to re-cognise. It is at that moment of pure affirmation, when all that is, is manifest. Unless there was mutual identity we could not know anything. It is why we are urged to evacuate the Platonic cave. Sadly most prefer the shadows rather than confront who they are. We already know – the basic condition that must exist for us to recognise. It is at that moment of pure affirmation, when all that is, is manifest. == Lost Shadows == The shadows move Lost in confusion Lost in despair Imagination shrouds the real Looking back Looking forward Is this the Centre? James Brines. = Basic Principles. = We are all in the business of living and attempting to understand the principles involved in that human process up to the end of life. The implicit principles necessary for life eventually disappear and all measurable criteria pronounce a body to be devoid of life. Throughout historical agreement we know what that means, and we act accordingly on *common knowledge*. We know that dying is a necessary factor of life. It is a Natural law that if we live - we also die. Natural law is Universal, for us to *know* that a body lives; we also *know* that a body’s life will end. Albeit that reports tell us that today millions of people die of disease, starvation, wars, we of necessity accept that as the ongoing reality because again we are universally connected and know the results of such carnage. Because it is in our realm of common knowledge we have graveyards, crematoriums, undertakers, doctors who pronounce bodies to have died. We understand the consequences of leaving such bodies unburied, the diseases that would prevail. Again, please explore the definition of principle (Universal principle) and try to go to the limitations of the definition without using mythical dichotomies. Principle: A fundamental truth used as a basis of reasoning. All of these questions are based on singularity (the Cartesian dogma)… If Descartes had only introduced inclusion into his musings (they were taken as conservative singularity) he may have realized the difficulty of addressing thought as reality. He then may have quite easily concluded that universally - *We are!!* That pronouncement is inclusive, and conclusive in every way, *we are - and we know!!* Because my knowledge is not a private, personal piece of property concerning principles, but Universal (Archimedes et al), then that innate knowledge has completeness we can share. Whatever identical resonance we may be able to share (concerning completeness), that can only be accomplished by understanding the principles involved and their constant relationship to each of us. Clearly the plethora of present and past discoveries establishes the existence of that which is fundamentally true, and the foundation for law. The principles are established, and always have been, we are in the business of making them transparent and complete. However tenuous the link we are all connected through communication, the air we breathe, the ground we walk on, the universe we live in, the common principles we live by. We all must have sustenance to survive, or we will not live. (See above) <nowiki>*</nowiki>There is nothing else to experience.* When we actively explore the reality of anything, all principles involved in that exploration are complementary, and honest, and we understand the wasteful divisive mythology that people attempt to attach to them. We cannot */partially know/* the truth, it must be complete. Dichotomies attempt to deny the existence of truth, and are misleading. How do we more reasonably completely know anything? === Naturaly. === How do we completely know? The complexity of language systems with contemporary usage requires new and creative structures to provide clear information. Internal and external reasoning capacities can only develop in concert with Man’s recognition of the principles that essentially form our lives. The inevitability of human consciousness rising beyond its historical beginnings posits a future outside our normal perceptions, and  a factual reality that points to the existence of new perceptions that are infinite.   It is natural to know when we are no longer trapped in any mythical ideologies that gives credence to dichotomies that stifle the recognition of simple principles. There was no cause for knowledge to be established - it has always existed.* Ask yourself - ‘how do you know to ask any questions at all?’ There is no hidden dimension or mystical world. The only philosophical reality is ‘that which is’. To access that we need a new structure to explore ‘knowledge’, a new transformational language. Real knowledge is not amassed information, nor is it the establishment of dogmas, isms, or mythical belief systems. Basic principles are the source and foundation of all /*knowledge*/e. Until that is recognized, extension from mythical sources only leads to a denial of one’s own senses. Trying to conceive of a contemporary world without principles is to posit a world without reason, or intelligence. The principle of pure knowledge could be said to exist in another dimension given the present state of human understanding. To progress that mistaken belief there would be strong support in the need for a comparative reality. Curiously it seems that philosophy (the seeking of knowledge) constantly discounts any knowledge that does not come within the sphere of established philosophy, and the comparative reality dictum. That consequence profoundly distils the purity of any experience and alienates the observer in their confrontation of that which is real. The measure of that ‘comparative reality’ knowledge bounded by dichotomies is so restrictive that it lies in a mythical dimension where denial of its very purpose is the order of the day. Evidently it will not allow doubt to undermine its own denials. A basic principle of Nature is /*knowledge*/ and it constantly communicates innately in every living structure. How to grow, develop, and disseminate. Knowledge is reciprocal truth that depends on our relationship and the recognition of principles operating. Knowledge (unlike information) is not stored in an individual box; it is ubiquitously manifest in everything we do. Knowledge is the experience of a positive reality, and its true construction is a dependable source of secure information (not to be misused). We daily have the opportunity of witnessing ‘knowledge’ in action as expressed by the ‘the principles of knowledge’, namely the principles themselves. The questions lie in a continuing mythical belief in a mystical unknown (the Cartesian stance) which because of its non-existence can never be known. It is a belief that is detrimental (because it attempts to deny all existing factual knowledge) to dealing with Nature and Reality and the fundamental necessity of our relationship with them. Making that relationship transparent is our basic obligation and the ongoing evolutionary activity. Although there is an obvious avoidance to address the definition of principle itself, it is a factor that must be paid attention to, to realize that ‘common knowledge’ is the only reality that exists. Amazingly although there is avoidance of principles - truth - reason etc, there is acceptance of the Cartesian dogma,. Paradoxically this determined acceptance of Descartes supports the reality of innate knowledge existing (I think - therefore I am) which establishes for him innate knowledge; however mistaken he is concerning the interpretation of his experience. = Leverage. = Long before I read of Archimedes and his various principle discoveries I was using the principle of leverage in a variety of ways, prying lids off boxes, moving articles with a lever well beyond my physical strength to do so without said lever, and I knew how to do it. Transferring that knowledge to a student or apprentice is relatively easy because innately they also /*know*/ how to do it. Every aspect of human industry uses the principle in a myriad of ways because it is our obligation to constantly progress the principle and confirm the constant utilization of knowledge. We wholeheartedly adopted Archimedes principles (et al) because we recognized their fundamental utility. It is preposterous to question the widespread /*factual knowledge*/ of all principles, more especially so when we cannot escape their ubiquitous daily existence in all our lives - Nature and Reality do not lie. One may abstain from admitting their existence. To do so is simply attempting the impossible, and is devoid of all reason. Real knowledge has been put into the realm of the mystical unknowable even to the point that knowledge practitioners go to the outer extremes and deny the gift of their natural senses. With their adherence to what they consider is knowledge they become captive automatons to any prescription for life that is expressed in that ’knowledge medium’, which then becomes the authority. When ’knowledge’ is addressed as having a collective source in Universal principles then we have the potential to experience its complete reality (microcosm - macrocosm) without any imaginary, or divisive comparative content. There is then a re-orientation process toward our true being and recognition of our own reality in relation to the natural processes we share. Real knowledge is elementary and Natural. We know, because that which is knowable is constantly expressed by the principles involved. We all share those principles and can correctly infer the most simple and obvious truths. All social life functions by our adherence to the implicit laws operating within them. Seeking experimental contradiction to a fact of life offers us nothing but proof which is the establishing of ‘complete knowledge’ however ‘more reasonably’ one wishes to extend the exercise. The construction of new philosophies must seek a mandate to fully explore the relationship between experience and innate knowledge as the foundation for pure knowledge to emerge. To repeat, knowledge or truth do not reside in any individual domain, nor are they the private possession of any human being. We know, because ’knowledge’ is an innate natural possession that we constantly share - otherwise civilization could not exist. == Knowledge: Evident facts about mutual standards that provide us with security. == We do not become human beings because we can ‘think’; we become more human because we learn to conform to the implicit principles in Nature and Reality. Denying them - denies our being. It is natural to know. There was no cause for knowledge to be established - it has always existed. Knowledge is an evolutionary process. Human beings developed from primitive innate instinctual knowledge to contemporary cultures. Some know more than others through experience, and make that knowledge transparent. Insistence on how we can ’completely’ know is an ephemeral philosophical question that attempts to deny that we can have ’knowledge’ at all, as you understand it. Knowing that we ’know’ the inherent completeness of everything through the existence of principles, is the natural catalyst to make ’that which is’ transparent. Heraclitus: "No man steps in the same river twice " He believed in the "Unity of Opposites (Absolutes). He cried for the needless unconsciousness of mankind.. “Exploration of a mythical dichotomy below for the purpose of establishing principles. Principles that are not a 'mind' construct, but the very essence of our being. Independence, is the curious and dangerous malady where humanity has lionized negative mythology in denial of its own reality. The human fundamental reality has at its base the simple natural law that we are dependent beings. From conception, the human embryo is entirely dependent on the health and well-being of its mother to provide it adequate healthy sustenance to enable its entry into the world. That form of innate dependability the human species carries with it throughout its spectrum of life. Every aspect of human activity is premised on the availability of air, food, and water without which the organism cannot survive (this would be an incontrovertible 'more reasonable' truism or an Absolute). In a perverse way, that which is our natural state has become the target for what appears to be open defiance of the laws that govern our behavior. When a basic premise is either used mistakenly, or deliberately, its consequences can be socially far reaching, for any deviation however far it is extended is a distortion of the truth, and a denial of who we are. The erroneous conclusion through exercising responsibility that we can confer independence to our actions has gained a distorting and ubiquitous influence which paradoxically undermines the very responsibility practiced. Within the context of being dependent we can correctly be responsible for our own actions but with the surety of knowledge that that responsibility is contingent on the measure of life giving forces available that we are dependent on. The mythical dichotomy 'independence' connotes with the myth of separateness, division, alienation, and the force of these particular myths is expressed in wars, genocide, criminality. Alarmingly the mythological term has become a residual in our lexicon and is used more widely with acclaim than its true counterpart. To uphold delusional 'independence' as a value to strive for erodes our human heritage by diverting useless energy toward a dubious goal, and consequences that leave us questioning our means of arrival. Sadly it is a loss of being with the paradoxical view that the energy expended will deliver up a personal reality. The cult of independent individuality with its mythical ideology based on personal intimacy is now taken for granted, which then passes into the acceptance of the spurious dichotomy as a tangible reality. This in turn disposes the adherents to discredit the very essence of their being, and in the process dehumanizes many cultures. The presumption of independent individuality leading hopefully to a superior future is in fact counter-productive to the purpose, and leads eventually to corrupt power, and subsequently the invention of immoral policies that continue the negative spiral, which in the end has no ethical base to extend from. The alienation from our substantial being creates inevitable tension anxiety, and the need to somehow relieve that anxiety with any artificial means available. = Responsibility. = Being responsible for any social activity would best be enacted with regard to the effect it will have (directly or indirectly) on the lives of those who are dependent on a beneficial outcome. To recognize with some significance the basic structure of our being in turn significantly increases the measure of our understanding of human relationships. Being dependent is not a mental construct choice - it is a state of being, and there can be no being-ness without at least one other being, there is then the possibility that the principle of true egalitarianism becomes the manifest reality. Being-ness can only be identified and expressed in relationship. This is why the cult of 'independence' is eventually so socially destructive, as it creates that alienation which attempts to deny each real human need, and leads to a depravation of honesty. Human relationships between children and adults where independence is the accepted norm is severely undermined when the educational process predominantly teaches an unnatural form of living (either intentionally or unintentionally). The educational process is then reduced to the adoption of a fiction, which in turn puts at risk any educational program. The effect of interpreting mythical dichotomies as described here is symptomatically ascribed to the existence of all other principles that govern life. The construction of any ethological debate should not be premised on comparative perspectives, based on human thought, but rather on the issues that we can recognize as being universally compatible, therein lies the common denominator point of extension. The focus of attention on comparative perspectives denies justice to 'what is'. To contemplate the 'right or wrong' of any circumstance is a deviation from the truth. The correctness of any debate (however minute the finding), is the justifiable extension, and the only true trajectory we are morally obliged to travel. Truth is not defined, nor experienced by comparison, but by 'what we are'. An orange is to an orange, what an orange is to an orange. To define correctly there should be careful and disciplined action toward establishing 'true factors' that we constantly use to promote reasonable standards. = Time. = In that moment of time, we have the potential to merge on an equal basis with the reality that exists, and to know what true interaction is. That is the point of 'direct experience'. It is then that we know the truth about ourselves and the beauty of this Universe which also reveals to us the folly of our present conditioning. In that experience, it becomes very clear that all so called cerebral activity has nothing to do with reality. The fundamental repository of our knowledge and relationship with life is our being-ness, which is not located inside a mental box to be analyzed, accepted, or discarded at will, but the very privileged natural gift of being. What some scholars deal with is the appearance of life prescribed by the illusion of comparative perspective which functions on the basis of dichotomous ideology. It is in effect a denial of our humanity to conclude that all things that exist [from our perspective] exist only in the mind, that is, they are purely notional. It compounds the denial of 'what is', and an extraordinary refusal to observe transparent life. It should be noted that there is ample contemporary exposure to the Cartesian doctrine, and in this regard, I would refer you to the works of Professor Gilbert Ryle, notably his publication The Concept of Mind. The heuristic principle applies throughout when establishing our connectedness with reality. It is only through our contact with reality that we can discover, and equate with the mutual structure of the principles that govern all existence. Have already noted that it is also a peculiar form of arrogance that presumes that life is only a notional existence beyond the boundaries of the 'mind in a box' assertion. It would be foolish of anyone to assert that ice cream has a cold smooth satisfying texture and taste on a warm summers day unless they had actually experienced it, preferably on more than one occasion. For anyone who has never enjoyed that experience, it would be foolish of them to discount the very numerous accounts of such an experience that is available just because they had not been party to that event. From a logical point of view, given the avowed experiences of ice cream eaters, we could reasonably ascertain the validity of each experience by documenting their separate opinions. Each participant would have 'direct experience' in the consumption of ice cream, which at that point in time has the potential for that participant to experience the full measure of that factual reality. We have the natural capacity to experience coldness, smoothness, which equates with the reality that exists, and the potential for those realities to unify. It is not a question of how to get outside of our minds (mind in a box position); we are constantly outside our so-called minds performing acts of transparency throughout our entire existence. The belief that our constant engagement with reality can never be based on a rational acceptance of 'what is', is at the least, very sad. The Platonic Cave shadows are a metaphor for the (mind in a box) syndrome. The need to reach simple, and obvious conclusions and accept them for the reality they are provides the opportunity to engage the complete reality of the moment. It is indeed going too far beyond the reality of the moment searching for philosophical profoundness which does not exist, that fails to establish the constant principles that always operate. Pure principles are not amorphous shadowy ideals; they are represented in everything that exists. The only way we can equate our inner knowledge of reality is through direct experience of its truth. Within that context then, life cannot take on a notional existence but is an existence that is very real, and that we continually share through our innate knowledge. That our so-called minds are defined by comparison - incompleteness - dualism would have extreme difficulty in pursuing the proposition that we are defined by our direct relationship with reality which is expressed in our innate ability to directly interact with 'what is'. The reality of interconnection, and interaction, are not idealistic concepts of a notional nature, but actual and consistent transparent realities. We do not live in a shadowy world that is hidden from our direct experience, but we are constantly engaged in the process of life, and we do not have the right, nor the choice, to deny it. The man whose book is filled with quotations has been said to creep along the shore of authors as if he were afraid to trust himself to the free compass of reasoning. I would rather defend such authors by a different allusion and ask whether honey is the worse for being gathered from many flowers. Anonymous, quoted in Tryon Edwards (1853) The World’s Laconics: Or, The Best Thoughts of the Best Authors. p. 232 == Create your own semantic template. == That will consist of an alphabetical list of Absolutes that are all interdependent and interconnected. Their unifying construction creates a ‘new’ consciousness meaning. That ‘meaning’ is yours specifically. The greatest knowledge you can ever have is your own! That meaning also creates its own moral construction that cannot be misused . The semantic template is available to everyone, and its dissemination is our responsibility. “Consider that the language structure, concepts, and definitions now in use no longer always deliver, accurate, reasonable, and responsible information. Indeed at times, they can be quite ambiguous. The statement ‘mutual agreement’, and its physical manifestation in whatever form, is its own dialectic, and will carry within it all other principles necessary for the activity to proceed. Given the Socratic assertion that if something is true then it cannot lead to false consequences no matter how circular any argument may be. Then extrapolating the statement into extended definitions must only lead to a better understanding of the inherent truths available. This can promote recognition of the underlying essence of all things, which can become more real than our conventional understanding of Reality. There is a contemporary need to find new definitions, new paradigms to explore the concepts that govern our existence. Where a circular argument is based on an untruth, then it cannot lead to a truth. The reverse of that is that when the truth is established, it cannot be denied. Establishing ‘mutual agreement’ as a center from which we can reach out for extended knowledge in its ever-evolving radius, is not a limitation, or a stop, it is only a beginning! When any concept is truly established the superficial exemplification ceases to dominate, and we can truly experience the apparent essence of ‘what is’. Paradoxically to resource innate knowledge, we must recognize our profound ignorance of Nature, and Reality. Completeness does not lie in individuality. This is an extreme form of monastic expectation. There can be no individuality (or completeness) unless there is at least one other individual. This is the true foundation of completeness. Whenever we are privileged to experience that instantaneous essence of one other, then we know in that moment that we experience ourselves. It is complete complementation with the knowledge paradoxically that it is an endless process. There are many paradoxes we live within that strain our conventional views of what is ‘more reasonable’. Any true relationship experience is not based on a causality premise, but on an experience that is necessarily complementation. Individuality in terms of completeness is a fundamental circular argument back to one, which in its form of denial excludes any form of reasonable argument to the contrary. It is a non sequitur, which denies the pressure of facts that are in abundance, despite the evidence of their reality. To observe ‘mutual agreement’ is looking at things as they are. True observation of ‘mutual agreement’ in action is observing essence transparency – it is knowing ‘who we are’. That form of recognition is essence duplication. The proposition that we can observe the Truth may well be the highest attainment of Realities properties, for Truth is knowledge. Consider the hypothesis of a human entity (an individual) being born in a black space with no other form of life in that environment. How could there be Agreement? How could there be Intelligence? How could there be Understanding? How could there be Recognition? How could there be Love? How could there be Law? How could there be Reason? All of the above principles are the transparent manifestation of Nature and Realities properties that are constantly evolving. They are ethical imperatives, and we have developed the positive properties of language to establish them for our use. We can only be defined through relationship principles for they offer us the best hope to recognize the factors that lead to complementation. There is a fundamental need to grasp simple common-sense essentials. The Here and Now is not a temporary transitional time phase that we move in and out of. It is a constant certainty that is essential to recognize, so that our focus of attention has a foundation. Centrism can imply a fixation, which also implies vulnerability, which can be perfectly true if it does not lead to extension. To understand who we are, it is essential that we recognize and become aware of the very principles that we operate from. They encapsulate all the measure of any human societies ethics, morals, and laws, which is a continuous evolutionary educational process within which the realization of its total essence is always available. To use the doctrine that reason is a reliable tool to discover Truth – therefore ‘mutual agreement’ in the context ‘correct information’ translates to the Truth to reason! Evolution is a constant dynamic process. The human phenomena of ‘who we are’ is only understood in our union with each other, and ‘what is’. The paradox again is that there never is any separation. Separation is a mythical non-existent. The principles that are our necessities have continuous expansion properties that as humans we are privileged to assist their propagation. The human constellation in its evolutionary march must use these fundamental principles to ensure continuity. To maintain coherence and consistency our source is centered in the principles and factors that we have interpreted from our association with Nature, and Reality. Whatever we write that is of any consequence, or at any other time, is written with the hope that stronger interpreters than us overtake what we present. To ‘see’ Reality as we have seen, and be intoxicated by it, as we have been, will ensure its progression.” == Discovery == The consciousness whole is the sum of all its parts and experiences. As we are all on an evolutionary path, our life and knowledge hopefully develop in the right direction. The exploration took us through a plethora of data and opinions about reality from authorities on science, religion, philosophy and metaphysics. Nowhere could we find a definitive conclusive argument, or agreement, that met our needs. For us, the question came down to “Is there anything at all that provides some form of construction, and certainty?” Something that has its own inherent ethical standards. The alternative proposition to that is a nihilistic “nothingness”. A pathological proposition that makes no sense. Heraclitus’s “unity of opposites” seemed the most promising. Our understanding of that now made dichotomies a semantic illusion. If achieved in a mindful way it is the act of uniting them, providing a conscious correct experience of ‘what is’. We live our lives with secondary knowledge that everything that is – is always interconnected and interdependent. Yet our illusory experience belies that form of knowledge. It is here that we understood Heraclitus and his “unity of opposites”. Mentally uniting opposites replaces the existing illusion of their existence – there are no dichotomies! Once the illusion is gone a new solution manifests that is peculiar to the mindfulness operator, and belongs to a higher form of consciousness. Heraclitus was known as “The Weeping Philosopher”. He wept for the needless unconsciousness of mankind when the ‘unity of opposites” was always available. He was also known as Heraclitus The Obscure. A title we suspect that came about because the successful conclusion to uniting opposites and replacing the illusion, opened a door to a new dimension. Semantic description at this time may not have been available. This brings us to the ancient Yin and Yang symbol of the ‘unity of opposites’. As separate entities in Chinese philosophy, they are complementary, and in fusion they represent the whole. So as dichotomies they don’t exist. The whole is the elemental answer to any fusion of opposites, whatever that may be. Symbolize a line as being infinite in the sense that any line can be categorised as being infinite. Apply an infinite number of points in any line. Intersect any line through any point by another line then we have a specific identifiable point at the intersection, which at that point in time has an infinite quality, yet constant and complete. Any such point has Matter, Energy, Space, and Time, the epitome of the microcosm. We may locate a Reality point that establishes the Truth. Conventional mechanical ‘thought processes’ deal with dichotomies that are based on a comparative perspective ideology, and consequently, skew any real experience of that which is real. We must use correct ‘measuring sticks’ to secure proper standards, but from the point of view that there is a belief in dichotomies, it will always be a compelling argument that aspects of reality can be contradictory. The element of denial within human historical memory accumulates to establishing dichotomies as being real. We are defined not by how ‘different’ we are, but by our commonality of existence. When we locate that Reality point we will then know that the definition in itself has a whole, and complete explanation of ‘reason” in all possible senses. All the reality we can deal with is here, and now. There is no possibility that ‘infinite regress’ (an imagined reality) is any part of our immediate experience. Infinite regress through thought processes, deals with questionable imponderables. It is a descending spiral, which further removes one from reality, which only produces illusion, and correct meanings are always deferred. It is making a holy virtue out of complexity. The epitome of completeness is the active realization of the operational principle. Conclusion: A brick – a house. Each complete in themselves. A house is not composed of one single brick, but each brick in its composition is complete, and whole in itself in that it has matter, energy, space, and time. In that context, it is a microscopic whole which has implicit within it the macrocosmic whole, a house. We cannot define that which isn’t. We constantly use negative dichotomous terms in language, which are in essence factually indefinable, and therefore non-existent, but they are used as though we can support a view as to their existence. At this time we constantly use mythical concepts as though they had real substance. That erroneous belief in turn diminishes that which is real and compounds the problem of recognition of Reality. The flat earth society no doubt had to be persuaded of the mythical nature of their beliefs. This dictates that we must research ‘that which is” to achieve an understanding that supports that reality. Separation is the mythical measure we use in an attempt to justify the real identity of either ‘relationship’ or ‘completion’, but it has no substance in fact.That we are connected, that we are related, that we are communicating, that we agree that ‘mutual agreement’ exists, all of these factors fall into the category of ‘that which is. There is nothing that is real that is not immediately available to us, there is ‘mutual agreement’. Attempting to view true relationships as having a necessary separation link, or dichotomy is a clear misunderstanding of the nature and completeness of all that we are related to. == Connectedness. == A relationship is defined as we are by the measure of contact (especially homo sapiens) that is apparent. It would be true to say that I have a measure of relationship with everyone who reads this material. No matter how tenuous the link we have a measure of relationship with all life – we are related! Depending on the strength of that relationship defines ‘who we are’. ‘Who we are’ is not defined by any spurious separation from life, quite simply because we cannot be separated from it, we are engaged in it at every moment in time. Any attempt to establish ‘separation’ as a reality is an attempt to deny ‘who we are’, and another exercise in futility! Again artificial interioralisation of concepts or principles leads only to a denial of their external reality. We are all connected by the very simple fact that we all exist on this planet. It is a very simple axiom that all life on this planet is supported by the conditional properties this planet provides. It is also a very simple, and more reasonable axiom to conclude that no matter how tenuous the link that all life in this regard has very concrete and definitive forms of relationship. We all must breathe, we all must eat, and we all must drink, and if you need any further certainty of ‘completion’ relationships, we certainly, all must die! [[File:Wikimedia|thumb|center|]Arcimedes] To set in qualifications from the premise that there is a ‘separate mind’ (a kind of Platonic cave) to get outside of. This premise precludes either in part or in whole the evidence and experience of Nature, and Reality, within which our beingness is located. It would all be beyond our grasp if indeed our conventional concepts of consciousness was adhered to, which in effect attempts to deny us that direct ‘relationship’ to ‘what is’, and the completeness of that experience. Knowing or being, despite solipsistic theories to the contrary, does not exist in penetrating one other mind, but in the democratic recognition that we know and have our being in relationship, and the mutual, and natural convergence of everything there is. Homo sapiens (race, color, or creed aside) necessarily conduct themselves in ways that extend recognition, and understanding at every level, without the constant need of ‘completion’ recognition that is inherent in all our interrelated actions. The notion of completion may be beyond what you call your ‘conscious grasp’, and therein I suspect lays the difficulty in recognition. The flat earth society eventually moved on to a realization that their visionary scope was shrunken, and severely limited. They were deprived of a planetary (never mind a universal) relationship that one can only imagine severely curtailed the very expansion of consciousness necessary for humanity to progress. We have evolved some little way because of our understanding of the natural relationship. The centered in the mind condition - which connotes with the separation ideology - screams to be released from that mythology, and engage itself directly in real relationship with everything that is. = Relationship. = Separation is the mythical tool we attempt to use to maintain a false continuity of an imaginary individuality that does not exist. The taking for granted conditional mythologies (the flat earth society) engage the victims in what can best be described as serious problems in recognizing the very limitations that restrict their development. We must learn to view wholes, which equates with viewing ‘completeness’. The whole is greater than the sum of its parts, but the parts are not necessarily separate conceptual parts. We can ‘see’ the whole when we are able to identify the factors that constitute their existence as a complete reality. That which is complete in Law = Agreements that produce secure and dependent outcomes. We know in essence the concept of ‘completeness’, and we demonstrate the evidence in myriad ways. Each act is a microcosm of the whole – view from the other end of the telescope! In the traffic analogy the driver, and all other drivers, conform to the law by driving off when the light turns green. There is an implicit agreement about the value of traffic laws, and traffic lights that control the flow of traffic. At that moment there is a complete relationship understanding of those values. The ‘complete’ or ‘wholesome’ activity of motorists waiting at traffic lights for the green signal to go, and they then move off, validates all the factors implicit within the properties of ‘mutual agreement’. Throw a ball from one side of the room to the other. The ‘whole’ or complete traverse of the ball is the instant it leaves your hand until it arrives at the other side of the room. You have already given credence to the concept of ‘mutual agreement’ as a reality. When there is a correct definition made in terms of ‘necessary factors’, then it has implicit within it the concepts of ‘completeness’ or ‘wholeness’ within the measure in which it is used. Whatever we communicate for the benefit of future generations should not be based on mythical assumptions, but should be based on necessary factors. It is ordinary life that portrays all the dignity, honor, and the complementary wonder of the human species. We are here – we are present – we are communicating. We have an obligation (which we necessarily fulfill) to make transparent the basic principles that govern our existence. That, which is factual, provides us with a correct motive for behavior, and we do a disservice to Reality when we attempt to deny it. We exist and live in a world where acts of ‘completeness’ expressed in one form as ‘mutual agreement’, are continually enacted. It is the form of expression, and continuance of processes that we constantly use to arrive at reasonable solutions, and we employ factors that are necessary to provide us with a clear, and unambiguous understanding. They motivate reasonable behavior toward activity that we can accept as being a logical process. == Natural Experience: == No form of life can exist in and of itself, it is brought into existence through a relationship with its own environment, or its species. The obvious egotistical monistic nature of oneness (if there was such an entity) could not leave any room for the realization of anything that might disturb its comfort zone. There is no real knowledge where any concentration is on the “I am” syndrome. "No man is an island, entire of itself; Every man is a piece of the continent, a part of the main." - John Donne, 'No Man Is An Island, Meditation XVII - Devotion Upon Emergent Occasion. “We are’ is an inclusive affirmation that deals with “what is’. There is no constructive dialogue, no real understanding, without a relationship. Based on personal experience, we are a distinct, and unique species born of Nature and Reality that has combined to provide us with the innate ability to recognize the very properties that created us, and utilize them through an evolutionary process toward ever-increasing transparency. That transparency can only become available through a matching process between innate knowledge, and the reality we share, a reality that is our heritage. We like all other forms of life are the product of Nature, and subject to its laws, and principles. Necessity created a language that evaluated Reality, and provided us with guidelines to emulate its constant properties. The measure, and quality of knowledge is dependent on the realization of ‘what is’. The crux of correct knowledge is to know the base that we function from. The principles that are implicit within Nature, and Reality we have now translated through the evolved language systems with symbols and definitions that we now use to share the experience. When principles are fundamentally true and recognized for what they are, misguided belief systems will evaporate. To ask what is the source of the principles we present is ipso facto to ask what is the source of Nature, and Reality, and we repeat, that is an exercise in futility, but that does not mean that we cannot recognize that which is natural to us, and express it, as best we can through language. The experience of ‘who we are’ is the Ultimate transparency that transcends all doubt, or denial. We can know with an understanding that is pure and indisputable, that is the motivational drive for evolutionary continuance. To understand who we are we must address Nature, and Reality, and ask ‘what is’ Here, and Now, with an understanding of the dichotomies that exist in language. The ‘Eureka’ moments, epiphanies, enlightenments, etc, are all evolutionary evidence of who we are, and when we can translate them into principles, and concepts, then the assertions of an Archimedes (and many others) are recognized, and properly used. Through Nature, and Reality we can establish what Truth is! Is it true to say that most people conform to the rules of the road? It would be more reasonable to assert that the answer is yes! Consequently, we can say that we have ‘Mutual agreement’, and ‘Co-operative Understanding’ as observable realities. There is no conceptual source through Time, or history where there is an end. There is only ‘beginning’. Here and Now is always the ‘beginning’, and a more reasonable platform to explore than any exploration into the past concerning our true identity. Contemporary terms like Absolute, Complete, End, we use to match our conditional understanding of ‘what is’. When we use contemporary conditional language to address concepts like Truth, Knowledge, Understanding, they are limited by the measure of our progression, but we use them all as stepping-stones. Language is a constantly evolving process. When we agree that there is ‘some certainty’ and ‘limited knowledge’ you have agreed to the concepts of certainty and knowledge as factors that are part of our natural reality. All of us function within the framework of certainty, and knowledge, to some degree. Given that we agree to their existence, these are the factors that can lead us to the experience of ‘who we are’. They are a part of us that can lead us to recognition of ‘what is’, and make a transparent reality of the very things we do on a daily basis. We do not need absolute and certain knowledge to perform everyday tasks, but those performances are structured contemporarily because of our advanced understanding of the things we do, based on our own innate reality. To honestly perceive the consistency of ‘what is’ (to be interdependent, and mutually connected) in interaction, can and does promote the visible reality of ‘who we are’. That visible evidence translates into the knowledge of our complete presence. We know with certainty that our beingness is of pure essence, and from that experience, we are obliged to formulate, as best we can, the structures that are responsible for making that transparent (witness the explosion of human progression, without the necessity in Time of experiencing fully ‘who we are’). To take a more reasonable stand please observe the multiplicity of human action where interdependence, and connection naturally proliferate. To realize that we are ‘interdependent, and mutually connected’ is the realization of a consistent fundamental truth – ‘what is’. Through identifying ‘what is’ as an internal reality we can make transparent the factors that are our natural construct. It is only through sharing this reality that we recognize it!! These factors are not based on any ideology, belief systems, or opinions; they are composed of the Reality that is available to us all. We are unique in that we have the opportunity to be able to use their value in the manner that is implicit within their structure. That use is evidence of our understanding of Reality. What could have happened without the assertion that traffic lights are a safe way to control crossroads, or the assertion that the rules of the road are beneficial for our safety? Any true experience, epiphany, enlightenment, etc, of ‘who we are’ provides fundamental, and indisputable knowledge of that Reality. Applying the recognized principles provided by Nature, and Reality consistently advances the evolutionary process, hence we have Science, Philosophy, Religion, Education, Art, and Law. When there is a Pure Realization of who ‘we are’ through relationship recognition, it is unquestionably the recognition of the encapsulated, and innate principles we all share, and there is no place for the dissolving of another Real identity through that recognition. Indeed it is a privileged insight into the epitome of purity. Nature and Reality can give us direction and guidance to our human existence, and we repeat, it is an exercise in futility to seek any cause to their beginnings. We have proffered the concept of two as a basis from which human reality can be ‘experienced’. To recognize through, and equate with the true substance, and essence of one other is to automatically experience the totality of ‘who we are’ in full measure. This does not mean that the terms ‘totality’ and ‘full measure’ convey an ‘end’ to ‘what is’. We have consistently offered recognizable facts (not assertions) that are part of our natural human activity, and give correct direction and meaning to our basic essence. We do function within the structure of ‘mutual agreement’, and we do communicate and ‘make known’ – basic obligations. These are evident simple examples of innate knowledge, and our understanding of ‘what is’ made transparent. To repeat we could not recognize anything without innate knowledge. All knowledge is a continual matching process 1 + 1 = 2. toward the realization of ‘who we are’ the development of language structures that correctly establish basic reality as it is, provides continuous knowledge that makes transparent the very nature of that reality. It is vitally important to recognize that we have active communal agreements concerning the existence of basic principles and concepts that form the very foundation of our lives. Constant change and movement in Space-Time - Energy - Matter are applicable absolutes to be recognised, which equates to evolution. There never will be a static property involved in the evolution advance. Evolution encompasses its own absolute properties to provide cognitive connection confirmation. The evolution of conceptual language exists to provide natural equality and to promote cognition between language states. No matter the languages spoken the concepts of air - food- water is the same and can provide a gateway to explore the future dynamics of human relationships. === Knowing. === Knowing what all the truth is is not some miraculous state of perception. Nor is it a high academic achievement of amassed information. It is simply an objective common-sense view of ‘what is” and in reality what must be. It is what must be for life to function within the principles that exist that are its natural foundation. We are always of necessity the living expression of a reality that must be experienced in the whole. Our recognition of the same principles operating universally is also our recognition of who we are. That proliferate ‘sameness’ is an evident easily recognizable identity. When a child is afraid of an imaginary monster in the dark, we generally do not accuse them of being absurd, or that they are lying. Appeasement comes with an explanation of reality at that level which is truthful. An explanation, which the child can grow up with, and find comfort. It is simply introducing a child to a level of truth that is more real to them. In every instance, the only reality that ever exists is truth. However, distorted it may be expressed. One of the major distortions as the result of thought processes is to consider that we can manufacture something other than the reality that exists. The ‘fact’ that you recognize contradictory or absurd statements is that behind them there is a measure of truth. The habitual liar lives in a world in which he or she imagines that truth is something they can manufacture. Where human ‘thought’ constructs its reality in terms of dichotomies it can never deal with the truth because it continually makes those comparative perception judgments. Those judgments are always in question because again they cannot deal with reality as it is. No matter how absurd or contradictory any statement is, that is the measure of truth expressed. Ergo whatever it is that is expressed, or made manifest, is the truth to some degree. Ergo everything that is, is the truth. It is our responsibility to recognize it for what it is. It may be appropriate to review previous observations on dichotomies and gradient scale. Consistently we have contended that there are no dichotomies, which then properly puts each principle into the category of an absolute. To identify ‘truth’ as an absolute in that category then everything that is must have a measure of truth. It is a very simple and sensible approach to establish ‘what is’. It is the means of identifying a reality that must have truth as a base – whatever it is, and however nonsensical it may appear. All principles have an elementary gradient scale that we must use to identify knowledge that is honest. That gradation scale knows no dichotomies. Dichotomies are always the imaginary properties of pseudo subjective reasoning. Necessary factors establish that gradient scale where only objective realities exist to furnish a healthy subjective reason with truth, and so we learn to apply the conjunction to address reality for what it is. Truth comes in an abundant variety of ways in its commonality – and therein lays its overall ‘complete’ power, despite any denial to the contrary. Embedded knowledge as we see it is neither experience nor knowledge without principle content. A person may be well educated in all aspects of the geography of a beautiful South Sea island, but have no practical experience at all of its beauty. Being clever about a subject does not necessarily equate to an understanding of the subject. Nor should it lend itself to posing as adjudicators on a proposition preset we imagine by the same adjudicators, or essentially the same school of embedded thought processes. Long before human evolution, the principle of leverage has always existed in all Nature (as have all other principles, wherever there is space – time – energy – matter). Our adaptation to the existence of principles has added to the sum total of ‘knowledge’ as we know it, including the concept of knowledge itself. That form of ‘knowledge’, and our ‘knowing’ is natural and not any personal or esoteric acquisition. Just as a fish knows what its natural habitat is, or a bird to fly in the air, the human species uses all available principles it recognizes to add to its knowledge (already said). Any valid theory of knowledge must have as its base constructive definitive principles to support it, and it is evident that our accumulated common knowledge equates to our common experience. No matter how erudite or convoluted any argument may be, if in the end it is reduced to inane observations that have no factual basis in principle, then it is time to abandon them. Do try to consider the sort of ‘mind’ processes that offers us up a world that knows nothing but separation. How can we possibly evaluate what ‘wholesomeness’ really is? How can we possibly evaluate ‘who we are’? How can there be any theory of knowledge without addressing Nature or our innate and biological relationship with it? Any attempt to debate ‘who we are’ and the completeness of that concept must have some sense of reality on the real meaning of ‘completeness’, and some understanding of the principles that are the nucleus of human society. To wrap any argument around a non-existent concept that can never be realized is apropos to attaching oneself to a system of belief in things that do not exist. One can make ponderous and convoluted statements about those beliefs but in reality, they are morally and ethically misguided. The ability to correlate correct definitions to the reality of life offers up that direct link to the truths that are common to us all. It corrects the presumptive notion that there can be ‘different’ perspectives on the same reality. There can be ‘differences’ but there can not be ‘different’ measuring sticks for the same reality. No matter the multiplicity of perspectives, they can never alter the core principle of ‘what is’. Historically evolutionary progress can best be measured by the adoption of recognized principles. Reality at whatever level we find it can only be understood by addressing ‘what is’. Nothing can be understood by attempting to relegate it to a non-entity through questionable theories of ‘knowledge’, which in essence negate the very content of knowledge itself. The perpetuation of any theory of knowledge, which cannot recognize the principles that are its foundation, can only be a shadow of its own reality. Construct the ‘necessary factors’ around the skeleton and a body will take form. If any particular theory of knowledge cannot identify simple truths, how can we possibly question how anyone ‘knows’? A dichotomy is the human attempt to deny the existence of a whole reality of a principle. We have the principle of leverage and its necessary gradient scale. Mutual agreements of a consistent reality, at a communal level, are a passive form of the Eureka moment, which recognizes fundamental principles that relates to truth. In every social structure, there are varying degrees of recognition, which determine social use. The mosaics of differences, which make up the rich pattern of life, are a testament to human creativity. Principles offer up a form of predictability in which our brain forms knowledge through the process of interaction. The experience accumulated through each moment, forms exponentially in use, or becomes transparent immediately in a Eureka moment, in which we know. Real knowledge is through the constant interaction with natural principles, much more than the transference of divided embedded information. The problem we face is that ‘knowing’ or ‘how we know” is never a personal possession. Any theory of knowledge no matter how in that respect, is true interaction. Peeling a potato and ‘knowing’ it, is rewarding enough! All principles are the repository of pure erudite knowledge. We recognize Universal principles in play at all times in Nature and its by-product – human societies. The correlation between determined interdependence to objective reality requires our intellect to ‘honestly observe ‘what is’, and assimilate that subjectively. Then the equality of the external and the internal becomes a reality and we ‘know’. Knowledge is the process of natural action, reaction, and interaction. It is nonsensical to ask how do we ‘know’.Every moment in time is complete because it must contain all the principles that form its nucleus. It can only be like that to facilitate the immediate experience of Eternity, or the wholesomeness of any of its principles. On the gradient scale of experience, we all exist somewhere on that scale. It is called life. === Relativism: === Relativism can be consistent with interconnectedness and a gradient scale of knowledge. What it cannot do is confuse the relationship that correct gradient scales of principles have in reality. Hot and cold would be on a temperature scale – no dichotomy! Leverage could only be measured by its own scale (say a child’s sea-saw to a high-rise industrial crane) – no dichotomy! The human being is a human being whether it is a child or an adult – no dichotomy! The domestic cat is the same animal species as a wild lion – no dichotomy! The domestic dog is the same animal species as a wolf – no dichotomy! How can we manage to classify these as dichotomies? Gradient scales are the natural human mechanisms used to recognize constant principles. They ensure the human perspective is aligned correctly to identify ‘what is’. The distorted human perspective is the result of human thought processes unable to establish constants that must exist in each moment of time. The problem with embedded information is that it becomes stultified and it can stifle healthy reaction. The injection of recognizable principles invigorates and brings new life that offers countless avenues for human energy to be released. More importantly, those energies are used to enhance the evolutionary process. We are collectively gifted with the potential to elevate life itself. We can correctly use such information by transforming its content so that its inherent truth is made recognizable. It would be impossible for life to function if it was composed of ‘different’ opposite realities. All theories of knowledge are in essence interconnected and can only contain validity when the principles that are the coalescent mechanisms are recognized. The unification of the truth that must exist in any theory needs to be harvested and used to offer up a body of ‘knowledge’ that has commonality of meaning. The identification of principles, truth, knowledge, and their subsequent establishment can only be achieved through direct interaction with Nature and life. Gifted with life we have an obligation to demonstrate its capacity to use every resource to sustain and nourish its own environment. We all know through the constant natural process of action – re-action – interaction. Depending on the quality of that process, knowledge will take its appropriate place on the gradient scale. That we ‘know’ is natural. It is not some extraordinary esoteric attainment, posited by a body of theories that, by their very nature, look for difficulties where none exists. Universal belief systems based on mythologies can have an entrenched view of good principles being established because of their beliefs. Indeed the perpetuation of the beliefs throughout history offers a dynamic that is counterproductive to the ‘realization’ of principles that are necessarily true. Principles used in this approximate way, paradoxically hold no real meaning, and in fact, impose unhealthy dysfunctionality. When there is a critical change toward establishing correct principles, it is axiomatic that the diffusion of mythologies becomes an automatic process. True interaction lies in the knowledge that correct action is its own reward Any other interpretation is less than tangible. The accuracy in interpreting basic principles, and the alternate knowledge implicit in the interpretation, will always establish the primary principle sought. Archimedes et al. Truth can be found in the oddest places. (Archimedes bath image here please) Archimedes cognition on how to weigh metals in water through displacement. === Truth. === Language is the construct of human action and the word “truth” seems to hold pride of place by the power of its usage and the meanings it evokes. It is preferable if we could turn our attention to the unity of principles (including truth) that are the construct of every language we use. By uniting the principal terms we can elevate the meanings we desire. Reasonable constructs and the correct duality of established principles always lead toward meaning. It is the only form of meaning that leads to its own extension eg. how to weigh metals – how else could it be? All principles have reciprocal value one to the other. No foundation principle can stand alone. They can only exist in a union, one with the other, the source of reciprocity. All absolutes are universal. There is no hierarchy beyond the meanings they evoke in their joint construction. The binary connotations, however, one may express them, provide a constant reality beyond conventional consciousness. That experience is the immediate reward through disciplined application of their use. That discipline takes the form in all human action (such as the bathing scene above) disposed toward the correct functionality of basic principles. The daily connections we make always include the distinct possibility of their recognition, when we make those connections in a mindful state. From any common sense, or ‘more reasonable’ position, it would be more productive to view reality as possessing at every level the same innate values or principles consistent with our ability to measure, or recognize them. To view reality as having ‘different’, or antagonistic properties, is simply a misguided view of ‘what is’. That form of perspective is counter productive when it attempts to establish mythical dichotomies as realities in their own right. When the realities of principles are made transparent, we can then ‘more reasonably’ make use of them to further their basic existence. Here we use reason to exemplify their necessary function, and once established it becomes (if necessary), ‘more reasonable’ to locate them in all things. ‘More reasonable’ seems to have the particular philosophical motivation, not toward simple, sensible, and reasonable evidence, but more likely toward that ‘immaterialism’ ideology, and continually seeking for an elusive protracted answer is hardly ‘more reasonable'. Since we are apparently confined to a human perspective, we must settle with the latter position: the apparent state of representation of the world. The de-materializing of any object through the practice of ‘perceptual illusion’ is an attempt to deny the reality that exists. Where perceptual illusions are concerned, innate direct communion with that which is, suspends the effect of such illusions. All the properties in a chair are recognized as the reality that exists. That is materialism. A chair does have the principles of form, design, structure, colour, substance, etc. However it is analysed – it is a quantifiable reality. === Truth and Reason. === The reason could easily be defined, and validated, as the correct application of common sense. More expressions of common sense can only endorse the completeness of any concept. A true experience of reality does not require endless explanations as to its ‘wholeness’. It just is. Truth is in reality a network of implicit principles in which it is the predominant energy in each of them. They are identifiable by their interdependent nature (see network below) not the least of which is common sense. Dictionary given definitions of ‘truth’ place it in a very common sense acceptable category. One of which is ‘accuracy of representation’. Note how the two definitions in this paragraph coalesce. The human drive toward recognizing and understanding the place of principles (constants) correlates to the energy we expend on questioning ‘who we are". The constant principles of action, reaction, interaction, are the automated natural impulse toward ‘establishing’ a human reality, and human identity. The process of evolving within that process has an egalitarian dynamic that powers it. In essence, it is a natural gift that we must accept. Each life and its identity contains all its personal experiences which can never be known to anyone else. In a sense, we can never “know” another person. Their life is sacrosanct. We can know a lot about them, and there it ends. Truth is at the top of the gradient scale that measures the veracity of all things that are complete and related and paradoxically all reality is the truth. It gets back squarely to ‘who we are” and where we exist on that scale. To view gradient scales as having no truth to their structures is denying truth itself. For the entire interconnected, interdependent network of principles, each of them has a gradient scale whereby each measure expresses truth in its own manner. All forms of leverage, from the minuscule to the lever that will move the world, are in of themselves, true and exact at that point. It is the only way we can recognize their existence, and use that complete truth at that time, to move up the scale. Time is the relative measuring stick that determines the amount of knowledge we can absorb. Consider the advanced extensions to the Archimedes principle of leverage throughout time. Network scale example. Truth Knowledge Common sense Responsibility Reason Understanding Intelligence All interdependent, and interconnected with all other principles and absolutes. No ‘thought processes’ or ‘mind’ constructs can create reality. All we can ever do to gain knowledge is ‘act’ react’ and ‘interact’ within the confines of our immediate reality. The quality of that action is determined by the nature of available information. When there is freedom from embedded thought processes, there is a natural human ability to relate to the existence of truth as it is expressed in reality, and our brain records it accordingly. Thus, the principles of civilized societies evolve. Where there are predominant belief systems, the implicit energy will naturally direct itself toward human standards that blend all ethics together. That implicit energy will find its true home in the principles it seeks. The connected strength of those principles offers sanctity of experience that demands no sacrifice. Everything that is, must of necessity, have a true comparative value (not a distorted dichotomy value) for honest recognition to be realized. which is to ‘know’. All things are relative but only within their own true scale. It is the process by which we can identify reality, as it is. Principles cannot operate on any scale practicing negative discrimination. Thus a healthy individual can be at the top of the scale and someone with various health issues can be near the bottom of the scale. But that is how Healthy they are. There can be no relativity when ‘mind’ or ‘thought processes’ believe in mythical dichotomies. When human perception is distorted by such beliefs, they create a false reality and deny access to the true state. Where there is a network of connected basic standards that are universal, then it is possible to use them and be nourished accordingly. The scale of natural human progression provides recognizable evidence that we are constantly developing. Reality is the direct and conclusive evidence of possibilities realized. Therefore, the reality is always the source of all possibilities where truth exists. When the truth is used as the universal measure of ‘what is" there can be no discord as to its accuracy. It can only measure the principles that are implicit in everything there is, its natural milieu! Truth can only deal with ‘something’, it cannot measure an imaginative negative counterpart. Truth is the constant implicit property in every universal scale of principles. Thus, reality becomes transparent. Have writ large on the value of distorted comparative perception judgments. The standard of correct knowledge always carries with it, its own appraisal. Where principles are concerned there is an obvious scale of identification (e.g., leverage and the numerous references) that is all-inclusive and provides us with evidence of its existence. We could say with some truth, that the industrial crane has more leverage than a child’s see-saw, but we cannot deny the truth at the lower level or the reason applied. Where principles are concerned, truth is not a possibility, it is a constant reality (e.g., leverage). When mythological dichotomies are recognized and established for what they are, the process of ‘ironing’ them out and experiencing their constant reality will translate into the reality, which they are, and used accordingly. To evoke that new sense of reality, the mechanisms of ‘selective immaterialism’ need to be dismantled. Where human experience is presented with something it does not understand and is unable to appreciate the principles involved, the reaction can invoke a sense of fear. That condition can be a primary breeding ground to establish a language of dichotomies and put a selective name to something it does not deserve. The diffusion of a false singular dichotomy into the natural healthy state of the common good puts responsibility into its proper place. Within the process of diffusion, there is the natural and equal absorption of our true reality. The transition between separation and inclusion will be a seamless process because it is our natural state. Objectivity and Commonsense: Explore the plethora of principles - truths - constants that are the mark and phenomena of Homo sapiens. We cannot have any doubt about our existence in this present reality. The truth of reality is and can be experienced wholly and completely by anyone at any point in time. All human progress is the result of such experiences, manifest in principles throughout time and their subsequent ‘use’ evolves exponentially. The overwhelming evidence is our reality, now. A simple analogy of objectivity and commonsense. Somewhere in our early development, someone put the ingredients of a loaf of bread together, somehow baked it, and hey presto, the first experience of a loaf of bread. It is now a form of sustenance, which feeds billions of people. We no longer need to experience that ‘truth’ that ‘knowledge’. It is unnecessary because it is subjective assimilation and the act of external and internal activity. Because ‘our’ brain functions in a manner that can identify the natural elements it exists in –space, time, energy, matter, we learn to ‘know’ and recognize ‘completely’ a child’s see-saw. Knowing is a natural evolutionary function. The quality of ‘knowing’ has its own natural determinants which of necessity contain the measure of principles required for universal recognition. An Archimedes insight (or anyone else’s) could not become a universal reality unless those determinants were in play. I know the very same way we all know – by experience via action, reaction, and interaction. Truth persists and what Archimedes experienced was true and complete. Any experience of any truth, principle, or constant can be as ‘complete’ within a grain of sand, or knowledge of a pyramid. All experience of that nature is an experience in ‘time’, and when it is the truth, we use it accordingly. There is no mechanical translation, or opinion of ‘necessary factors’ as they are constant universals. That, which is ‘complete’, is transparent universal knowledge e.g., the principle of leverage. The precision of terms must include ‘necessary factors’. Necessary factors translate into a common universal language so there is no loss of meaning. All truth – principles-constants – absolutes, that stands the test of time we use accordingly. Thus, human societies evolve, and we evolve without the necessity of having to re-experience any of the principles we recognize and establish. That simply would not be a ‘natural’ commonsense proposition and an entire waste of unnecessary energy. The truth of any principle at any point in time, and at that point in time, can be experienced completely by anyone. Whatever it may be if the principle is established – from then on, it will evolve. There are some misguided notions that ‘truth’ ‘knowledge’ and ‘experience’ have some secret value that is unattainable by ordinary experience. It is a ‘natural’ evolutionary reality that what we ‘know’ becomes useful. We have a mutual responsibility to recognize, use and honor the principles, which are the common property we share. = Human consciousness. = [[File:Lane past Coombshead Farm - geograph.org.uk - 589598.jpg|center|thumb|369x369px|"The roads half travelled when you know the way"]] [[File:Sombrero Galaxy in infrared light (Hubble Space Telescope and Spitzer Space Telescope).jpg|center|300x300px]] === The Universe === ''<big>We do not own Space</big>'' ''<big>We do not own Time</big>'' ''<big>We do not own Energy</big>'' ''<big>We do not own Matter</big>'' The human capacity to understand the question of sovereignty or ownership of Space - Time - Energy - or Matter can only be accepted when any basis of dispute includes two dispositions - human and spiritual indigenous ties throughtout history. With the evolutionary appearance of indigenous peoples throughout this planet, their way of life should make it paramount that their existence be recognised as a natural law that has providence! Their culture and way of life has its own identity in which Space - Time - Energy - Matter is expressed as they experience it. That proof also lies in the existence of caves thousands of years old, and the existence their art and culture. So the constructs defined below are a new approach to understanding the concept of ‘knowledge’ and its proper place in an evolutionary expanding universe. Knowledge acquisition requires appropriate recognition through action, reaction, interaction, in which proper perception and comsciousnessvalues are applied. That form of construction requires the dismantling of previously embedded information. This requires a new direction to formulate a sound basis from which to extend. Construction of an analytical methodology to establish a form of ‘knowledge’ that is best suited to distinguish in a contemporary reality. A reality that adequately conforms to common notions of that which is true, and can only exist without any false relation to that which does not exist. All science needs the certainty that established absolutes provide. A pragmatic construction of real knowledge would propose that all reality expresses a form of evidence or proof and that the observer and the observed contain innate properties necessary to establish a foundational agreement. That form of agreement would necessarily function on the basis that everything that is – is truth. Any other interpretation would be a disconnect from reality, and the interdependent correspondence that must exist for true recognition of any absolute. Controversy will always emerge when the discourse on bifurcation and the introduction of non-existent dichotomies are used as arguments to be explored. Pragmatism would say that the human species would need basic properties to formulate any form of reliable epistemological analysis to explain and simplify the reality that forms their existence. That reality could well be recognized initially as the absolutes of space-time - energy-matter, through innate perceptual data that corresponds to an outward structure that is constantly evolving. That which is absolute is the determinative factor in establishing the existence of truth. When a chair, is a chair, is a chair, its recognition is established when we ‘commonly’ apply subjective and objective measures to that which is truly external to that which is truly internal. That which does exist becomes ‘common’ knowledge, and accepted as being true. Thus, the distinction between a priori knowledge and a posteriori knowledge no longer exists. Where there is consensus, everything is. The formulation of absolute criteria that offers ‘simplicity’ as a tool to measure all and everything, dispenses with the confusion of ‘difficulties’ historical philosophies engage in. It also offers an observable synthesis that clarifies the confusion. Within the combined properties of those absolutes then everything potentially would be. Everything that is by that definition is original, ipso facto, everything that has no false relationship, and therefore true (no dichotomies). Given contemporary human development, it would be ‘common’ pragmatism to accept the proposition that we exist within the absolutes of space-time – energy-matter. Within those absolutes and their innate properties, it would also be pragmatic to assert that ‘everything then is’ (whatever everything may be). Pragmatism would also dictate that ‘everything that is, is its own form of truth, and must contain available constructs of meaning. Therefore knowledge and understanding would be obtainable to that common experience, and at whatever level that experience is activated it is in interdependent unison with the source. The continuing establishment of basic absolute principles (their generalities and their specifics) forms our reason. 1+1= 2 is a ‘simple’ but true universal constant generality. That form of generality is used because we recognize the specific principles of a balanced equation that adds up and makes sense. Simple generalities with their inclusive specifics form the foundation of human reason and its constant evolution. Simplicity is the bane of a ‘mind’ that must have difficulties. = Availability. = The unifying feature that makes ‘knowledge’ available to us all, are the innate universal principles in all things. Archimedes established the principle of leverage. To paraphrase - ‘give me a lever, and I will move the world’. Once the law is established it can then be put to good use. The principle of leverage is manifest in countless ways, and put to good use! The principle of leverage is a constant available to us all, and always has been. Through his application Archimedes conveyed his insight in practical terms, and made aware of the principle and the laws governing it. We now use those laws. We no longer need to philosophise on its existence as a truth. Similarly whenever 2+2 = 4.  Whatever we use to make the equation – apples, oranges, bricks, the mathematical equation is a constant, and the principle of correctness applies. Here 2+2 =4 is empirical proof that the principle of correctness and agreement exist. Principle, or law: ‘a fundamental truth used as a basis of reasoning’.           If it were otherwise we could not measure anything. That which appears abstract then, requires no implausible argument as to its non-existence. Such is the nature of all universal principles, they exist whether the notion of a ‘mind’ can perceive their existence, or not. Because ‘thought’ does not create the reality of principles – universals- truth, it cannot from its mythical standpoint understand the simplicity of a Reality where ‘everything is’, nor the simple and factual conclusion – how could it be otherwise! Where ‘everything is’ evidently encompasses the whole evolutionary dimension. It is not an ideological enclosed static that stultifies expansion of an unfolding Reality. Any pure knowledge experience that ‘everything is’ ,(quite apart from the common-sense truth of the statement) is to experience the Absolute in any immediate part of anything that exists, which establishes its own truth forever. To examine a road code of law with that knowledge, and view the actions of drivers at traffic lights, it is more than reasonable to conclude with some conviction that there is  to some degree, Agreement – Knowledge – Understanding, and Conformity to that code of law. It becomes a ‘more reasonable’ proposition within Reality to understand that that code of law is multiplied exponentially, and the principles practised,  wherever drivers, motorcars, and traffic lights exist. It is the nature of the type of knowledge we are measuring that determines the measure of reason that can be applied to any given form of Reality. We can conclude that 2+2 = 4 is a reasonable mathematical calculation that contains the principles of Agreement – Understanding – Conformity. Given the accepted knowledge of these innate principles we can with more reason apply such a calculation Universally. Knowing is agreement with ‘what is’. Knowledge is not the attempted denial of any existing reality. That is a contradiction in terms. Mechanistic observation is akin to viewing from the outside, a straw in a glass of water. The straw always looks bent, but when removed from the glass we realise it is straight. To claim an experience of that which is Absolute, is not a claim of an experience from a higher domain, or an  isolated incident – it is common-place, numbered by just how many we are. Attempting to denigrate such experience is denying the everyday actions that contain the innate principles of a constant Reality. All life functions within the constraints of the laws that are the constructs of Nature and Reality. All life is an expression of the Absolute. It is when that expression is realised, not only in an instinctive sense, but in a real sense, that we penetrate reality beyond a comparative framework of mythology. To claim that you ‘know’ intrinsically what the principle of leverage is, or the principle behind the mathematical equation 2+2=4 is to claim experience of the Absolute. The Absolute is not some abstract esoteric truth – it is that which is immediate. Whether in awareness or not, we constantly comply, to some degree, with the laws of a constant reality. Therein lies the difficulty for a comparative framework mythology – the Absolute is everything! = Basic Equation. = However much the simplicity of the equation is, it contains the properties of correctness – balance – equality – mutual identity – meaning, which in its ‘simplicity’ presages all future mathematical equations. In that universal meaning, there is particular knowledge of consistent truth. That form of ‘simple’ consistency, creates its own natural equilibrium, and its ‘usefulness’ evolves exponentially up the reality scale. Here is where we need to give proper credence to ‘that which is’. All generalities have profound and specific principles as their ‘common’ identity, which are absolute. Unless those components are recognized, both objectively and subjectively, they are reduced to a comparative value spectrum (using dichotomies) as a misguided ‘simplistic’ factor. Deductive reasoning is then deprived of all value, and leads to the inevitable spurious question ‘how do we know? Given the above criteria to establish a correct basis for knowledge that is recognizable, and of a kind that can be used universally, ‘simplicity’ can be recognized as a tool that promotes its own established formula. That which we constantly use. Everything is the truth with regard to the methodology. How that truth or generality is expressed denotes the measure of the principle that is at its core, and forms that measure of reasoning we enjoy. Fortunately, although the truth is an innate property, it is not a ‘personal’ property per se, nor is the ‘experience’ of its reality. Its natural evolution is progressive. In that progression we are in common, the beneficent recipients that ‘evidently’ conform to its constant existence. Philosophical dissertations have become a monopolistic form of opinions that always seem to presume the ‘rightness’ of difficulties in establishing the source of our being, and are unable to put in ‘simple’ terms the question of ‘who we are’. There comes with that the denial of evidence that permeates human history, which establishes the principles of our ‘common’ reality. Those opinions carry with them a colossal library of questionable erudition that becomes embedded, using questionable values to support their argument. The most distinguished opponent of such arguments (Ludwig Wittgenstein 1889-1951<ref>{{Cite book|url=http://worldcat.org/oclc/1203018418|title=Tractatus logico-philosophicus|last=author.|first=Wittgenstein, Ludwig, 1889-1951,|isbn=978-1-78527-656-9|oclc=1203018418}}</ref>) proposed that language logic was a necessary tool to dismantle the convoluted ‘mind’ propositions that have permeated the philosophical hierarchy. Those ‘mind’ propositions only served to construct meaningless concepts as to ‘who we are’. Although he gained prominence in philosophical circles, his work was directed more toward academia. In his Tractatus Logico – Philosophies he quotes: “The limits of my language mean the limits of my world - What cannot be shown cannot be said”, and “There can be no representation of the logic of facts”. Having a belief in mystical truths that were inexpressible, his statements above are indicative of the embedded language of dichotomies. His form of ‘knowledge’ ‘philosophy’ gave (without question) the concept of ‘mind’ credence to formulate its own logic to clarify its own form of reasoning. Wittgenstein, by not recognizing within the language the distortion that dichotomies create, was unable to approach the evident constructs of true meaning that lie within the interdependent relationship of absolutes, and their constant existence. Evidential reality is all there ever is. The ‘meaning’ or ‘knowledge’ that becomes evident in reality is ‘commonly’ accepted and used accordingly. To repeat, the evidential reality is all there ever is. In that regard, the exponential drive toward ‘difficulties’ amassed a historical discourse of misinformation, which is used to address the very ‘difficulties’ created by spurious value systems. In effect, dealing with ‘nothingness’. We have managed to turn ‘truth’ into a problem, into a difficulty, when the truth is simplicity itself. An oxymoron of gargantuan proportions. This contemporary malady solidifies a diversion that discounts the reality of continuous progression. Progression in which ‘common sense’ is a motivating principle that promotes human evolution. The consistency of specific principles allows us to achieve correspondence. Real knowledge is not a fabrication of convoluted prescriptions. It is the ‘coalescence’ of ‘what is’ to internal reality. That established, evolutionary progress is assured, and reality factors are recognized for what they are. True meaning can best be attained by the interaction and interdependence of natural principles and so recognized as such. That meaning which contains all the specific components of reality is experienced as ‘true knowledge’, measure by measure by anyone. That form of ‘experience’ is not a ‘mind’ process, but a very natural state of realization consistent with our level of action, reaction, and interaction. A correct and pragmatically form of knowledge-seeking foundational answers to perennial questions would seek a direct passage to our ‘commonality’, the beacon that offers guidance. Consider the quantity and quality of knowledge we all pursue that has meaning and usefulness. To posit the notion that there are no dichotomies is a cataclysmic proposition that seems nonsensical to established embedded constructs of knowledge. Constructs of knowledge that offer only confusion, and continually pose impossible questions, whose absence would provide clarity. That absence of confusion would dispel and dismantle a reality of ‘mind’, which functions on its own selection of problems. To address reality as having only absolute constructs dispels the confusion of duality and its inability to ‘use’ relativism in its proper fashion. All general absolutes contain specific principles representing facts; the essential properties that confirm reality. This reference directs the observer to observe, and go beyond the restrictions of a ‘mind’ governed by dualism, monism, or any other spurious form of philosophy that distorts the very reality it exists in. Where there are no dichotomies, all we can deal with is ‘what is’, and the logic of ‘necessary factors’ thus destroying the possible inclusion of anything described as a “paradox”. = Pure Experience. = To design a chair our brain requires to exercise the qualities and properties necessary for its manifestation e.g., strength, balance , design, functionality etc, etc. A chair, is a chair, is a chair, the product of innate knowledge.   If we did address any tentative agreement that ‘I’  is a ‘fiction’, could not our ‘conscious experience’ of that ‘fiction’ be just as fictitious. It would follow that whatever perception of Reality we experience must also be fiction. Our contention, as always, has been that ’I think - therefore I am’ by Descartes is the greater fiction for reasons already explained. To consider to whatever degree that we can function on the basis of a fictional ’I’ precludes any attempt to honestly address ’who we are’. Saying that ‘conscious perspectives’ are limited and inconsistent with apparent reality are quite correct. It then brings into question the validity of ‘conscious perspectives’ to guide us toward ‘what is’. The entangled fictional relationship between ‘mind’ ‘I’ ‘thought’ ‘consciousness’ impose formidable barriers to that which is evident. Base observations on the construct and interpretation of what ‘knowledge’ is. Human experience is limited by its mechanical interpretation of Reality, especially ‘cogito ergo sum’. If everything that is, is its own measure of Reality (the differences) then everything must be measured, at whatever level, as being that part of the whole with all principles intact, making that measure available to be experienced as the Absolute. With absolutes there is no antagonism. We cannot exist or experience anything without a Universal complementary source of identification. It is notable that within the structure of Cartesian dualism, Descartes' personal address to innate knowledge he attributed to ‘thought’ which he identified as being distinct from his body. How different Western philosophy may have been if his attribution had been toward his brain and the existence and evidence of other physical entities that functioned every bit as efficiently as he did. The premise that Descartes operated from ‘never to accept anything as true’, was simply a wrong ended approach which brought him into conflict with his passing acceptance of innate knowledge, that the idea of God was innate to his being. To view the proposition that ‘everything is true’ allows reason to seek and identify that measure of truth. No quest can be productively based on cynicism or denial, nor adherence to belief systems that separate experience, knowledge, and Reality. We have the obligation to question whatever reality has placed before us , but if we constantly deny its existence and attempt to ‘disappear’ it from our experience, then we are in danger of never experiencing that reality. Not experiencing Reality as it is, is equivalent to not experiencing ‘who we are’, and is indeed the only human source and validity of truth, although Descartes held the erroneous belief that such knowledge was independent of any experience. That belief we suspect was some form of impetus toward his ’cogito ergo sum’.                   Knowledge and experience are co-existing ‘necessary factors’ So long as anyone believes that human experience is based solely on indirect conscious interpretation (mechanical disposition), therefore any ‘knowledge’ derived from experience will be incomplete. Clearly it is the quality of ‘knowledge’ that one experiences (e.g., Archimedes) that leads to a common certainty of evidence realised through direct experience. That quality of knowledge can be available when we observe directly the activity of drivers at traffic lights with the knowledge that it is a very common activity recognized internationally. In every case we can logically pronounce the premises to be true, therefore we have a conclusion that is also true - whether that conclusion is defined as Mutual Agreement, or Common Acceptance, it does not matter - they are mutual principles. It is this form of logical knowledge of innate principles that is the precursor to knowledge of the Absolute logically defined within all reason for ‘what it is’ . Where the basic premise is true that there is ‘Mutual Agreement’ between a multiplicity of drivers at traffic lights then we can with certainty conclude that the same principles exist Universally. We can also draw concrete conclusions, and establish knowledge, that it is not ‘absolutely necessary’ to experience by observation the multiplicity of drivers conforming to their particular road code. We have already established that knowledge. Knowledge and experience are not separate philosophical theories. One cannot be without the other.. Everyone has the potential to experience the Absolute paradoxically, in part or in whole. Everything that is, must contain the properties of the Absolute, otherwise nothing could be. To ask questions about human experience based solely and inevitably on our interpretation of ‘knowledge’, and co-existing with that, its particular meaning in human existence. So long as we can only deal with our conscious interpretation as representing Reality then we derive functionally less meaning than we are entitled to. When we see other humans consistently using levers to open crates then we can recognize a ‘social intelligence’ operating which equates to understanding that is not based on opinion, but is a clear expression of human activity that has correspondence. All of the principles involved in that experience can coalesce to provide that form of Reality that requires no interpretation. It becomes recognizable knowledge. How we understand that knowledge is through the realisation and identification of the principles involved, which become immediately transparent. The Absolute could be categorised as a knowledge experience that encompasses all and everything. Whatever is manifest is that measure (complete in itself) of the Whole with all its principles intact. Where there are at least two actions that are identical we can reach a common-sense conclusion that a definitive principle is operating. When that corresponds with innate knowledge then we have the complete cycle. The definition itself is language opening the door to an experience of Reality. No one can know in isolation. An imaginary ‘I’ restricts any experience of who “we are”,  and is not a necessary part of human experience.   Explaining experience beyond imaginary thought processes requires a definitive language that deals with the principles of Reality itself. Pure experience. The world-wide disposition that has no grounding in Nature and Reality becomes captive to any mythical fear that  offers a target to give some form of direction or stability. There is nothing more simple than to make Reality transparent - its evidence abounds. We can pronounce the principle properties that provide guidelines to its existence whereby the reality is made apparent. = Expansion. = To address concerns on ‘negativity’. Negativity is in essence the inability to establish a measure of Reality. Mechanistic processes of denial are the attempt to understand and make transparent that which is apparently unexplainable, and resolve a condition whose energy is driven toward finding that core of affirmation. The evolutionary principle from all available evidence is that human beings as a species progress. This seems a paradoxical contradiction to the embedded proposition that we can never know the ‘truth’. The consequence of such a traditional premise is that denial and negativity both hold paramount positions. We are conditioned to accept the premise that there is in fact no premise that will enable us to go beyond presently accepted norms of experience. We are conditioned to accept that the ‘truth’ is inexpressible. The evolution of the human species is constantly subject to contemporary ingrained social habits, which give some kind of credence to that particular point of existence. Indirect conscious interpretation classifies itself as a solid perspective to govern and justify human activity, which in many historical ways has proved disastrous.                   Our continued intention is to expose detrimental barriers to the realisation of ‘who we are’, and in that process establish a smoother, more realistic approach to ‘who we are’. The Archimedes legacy. When we establish knowledge of something that exists through a multiplicity of experience and evidence, then from every reasonable standard we can establish that it is true, ergo that which is true is Absolute. The principle of leverage is well grounded in social intelligence, and our natural knowledge of that does not need erudite explanations of its presence, nor any ‘conscious interpretation’ to realise its existence, or its practice. Evolution eventually removes restrictive passages to direct experience, the very purpose of evolution. The principle of leverage is not a matter of opinion, it is the realisation of actuality and our continued ‘more reasonable’ response each time the principle is applied. Children learn to speak their language primarily through experience without any direct, or indirect conscious interpretation, and so, universally we ‘know’ the most powerful means to communicate. Were we to move 50 miles in any compass direction from the town we live in, there is a certain predictability that we will meet others who speak the same English language that we do. If in that experience we find that these premises we have drawn about our travels were true, then the conclusion we would come to in particular, is that when we communicate we make known what we know. …………………………………………………………. '''Please note the date:''' '''Oct 2005.''' I am offering up this older material below to provide insight as to the progression of this work. There may well be some duplication to date. During this period my wife and I worked in collaboration to ensure an equality of experience. = Stepping Stones 1. = There is nothing other than what is – there is no hidden Reality that we need to seek, it embraces us at every turn. The Archimedes experience is the pure experience of Ultimate Reality, which provides indisputable knowledge. Reality is the source of complete knowledge, it is the constant source that has provided us with all human development, from the writings of William Shakespeare, to the computer development of Bill Gates. What they have produced is now an evident part of our reality that we can engage in. We can experience ‘mutual agreement’ through epiphanies, insights, enlightenment , understanding, Eureka moments etc, they are all one and the same. Implicit within the macrocosm is the microcosm – it cannot be otherwise. The more we conform within the microcosm the more we begin to appreciate that Reality contains everything, and that we can realise through experience its manifestation. Each Eureka moment is that personal point of experience that connects us with the Truth. The principle of leverage was always available, it took an Archimedes to explain it to us. Each Eureka moment necessarily engages with the reality of complete knowledge, and utilises its share at that time. When we have complete knowledge of who we are in that personal moment, then we understand that these, egalitarian properties, are rightfully shared by everyone and that we have experienced that which is infinite. It does not mean that the process of evolution is over – it has only just begun. It does mean that we can no longer continue coasting through this existence in a near comatose state. If there is a hypnotic fixation in holding the principles of Reality as being separate, and different, then the potential realisation of their immediate unity, and communion, becomes problematic. Knowledge, and experience are one and the same – they are not different!! Experience = Immediate knowledge of basic reality that is factually correct, and that we can reasonably use. Knowledge = Immediate experience of secure, and accurate information that is constantly stable, and sustains principles. Reality = Complete Knowledge. As the microcosmic part of the total macrocosm we are immersed in reality. The real question should be, ‘how can one not know Reality, or ‘who we are’. Mutual agreement is evident when we know we can go to the bank, and deal with money transactions. Mutual agreement is evident when we know we can go to the supermarket and exchange money for goods. Mutual agreement is evident when we know we can send our children to school to enhance their education. Evident proof is validation of what is – it is not a matter of anyone’s opinion, nor is it an assumption of ours. Neither do we assume, or offer any opinion, on the Universal Reality that there is ‘mutual agreement’ that we need air, food, and water to sustain us. Evident proof is also the basis for the mechanics toward realisation of ‘complete knowledge ‘ of who we are. Reality can be realised through concentration on its basic principles. We use language to express our understanding of who we are.It is relatively easy, it is reasonable, and it is responsible. We convey through language our measure of intelligence, and to the best of our ability conform to the basic rule of communication – ‘we make known’ Implicit within that exercise is ‘mutual agreement’. We may differ in some specifics, but we meet the basic obligation of communication – ‘we make known’, and always we progress to some degree. Simultaneity is one of the constant principles that we all share and they come from Here, Now, the Present, where they have always been. Everything is. Our being is always engaged in the present, and we each have an obligation to understand our relationship to what is. The present is the only point of contact we can ever have with Reality. To some degree or another, each one of us is directly connected to Reality (we do not have any choice in the matter), and we can potentially evaluate ‘what is’ through the utilisation, and examination of factual reality. We are the microcosmic part of that Universal Macrocosm, and because we already have that innate information it is a matching process when we have a Eureka moment, an epiphany, an understanding beyond question. Nothing enters our minds - we already know! Everyone has innate knowledge of the principle of leverage. It requires correct examination of ‘what is’ for realisation to occur. It is then a relief to have ‘mutual agreement’ on the things we would wish to make transparent to others. To use a traffic analogy, it is evident that there is ‘en masse’ mutual agreement when we know to drive off when the traffic light turns green. Mutual agreement is translated into people obeying traffic rules (otherwise chaos). Two cars, two drivers, sitting directly alongside each other at traffic lights, discuss their understanding of their Road Code in this particular position, and what they should do. When the light turns green there are a myriad of principles that apply when they drive off simultaneously. They have both demonstrated their ‘complete knowledge’ of the significance of the green light from this perspective. There is Mutual Agreement. There is Predictable Conformity. There is Common Ground. Each one complements the other. They are both right. One more remove: From an outsider’s point of view – they both know! The green light could be categorised as a Eureka moment, it sets in play all the above principles, whether the drivers are aware of it or not. From the perspective of two outside objective observers who know the traffic rules, if asked, did the two drivers at the lights obey the rules – the answer would be yes, there would be mutual agreement. They have complete knowledge of this particular circumstance concerning drivers, and green lights. Could it be that certain schools of thought are curtailed by a questionable refusal to recognize what is, and have a preference for creating a difficulty where none exists! No one can examine what isn’t! There is no such thing as ‘nothing’. Something is – what is it? A Scottish engineer functions on the same principles as an Italian Pope. Because Archimedes was prominent as a mathematician, his realisation of the principle of leverage, and his understanding of the difference in water displacement between silver and gold was widely reported. This does not mean that realisation of ‘what is’ is an exclusive experience. As said previously Archimedes did not realise something new – it has always existed, and all forms of life would have utilised the leverage principle to some degree or another (watch a bird build a nest). At that time there were probably many thousands of people who had some understanding of the principle, but Archimedes was the one who made statements about it. As in any Eureka moment, we can experience infinity and who we are. It is mutual agreement (an understanding) between the part, and the whole. It is when the principles are in unison Eureka! Reality is there to be examined, and experienced, it is not separate from us, nor should we try to make it so. = Stepping stones 2. = Knowledge is not conditional by the activities of what may be called ‘thought’ or ‘consciousness processes’. Real knowledge is that which is available to all, and to be shared by all. It cannot be contained by the ‘experiencer’ and then not ‘known’ by the accident of experience. It is the actual innate experience itself which conclusively establishes the truth. It can only deal in the truth which is its modus operandi of dissemination. For me to say that ‘everything is’, is a statement of fact which cannot be denied, and an intellectual dishonesty to attempt to deny the evidence by philosophical machinations. Hostility toward the truth leads inevitably toward attempted negation - looking for nothingness! ‘Being here’ demands its own recognition - attempting to deny it is simply perverse. Knowledge is the realisation of ‘what is’.. Rene Descartes ‘I think - therefore I am ‘did no service to human evolution, or education. It established in Western societies especially, the culture of individualism, with the precursor that so-called ‘thought’ was the inward evidence for existence, and for the following unfortunate claim that we have a ‘mind’, or to use the euphemism, a soul! Experience is true knowledge. When that experience marries up with its innate counterpart then recognition is realised (cognition). In simple terms, a light goes on in the brain. There can be no real knowledge without truth. All thought qualifies experience and attempts to reduce truth to near nothingness which is a widespread conditional activity. We cannot manufacture knowledge, or the principles which are its properties. No matter the amount of correct information anyone can ingest, it does not become knowledge until there is tripartite coalescence between inherent knowledge - ingested correct information - and ‘what is’. Then we truly recognize that which is Absolute. Within Nature we have the distinct privilege of evolving in a Universe that can only recognize the attributes of social cohesion. Knowledge is not anyone’s personal possession. Whatever measure of experience we may have of it, it is only available as a Universal sharing experience to be beneficially used. Human activity whereby we witness people using tools for leverage, or drivers at traffic lights obeying the rules of the road, are observable markers that contain the properties for understanding our own reality. Unless seen for what they are, they are only mechanical platitudes with an equally mechanical response. We could rightly claim that that at least is some response, but of no real value. The natural process that operates when we see that which is innate, overrides any erudite explanation from an academic base however intellectual its original source. ‘Thinking’ for oneself cannot make judgments about a ‘natural’ experience. When we see human duplicate functions in operation then we are in communion, and at another level we recognize who we are. When we actively see the activities of the human brain in action we are not dealing with any internal ‘will - o’ - the wisp’ that no one can ever experience. We exercise that prerogative (human activity) at every moment in time, but quite apparently without that focus of attention that denotes realistic recognition. To seek identity in sectarian, or secular belief systems to overcome the contemporary feeling of loss of identity leads to the acceptance of anything that offers some form of stability. That is then used to strengthen that which is euphemistically addressed as the ’self’. To retain that security the acceptance of information transmitted throughout generations, is absorbed into the culture, and defended to the death against those who would question that belief system. The greatest knowledge we can ever have is our own and it has the potential to transcend all else and provide insight into infinity. The most tragic human condition is the lack of experience of identity in a multiplicity of identities in which we all share. The real problem is not one of ‘identity’, but a lack of ’communion’. Whether we like it or not, whether we are aware of it or not, the principle of ’communion’ must always exist to some degree for evolution to proceed. It is within the experience of that principle that we understand the fallaciousness of that much heralded ‘self’ which draws down so much energy in an attempt to establish itself as a reality. Within positive language structure possibilities (no dichotomies), there should be the disposition toward the realisation that our relationships to cognize into ‘communion’ must be addressed as specifically dependent. Social attempts to be ‘independent’ are the very remove from reality and signify reduction attempts toward nothingness. Adherence to, and the cultivation of faith and belief systems give little elbow room for any factual occurrence to be anything other than a comparison to the myths that are held. The cultural and educational socialisation of generations of children must carry with it, its historical belief systems that overwhelm the natural instincts. Observe an animal out of its natural habitat and locked in a cage for its entire life. It would be a salutary exercise if we could dispense with the term ‘mind’ from our vocabulary and magnify the use of the word brain to promote a realistic discussion on ‘who we are’. My action of levering open a wooden crate and knowledge of it is one and the same. Our remarkable brain functions like that, the purpose of a brain, the natural repository of innate knowledge. The assertion of principles is critical to avoid all activity being submerged by questioning their very existence, and being unable to see directly. It would be a rarity today, if anyone using a lever to pry open a wooden crate would have the same enormity of experience that Archimedes had, nor the need to make pronouncements about it. It has all been done prior to our awareness of its value with the accompanying data attached. Our brain knows the value of a lever and activates our body accordingly when needed. It could be categorised as evolutionary transmission. The observance of someone prying open a crate with a lever, or drivers conforming to the road code at traffic lights, is a function of the brain in action, not a mythical entity in a singular locality that denies its own senses. When the brain is not burdened by distorted belief systems it then has the potential to experience ’that which is’, which is always constant. When we understand the function of a lever, or the presence of traffic lights, then we can activate the principles involved because we already know how! The negative impact in the use of dichotomies in language lies in their distraction from the truth, as our brain processes the words we use in relation to Reality. The tendency to attempt to separate inherent truths through the words we use disrupts that natural correspondence necessary for identification. A chair, is a chair, is a chair. = Stepping stones 3. = Where principles are concerned the constituent linkages in language are identity markers to that which is real - reference points. Without dichotomies there is no separation, or ambiguity between what we experience, and ‘what is’. Philosophy in its attempt to address something through denial is an elementary confusion. To say that that is a chair, and then attempt to deny it invoking philosophical theorems concerning the human ability to experience it, is a severe contradiction on the existence of the object , and the observer. When this form of contradiction is then taken as a constant, it then precludes any common-sense and definitive answer to the existence of a chair. '''For philosophers, George Orwell’s ’to see what is in front of one’s nose needs a constant struggle’ would be apt.''' Real concepts cannot exist in any mythology, therefore all that we experience is inevitably the truth that is there to beproperly categorised for what it is. The proper use of language in this context will identify whatever it is to correspond with present reality. Misuse of language (dichotomies and mythologies) leads only to the acceptance of a fractured state where nothing is whole and represents confusion. The dissipation of the supposed problem is never realised. Fiction has been elevated to the status of an accepted reality. Very early evolutionary physical dangers allowed the development of fictions that offered some form of imaginary protection beyond limited physical ability. That contemporary humanity endorses the mythology of ‘I’ is testament to the psychological fear that still exists and requires its proper recognition. Emphasis must be placed in the relationship between language and reality for understanding to proceed. The persistence of dichotomies has their own persistent confusion which then promotes a false reality through misleading information. Microcosm and macrocosm are one and the same in a Universe where ‘everything is’. Isolated viewpoints are exactly that, and are unable to view the expanse in which we are encompassed. We must learn to view reality through both ends of the same telescope. When we understand the extensive scope of ‘truth’, then we know that its values and properties do not change - which relates to ‘completeness’. Philosophical, ideological, and intellectual endeavour , try to shape the structure of ‘what is’ based on pre-dispositional knowledge, which can only ask the same questions, and look for the same answers. Not to experience that which is absolute or whole is the normal result of the confusion of language which has no correspondence to that which is real. To discuss with a philosopher the possibility that ’mind’ per se does not exist, and to dissolve it as a concept would place them in a realistic position, would indeed be a difficult proposition. The strength of that difficulty lies in another imaginary concept, that that ‘mind’ represents ’I’, and it is anathema to that fiction to consider its own demise! = Stepping stones 4. = There are no dichotomies. Everything is, and everything that is, is complete, everything is an Absolute complete Reality. You are experiencing your measure of that reality. It cannot be otherwise that you are experiencing that measure of completeness. When we come to terms with it we have the innate capacity to see the Absolute in a grain of sand. That is knowledge. Belief in dichotomies is the mythical barrier to that particular experience - which is only denial, supported by erudite protestations that human construct dichotomies exist. At a mechanical level Intelligence and Stupidity appear to be separate identifiable conditions, and they appear to be antagonistic. Stupidity is in Reality a measure of the Intelligence which is always constant. If someone was in a state of mythical utter and complete stupidity we would not attempt any form of emancipation from that condition. We know that that is misguided and proceed with techniques to advance intelligence. Consider the proposition that there are no dichotomies, and within that possibility all questions become irrelevant. Presuming that there are no dichotomies allows the process of establishing ‘necessary factors’ to proceed, and allows each measure of wholesomeness to be realised. Experience is the criteria for knowledge. Some Reality experiences were simply transposed into particular belief systems and elevated into a pseudo spiritual dimension, or a philosophical conundrum. Where there is a belief in a divisive fiction (dichotomies) there is automatic mechanistic restriction to that which is Real. There is a capacity beyond ego and intellect which can commune with ‘what is’, and recognize its properties. Reality is constant. Within the accepted comparative framework there is the view of principles as having different divisive categories e.g., as above, Intelligence and Stupidity, and classify them within ‘thought’ structure as dichotomies and give credence to them as being an antagonistic reality. The consequence of that, is, that one is always a remove from recognizing the structural properties of immediate existence. Any construct of knowledge necessary to evaluate ’what is’ will address the properties (principles) that are the constituent constant markers available in that which is the microcosm and the macrocosm. That identity (the Absolute) is found in any sphere of Reality.Everything is - and everything that is, must be experienced for what it is, and not for what anyone denies it to be. There is no mythical human construction that can deny ’what is”. Everything is - without dichotomies. To repeat, we do not have the ability to create ‘nothingness’ - ‘that which is’ has no imaginary comparative human construct. To attempt to deal with such constructs, and give credence to them is always the denial of ‘what is’, and adherence to ‘thought’ processes whose only purpose is to cement that activity. Indeed realising that the concepts of dichotomies are human mythical constructs, denying true perspective, is the beginning of insight. The dissipation of such processes through addressing the principles of Reality allows us the potential to experience directly ‘what is’, in simple terms -the truth! Intelligence is a ‘necessary factor’. Addressing stupidity is a denial of reality at whatever level we find it. Intelligence and Stupidity are not antagonistic, they are one and the same principle with measurable degrees of existence. Only from a comparative framework standpoint is credence given to any mythical form. The above observation is not negating the process, it is questioning the markers which evolve into imaginary separation (trapped in a comparative framework mythology). That particular process can and does create a false mythical reality that appears divisive. We cannot exist within a divisive reality! Reality must be complete for us to recognize its existence. Where there are no dichotomies within the premise that ‘everything is’, there exists no antagonistic position. The distinction between human constructs of positive and negative are matters of mythical perspective wherein no experience of the Absolute is available. It is because the human ’mind’ per se places its own construction on its immediate experience, and must have its particular interpretation based on what it considers ’knowledge’. There is a difference between ’mechanical knowledge’, and ’pure knowledge’. From the mechanical knowledge standpoint which can only deal ‘in indirect conscious interpretation’, it is quite correct to say that that form of knowledge is incomplete, and it always will be. Pure knowledge experienced via our brain knows no separation, nor antagonism, and is responsible for our ability to recognize the actions of others who may pry open wooden crates with a lever, or drive off uniformly at traffic lights. Within that cohesive activity it precludes ’a matter of opinion’ and by themselves can become subjects of a pure knowledge experience. To repeat, it is a form of ’communion’ with ’what is’, and available to all. Where drivers at traffic lights universally conform to their particular road code, and where universally there is a language which identifies their activity as Mutual Agreement, or any other logical definition, we can concur with the common-sense conclusion that we have universally established that within language and common activity, there is indeed a truth formed. The coalescence between universal language and universal activity are the logical constructs that create civilizations. There is a vast social network of common activity that solidifies the logic into an honest and persuasive conclusion that confirms innate common principles –knowledge. = Stepping stones 5. = The Art of making sense of everything. How to understand principles. # Principle. A fundamental truth or proposition that serves as the foundation for a system of belief or behaviour or for a chain of reasoning. # All principles are interdependent, interconnected, and infinite. # Each one is dependent on the other two. Examples of a principles template and how to define them without dichotomies. Communication. Truth. Standard. Proof. Express. Contribute. Mutual. Direction. Advance. Comfort. Organize. Certain. Immediate. Interest. Improve. Present. Constructive. Gain. Trust. Progress. Source. Knowledge. Basic. Original Reality. Awareness.Freedom. Purpose. Connect. Understand. Support. Peace. Cause. Unity. Ability. Rights. Honest. Discover. Positive. Energy. Balance. Good. Courage. Willing.  Control. Use. Association.  Observe. Reason. Easy. Wealth. Simple. Law. Increase. Order. Flow.Co-operation. Exact. Quality. Accuracy. Strength. Responsible. Operating. Creative. Measure. Recognition. Accept. Constant. Obligation. Include. Dependence. Relationship. Value. Success. Principle. Equality. Stable. Share. Love. Sustenance. Action. Identity. Intelligence. Education. Secure. Facts. Agreement. Information. For. Rules.Clear. Yield. Example: Success = Securing facts                = Responsible co-operation               = Constructive knowledge So success by definition is : Securing facts through constructive knowledge and cooperating  responsibly. All definitions of success from your template are infinite. You will find your own suitable definition. ==== There are no dichotomies! ==== Any principle is correctly defined by any two other principles. You create a new language of Absolutes. Using conjunctions you can write your own book. The man whose book is filled with quotations has been said to creep along the shore of authors as if he were afraid to trust himself to the free compass of reasoning. I would rather defend such authors by a different allusion and ask whether honey is the worse for being gathered from many flowers. Anonymous, quoted in Tryon Edwards (1853) The World’s Laconics: Or, The Best Thoughts of the Best Authors. p. 232 Amen to that! “One is not born, but rather becomes a woman” Simone de Beauvoir. “Time does not change us. It just unfolds us” Max Frisch. We experience ourselves our thoughts and feelings as something separate from the rest. A kind of optical delusion of consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Albert Einstein, in One Home, One Family, One Future. = Who we are. =   It is notable that within the structure of Cartesian dualism, Descartes' personal address to innate knowledge he attributed to ‘thought’ which he identified as being distinct from his body. How different Western philosophy may have been if his attribution had been toward his brain and the existence and evidence of other physical entities that functioned every bit as efficiently as he did. The premise that Descartes operated from ‘never to accept anything as true’, was simply a wrong ended approach which brought him into conflict with his passing acceptance of innate knowledge, that the idea of God was innate to his being. To view the proposition that ‘everything is true’ allows reason to seek and identify that measure of truth. No quest can be productively based on cynicism or denial, nor adherence to belief systems that separate experience, knowledge, and Reality. We have the obligation to question whatever reality has placed before us , but if we constantly deny its existence and attempt to ‘disappear’ it from our experience, then we are in danger of never experiencing that reality for what it is..   For anyone to say that ‘everything is’ is a simple linguistic absolute that no amount of ‘more reasonable’ requests (above) can deny. Those requests only appear to be governed by the difficulties of ‘mental complexities’, and embedded ideologies. To accept that ‘everything is’ as an absolute, is a realistic basis to establish any reality, and comprises the basis for reason to be activated. If there is ‘nothing’, nothing can be achieved. Within that which is Absolute there are no dichotomies. Therefore there are no antagonistic positions available. Everything that is, is a measure of the Absolute.  We are always in the present, everyone and everything. Instant elementary ‘knowledge’ which we all share, and must admit to. In being alive, we do not have the ability to not be here, and we do not have the ability to not know! To have a problem in addressing what Truth and Knowledge are, to the point of denying their existence, then that problem exists well below the scale of Reality. To repeat knowledge is not the proprietary right of any individual, it is enshrined in the principle of agreement that we mutually exercise to establish its own reality. The reality of experience is not, nor ever will be, a personal possession that we can have and hold. Its reality becomes more alive when we see the same activity being practised by others. Then we know we are sharing that reality, and that experience. We cannot "have" the principles that exist, but when we undrestand them then we are obliged to use them with integrity. That form of integrity in any language, is an added foundation stone to any belief system Mechanistic ‘I’ has no concept or understanding of ‘pure knowledge’. Only when we break free of the myth that some clarity becomes apparent, and we have the opportunity to engage with what is real. Philosophy it appears to me is constrained by individual ‘thought’ processes, which (without experience) cannot escape from that individuality. Those ‘thought’ processes conjure up a human history of inflexible, and impossible propositions which only serve to protect that individuality. Descartes ‘cogito ergo sum’ has compounded the difficulties by strengthening the incorrect premise of a false individuality. ‘I’ is a phantom consciousness much like a phantom pain experienced after a limb is amputated. The brain registers the pain signifying that something should be there. Likewise our brain has that same relationship with Nature and Reality. It is analogous to our brain dealing with a ‘phantom reality’ knowing that something is missing but is continuing to evolve to establish the whole. There exists a ‘phantom chasm’ between our brain and Reality and an understanding of its properties. We are robbed of real meaning. Evidence, recognition, and the truth are the principles it uses to reform. Within their structure is the meaning of reforms. Nature does not impose any morality on us, the principles implied in morality are there for us to understand and use. Our brain has the capacity, once reality is correctly examined, to recognize ‘that which is’. Once realised it becomes embedded. That ‘phantom consciousness’ is an experience removed from its proper environment. It takes its proper place when we experience reality for ‘what it is’, which provides the totality of meaning. True experience allows us entry to the quality of knowledge that is a continuous reality. So long as anyone believes that human experience is based solely on indirect conscious interpretation (mechanical disposition), any ‘knowledge’ derived from that experience will be incomplete. That form of philosophical negative conclusion can come down to not believing that Reality exists (a chair is not a chair, is not a chair etc,), or that our experience of ourselves and others is real, and discount any other form of knowledge that threatens that belief. There is an intellectual dishonesty in denying the existence of principles. Hostility towards the truth leads inevitably to negativity. Being here demands, not denial, but the right to be recognized. We are the recipients of a ubiquitous communication system – making known. We can only understand that which we know. We make known all the time. Knowledge of Reality – Truth – the Absolute is a collective inclusive experience of the principles we share, and never the property of any individual. To ‘know’ ‘who we are’ is an inclusive experience of the principles involved. Never ‘cogito ergo sum’.   Philosophers in investigating the nature of knowledge and the Universe, firmly established for themselves that the source of reason and logic was located in a mythical concept ‘the mind’. From the wrong basis evolved elaborate and metaphysical constructions which removed the investigations further, and further, from the truth. To comprehend the material world, and give it credibility, the recognition of implicit principles is paramount. We need to construct a language that provides that form of recognition. Any philosophical theory of ‘mind’ that will deny the evident structure of solid objects is misguided by the injection of a mythical entity (mind) that determines that seeing solid objects is a ‘perceptual illusion’. That form of determination is singularly narcissistic, empowered by the self-induced threat that venturing into a ‘materialistic’ world is a loss of that illusory self, and all the belief systems it has constructed to protect it. That erroneous established view that not addressing ‘materialism’ as a profound Reality, and as only a ‘perceptual illusion, is compounded by the belief that that form of illusion is implicit in every human view available. We cannot manufacture knowledge that leads to a mechanistic understanding of ‘what is’ , nor the principles which are its properties – however much dogma is practised. We can only aspire to relate to ‘necessary factors’ that are the implicit fundamentals of existence. For me to use a lever to open a crate is a form of communion with Archimedes through the principle he enunciated. It is now not ‘necessary’ for me to go through the same experience as Archimedes to establish that ‘necessary factor’ or ‘principle’. It is now common-place, and common-sense to utilise the principle. The extract below provides some explanation of the brain processes in action Universally, and coincides with any reasoning on the observance of the leverage principle, and the actions of motorists conforming to the Road Code wherever traffic lights exist. ''The right-to-left shift of mental control looked increasingly like a universal phenomenon, capturing the essence of every learning process on every time scale, from hours to years. An individual faced with a truly novel situation or problem tackles it mostly with the right hemisphere. But once the situation becomes familiar and is mastered, the dominant role of the left hemisphere becomes evident. It looked like the empowering patterns capturing the essence of the situations (or rather the whole class of similar situations) were, once formed, stored in the left hemisphere. (The Wisdom Paradox. Professor Elkhonon Goldberg. P202)<ref>{{Cite journal|last=Vandermeulen|first=Jo|date=2008-08|title=Verstand komt met de jaren|url=http://dx.doi.org/10.1007/bf03077135|journal=Neuropraxis|volume=12|issue=4|pages=137–139|doi=10.1007/bf03077135|issn=1387-5817}}</ref>'' = Limitations. = However limited our view of connectedness is, or however tenuous the reality our experience is, ‘everything is’, and everything is connected. Innate knowledge and the fundamental nature of Man is the prior source of knowledge that seeks and identifies that connectedness. Attempting to address what we don’t know is that mythical infinite regress toward that parallel mythical ‘nothingness’. To always address what we know establishes Reality. To establish knowledge of principles, start from ‘we are here’. The natural principles within the diversity of human culture and activity when recognized as mutual ‘necessary factors’ will have the effect of enhancing and directing vital energy toward the very process of  the communion we seek, and the gradual penetration of a reality that always exists. Knowledge of Reality is not ‘different’ in other locales. The fundamental principles are the same. Addressing ‘what is’ instead of denying ‘what is’ is the basic construction of real knowledge. Within the structure of the Absolute we are all the same with a magnificent differential in our expression of the principles of necessity. That expression is our ongoing effort toward its own experience which gives it life and meaning. That experience in turn exposes us to an immediate Reality that is in communion with the fundamental structure of our being. All that we can contribute toward that is 50%, the other half is in our momentary relationship with Reality – then we know! That form of knowledge is always available through that form of experience, and it always comes in the form of confirmation which reforms. Until that experience our prior condition appears mechanistic, without direction, or understanding. Reality, life, is not mechanistic. We are the recipients of innate principles with the constant potential to experience those principles in action (Archimedes et al). Dogmas, ideologies, are the restrictive practices used to blur the recognition of principles operating to a level that understanding of that common and constant activity is virtually denied. Our natural capital (principles) is degraded to the point that their factual evidence is reduced even to the point that they are categorised as a ‘perceptual illusion”. We can trust facts 2+2=4. Simplicity has its own majesty. Anything circumscribed by reason requires control of our emotions. To comprehend the material world, bring it alive, and give it credibility, the recognition of its implicit principles is paramount. We need to construct the language that provides evidence of that Reality. Any philosophical theory of ‘mind’ that will deny the evident structure of solid objects, is misinformed by the injection of that mythical entity (mind) that determines that seeing solid objects is a ‘perceptual illusion’. That form of determination is singularly narcissistic, empowered by the self-induced threat that venturing into a ‘materialistic’ world is a loss of that illusory self, and all the belief systems it has constructed to protect it. The erroneous established view that not addressing ‘materialism’ as a profound reality, and as only a ‘perceptual illusion’, is paradoxically compounded by the belief that that form of illusion is implicit in every other human view available, thus it then makes its own sense, form and justification to the illusion! The evident question we must ask, ‘how does a ‘mind’ conclude that ‘immaterialism’ exists universally? Surely it is a simple but massive contradiction in terms. If there is nothing there but ‘perceptual illusion’, how can you attribute it to other ‘minds’. The oxymoronic effect of narcissism is that it is the very denial of ‘who we are’. Man is not composed of an overwhelming self-love. That mythical embedded belief cannot consider the possibility of underlying principles that are the real life force of Man in his relationship with Reality. The truth of that, is that humankind (in spite of itself), evolves towards its own Reality. The only human values that exist, lie in Man’s recognition of the principles involved that provide human direction. Our ‘material brain’ is a product of Nature's evolutionary process, and has innate within it the same principles that exist in all matter. That ‘which is’, is the truth, and our brain evolves to process that at every level, and we constantly manifest that in every action we take – whether we like it or not. The fundamental similarities between human beings is that we are not only evidently human, but that we also function and construct societies that we recognize as beneficial to our immediate well-being. All social function is determined by our brain capacity and its ability to postulate the relationship it has with Universal principles. = Illusion. = Considering that we can contradict things is an illusion. We can never contradict the truth. We do not have the ability to create proprietary constructs of reality. That ‘which is’, can only make its basic properties transparent to us through direct experience. Imaginary concepts must in the end conform to a measurable construct that we can identify. Within the structure of any philosophical theory of ‘knowledge’ it must contain the basic elements of truth at all times, or there is nothing!! To say that ‘everything is’ is motivated by pure reason experience as an objective, and subjective reality and as an axiomatic grammatical premise that no amount of mental acrobatics can deny. We can only deal with ‘something’, whatever it may be. There is no metaphysical construct that can provide evidence that ‘nothing’ exists, outside a mythical mind. Explaining experience beyond ‘thought’ processes requires a definitive language that deals with the reality itself. We all Know. It is innate. The ‘difference’ between us is only the measure of the knowledge that is made manifest, and that knowledge continually proliferates. The ‘individual perspective’, and the illusory ‘I’ which dominates, is the barrier to any relation to ‘what is’, and the malady of never experiencing the truth directly!   Truth, knowledge, agreement are the abundant and embedded Absolutes that form the structure of human evolution. That we constantly utilise and improve on their use is evidence of their reality, and the material transparency within every social structure. The survival and proliferation of such realities should be the evidence to establish that ‘that which is’ is Absolute. When we focus our ‘perspective’, opinion, or a hypothetical consideration of a space, time, or identity to question a Universally accepted fact, it is hardly a categorical argument to dismiss that which is true as nonsensical. Any denial that 2+2=4 is a fundamental truth hardly takes into account that the reality of such basics are vital to the success of higher mathematics. Unless the basics are continually correct, and evidently so, then no correct solutions could evolve. We know that within any basic structural ‘use’ that the calculation is correct. We commonly accept its correctness as an embedded reality. All forms of lower or higher mathematics would have the axiomatic principle of ‘correctness’ as their basis to extend from. Also, they would have as an axiom that the reverse is true. The 2+2=4 is, in its reality, the epitome of balance and construction. The 2+2 reality forms its correct conclusion when the principles of mathematics are propounded and they conform to transparent truth and arrive at 4. Only when it ‘adds up’, does it become a truth that we all recognize. Our greatest ignorance is taking for granted the proliferation of such truths through an ideological blinkered perspective. Because truth takes a commonplace form it is no less fundamental. Unless there is correct knowledge as to the existence of fundamental truth, that ‘which is’, goes unrecognised. That form of truth must be applicable to all. Truth exists in everything – it is an evidential reality. Searching for an esoteric truth is chasing shadows. Every truth is a ‘necessary factor’, and fundamental to our existence. Because of the imposed limited perspectives (via education, ideology, beliefs) that which is evidently true, and transparent, is delegated to a position of simple practicality with conditions placed on it which further deletes its substance, and we have the awful predilection of conforming to the attempted destruction of that which is true. Do we have a problem with seeing something, which is correct, as also being true? All truths are fundamental. They are not subject to attempted denial because of any diminished realisation at any point in time. Where there is reasonable evidence of balance, equity, and agreement we can conclude that a truth exists. Once innate information of that truth becomes transparent, it becomes an embedded useful human utility that must have some measure of fundamental truth as their starting point. From any common-sense, or ‘more reasonable’ position, it would be more productive to view reality as possessing at every level the same innate values or principles consistent with our ability to measure, or recognize them. To view reality as having ‘different’, or antagonistic properties, is simply a misguided view of ‘what is’.  That form of perspective is counter productive when it attempts to establish mythical dichotomies as realities in their own right. When the reality of principles are made transparent, we can then ‘more reasonably’ make use of them to further their basic existence. Here we use reason to exemplify their necessary function, and once established it becomes (if necessary), ‘more reasonable’ to locate them in all things. The dematerialization of any object through the practice of ‘perceptual illusion’ is an attempt to deny the reality that exists. Where perceptual illusions are concerned, innate direct communion with that which is, suspends the effect of such illusions. All the properties in a chair are recognized as the reality that exists. That is materialism. A chair does have the principles of form, design, structure, colour, substance etc.  However it is analysed – it is a quantifiable reality.   = Human representation. When we understand the validity and existence of principles in all things, it is easy to understand that ideological dogmas are never the foundation for real knowledge, or that direct experience of ‘what is’. Our real perspective is not some individualistic experience that confines us, it is that expanse in which we exist that offers us the view of that expanse. Everyone has the potential to go beyond their ‘apparent’ human perspective limitations. Shifting our sense of perception toward that which is basic, paradoxically extends the experience of that which is true. Let general knowledge be directed toward the performance that identifies the measure of principles that are enacted. Therein lies the production of knowledge that offers a sustainable growth of that vital universal aspect of knowledge, where, reason and truth, can prevail. Any correct definition is language itself, opening the door to that reality experience which is critical. Only when we know and experience that the same reality (with all its principles intact) exists for all of us can we then recognize the mythical distinctions that are taken as being real. The majestic experience of that reality goes well beyond historical beliefs. Exploring simple ‘necessities’ is not based on any sacred text, but the privilege of recognizing a sensible evolutionary path through life. Whatever may be in the future, is implicit in the material world now, and it has always been so. hdf2waxobozwxnekja966r0xk388psl 0 281651 2410846 2391121 2022-08-01T22:50:44Z Dave Braunschweig 426084 Dave Braunschweig moved page [[User:Jacsaw/sandbox/Who's Online? Building the next 30 years of the Internet]] to [[Advocacy in Technology and Society/Who's Online? Building the next 30 years of the Internet]] without leaving a redirect: Moving to main space wikitext text/x-wiki === '''Topic Summary''' === This week's topic was broad overview of early internet ideals, the state of the internet today and who has access to it, and whether or not internet should be a public utility. We discussed what percentage of the world and of the United States is online, disparities in service in New York City specifically, how "healthy" the internet is, and what it would take to make the internet a public utility - meaning everyone could access the internet. For this class we read: * Hacker Practice by Gabrielle Coleman and Alex Golub, which takes a close look into the world of hacking and dispels some of the prevailing notions attached to hacker profiles.  A cultural, ideological perspective is offered.  Keywords of the article included: computers; ethics; hackers; law; liberalism; moral genres; and technology. Hackers were described in broad terms across multiple genres and not restricted to the limitations ascribed by society. The traditions of liberalism as applied to the computer world were examined carefully. '''The article gives three examples of hacker ethical practice.'''  The first was '''crypto-freedom''' rooted in principles of privacy and self-reliance. '''Free software''' was the second and a natural extension of the first’s principles.  The last is the '''hacker underground'''. This refers to ideals that realistically cannot be achieved and are considered a show of defiance to established institutions. Government crackdowns on hackers surged in the 1990’s for computer related fraud. The FBI made some of the subjects a top priority. This became a time of turmoil. Protests and conventions like '''HOPE''' (hackers on Planet Earth) were born. In the hacker community, a culture developed from diverse personalities, practices, moral codes, and ethics. Questions of connections between property rights and human freedoms, individualism and collectivism, and self-development and human flourishing. These issues will continue as '''technology is central to a sense of self and existence'''. This is at the heart of hacker culture.   * [https://certificates.creativecommons.org/cccertedu/chapter/1-1-the-story-of-creative-commons History of Creative Commons], which talks about copyrighting. “Creative Commons was created to help address the tension between creator’s ability to share digital works globally and copyright regulation.” They share that in 1998 the Sonny Bono '''Copyright Term Extension Act (CTEA)''' was passed that makes a copyright term equal to life of creator plus 70 years. They argue that copyright ending is important for works to move into public domain - especially in age of internet. “All creativity and knowledge builds on what came before, and the end of a copyright term ensures that copyrighted works eventually join the pool of knowledge and creativity from which we can all draw to create new works.” In 2002, a non-profit organization called Creative Commons was formed and created '''Creative Commons licenses''' - "a set of free, public licenses that would allow creators to keep their copyrights while sharing their works on more flexible terms than the default “all rights reserved.'" '''This basically made copyright automatic.''' The idea behind Creative Commons was "to create an easy way for creators who wanted to share their works in ways that were consistent with copyright law." * The [https://2020.internethealthreport.org/ 2020 Internet Health report], which provided an analysis of multiple subtopics like a physician performing a battery to assess a patient.  A sample of some of those topics are spotlighted as follow: Human health was a big part of the internet health for 2020. The country was shut down and the internet was the lifeline for all America during the peak of the global pandemic. But not all had access, even to a cell phone. The big seven tech companies consolidated more power and their profits soared. But it was surprising to see a large number of tech employees speak out concerning the injustices existing across all segments of the tech industry. The algorithms were of major concern. Digital rights advocates voice major opposition to the internet shutdowns in authoritarian countries—the splinternet. A people’s movement evolved led by protests around the world. There were calls for democratic freedoms and protections for Black people. The year 2020 was the year of racial reckoning. This demonstrated the connection between the health of the internet and that of the people. There was a need for balanced regulations. This balance would preserve liberties and affordability. The world began to learn what was already known; technology is not colorblind. Searches for people of color displayed negative information such porn, different from that for whites.  The internet was white centric. The MIT Media Lab demonstrated that Amazon’s facial recognition software was extremely biased. The report goes on for pages highlighting disparities. It provides a good foundation for in-depth case studies and analyses. During class, we talked about early internet ideals, who is online today, and the internet as a public utility. '''Early Internet Ideals''' "Had the technology been proprietary, and in my total control, it would probably not have taken off. You can’t propose that something be a universal space and at the same time keep control of it." - Tim Berners Lee (1991) * Tim Berners-Lee outlined protocol for the first public "internet" plan in 1990. Early internet ideals revolved around decentralization, net neutrality, bottom-up design, universality, and consensus. * Decentralization is the idea that there is no single, central authority of the web or central controlling node. * Net neutrality means that if two people pay to connect to the internet at similar quality levels, they will be able to communicate at the same level. * Bottom-up design means that code was not written and controlled by a small group of experts but was developed publicly with maximum participation and experimentation encouraged. * Universality means that all computers must speak the same languages to each other so that everyone, regardless of identity or location or resources, can publish on the internet. * Consensus simply means that everyone has to agree to use universal standards for them to work. These early ideals gave rise to current approaches such as Open Data, Open Government, Open Access, education, and Free Culture & Creative Commons. '''Who's online now?''' According to data from the International Telecommunication Union ( ITU ) World Telecommunication/ICT Indicators Database, made available through the [https://databank.worldbank.org/home.aspx World Bank DataBank], in 2019, '''56% of the world''' was online (defined by having used the internet in the past 3 months). As of 2021, '''37% of the world had no connection''' to the internet. Using the interactive data visualization tool, you can observe a steady increase in the percent of the population online beginning around 1995, with a more steep increase from 2018 upwards. In 2019, '''89% of Americans''' had used the internet in the past 3 months, logging on from various types of devices. We expect this number to continue to rise as the world went online during COVID-19 to learn, work, and connect with each other. '''Internet as a Public Utility''' As a class, we discussed the reasons for rapid or slowed connection to the internet and the noted on the timeline when smart phones and internet-connected devices became common and available. We focused specifically on New York City as a case study and discussed the disparities in service in NYC. For example, according to the [https://www1.nyc.gov/assets/cto/downloads/internet-master-plan/NYC_IMP_1.7.20_FINAL-2.pdf NYC Internet Master Plan], '''40%''' of NYC households lack the combination of home and mobile broadband. '''18%''' of NYC residents lack both home AND mobile broadband. '''46%''' of households living in poverty don't have broadband at home. In groups, we explored four different organizations/plans to create universal broadband service in NYC: [https://redhookwifi.org/ Red Hook Community Wifi], [https://www1.nyc.gov/assets/cto/#/project/internet-master-plan Internet Master Plan], [https://www.link.nyc/ LinkNYC], and [https://www.nycmesh.net/faq Mesh NYC]. === Insights === Early internet ideals centered the idea that everyone should be able to access the internet equally but as technology has progressed, this has not been the case. As the data from above shows, there are large percentages of the population who are still not able to access the internet. It is important to note that access does not signify quality. This means that even though a majority of the world's population has '''access''', the '''quality''' of internet service people are accessing varies greatly. Many of view the internet as a given, especially with such a high percentage of the population using it consistently. We know that the internet is necessary for things like applying to jobs, paying bills, and connecting with loved ones. So, the idea that not all American's have access to the internet might feel foreign to some. We can, and must, address this by making the internet a public utility. Even for folks who do have access, the quality of that access differs depending on a variety of variables (see the [https://2020.internethealthreport.org/ Internet Health Report]). This is an issue of power and privilege; those with enough money to afford the internet are able to access it, and those without the income to afford the internet cannot. Who holds money and wealth is directly tied to white supremacy and oppression, which means that folks with the most privilege (i.e. white, cisgendered men) are more likely to have more wealth and income and therefore will have the best access to the internet. These are also the same individuals who control the internet. According to the [https://2020.internethealthreport.org/slideshow-internet-health/#3 2020 Internet Health Report], there are '''seven''' companies that control the internet and these seven companies are part of the '''ten''' largest companies in the world; most of these companies are controlled by white, cisgendered men. <blockquote>"It's understandable that many people feel afraid and unsure if the web is really a force for good. But given how much the web has changed in the past 30 years, it would be defeatist and unimaginative to assume that the web as we know it can't be changed for the better in the next 30. If we give up on building a better web now... we will have failed the web." -Tim Berners Lee (2020) </blockquote>When imagining the next 30 years of the internet, we must push back against the ways that the internet has been used for control and violence. This means that the internet must become a public utility to expand access and ensure quality. We must advocate to combat the specific ways that the internet is used to harm marginalized groups of people such as women, queer folks, Black folks, Indigenous folks, and all people of color. === What does advocacy look like here? === Surveys of the advocacy work indicate that '''access''' is the issue of the day. Within this space, supporting guaranteed access to broadband services seems to be a common interest developing from multiple sectors ranging from environmental justice organizations to housing rights initiatives. This evolving common theme among multiple types of social justice organizations provides an opportunity for a united campaign platform to push for broadband access rights. Access to information is at the heart of social justice movements. Class readings provided detailed ideas to increase internet access. They included a call for municipalities to provide internet access and the FCC to declare broadband as a public utility. Both avenues could be promising. Another option is for government bodies to play a role in compelling broadband service providers to deliver access to all within controlled jurisdictions. For example, the City of New York licensed Verizon as a vendor to roll out broadband cable in certain parts of the city. As part of that agreement, the city required that provider to guarantee connections to remote neighborhoods. The bottom-line is that broadband access is '''essential for government public policy''' given the ongoing of the pandemic. Thus, it seems like the government has a key role and an important interest in ensuring broadband for all, at least on this basis alone. There are however many more reasons for the government to guarantee access, like an educated electorate. Following the works calling for access should prove to be interesting.   === Additional References === 1. [https://publicknowledge.org/we-already-knew-broadband-should-be-a-public-utility-the-pandemic-made-it-obvious/ Public Knowledge: We Already Knew Broadband Should Be a Public Utility. The Pandemic Made It Obvious.] * This article explains that during COVID-19, the rapid migration of services like healthcare, vaccine coordination, education, and social connections has made Broadband an increasingly essential part of everyday life. * A survey showed that '''eighty-seven percent''' of people found the internet important during the pandemic; '''fifty-three percent''' found it necessary for critical and everyday tasks. * The FCC has ample justification for instituting telecommunication regulations (under Title II) of broadband services similar to those of water and electric services, which include moderating profit incentives. The FCC has broad powers to prevent digital redlining, unreliable services, inequitable practices. * Congress’ fifty-dollar, monthly subsidy to low-income families for broadband services during the pandemic acknowledges the need for government action on this issue.   2. [https://www.aclu.org/issues/privacy-technology/internet-privacy/public-internet-option ACLU: The Public Internet Option] * This article by the ACLU argues that the public should be guaranteed equitable, trusted, and accountable broadband service providers that respect free speech and net neutrality principles. They make it clear that municipalities are arguably better positioned to fill this role as they do with water and electricity. * They argue that within this advanced information age, it is critical that the internet is established as a public utility and not subjected to the FCC’s politically charged regulations of private companies. * Low-income and rural communities should have the same access as other communities; households with incomes below $20K, African Americans, and Hispanics had the lowest broadband utilization rates. Municipal systems could ensure equity in the distribution of broadband and public Wi-Fi services across all income levels and communities. * They state that the public should organize, demand, and create this pathway. 3. [https://datareportal.com/reports/digital-2022-global-overview-report DataReportal: Digital 2022: Global Overview Report] * This report was published January 26, 2022 by Simon Kemp. It provides an incredibly comprehensive overview of the state of the global digital world as of January 2022, in both SlideShare format and as a written report with notes on how they acquire their data. * Some key takeaways from the report are: As of January 2022, worlds population is at 7.91 billion. 57% of the worlds population lives in urban areas. 67.1% of the world’s population uses a mobile phone. The global total has grown 1.8% over the past year; 95 million new mobile users since January 2021. Global internet users are at 4.95 billion. Internet users have grown by 192 million over the past year, but because of research and reporting restrictions due to COVID growth trends might actually be higher. 4.62 billion social media users in the world as of January 2022, equal to 58.4% of worlds population. This has grown by more than 10% over the last year; 424 million new users joined social media in 2021 and they include in-depth analysis of different social media platforms and their usage. Internet users have more than doubled over the past decade. * The number of folks without internet (“unconnected”) has dropped below 3 billion for the first time - which is a milestone - but Kemp highlights there is plenty of work to do. More than 1 billion people are offline across Southern Asia. Almost 840 million people are offline across Africa. More than 400 million people are offline in China, even though China accounts for 1 in 5 of the worlds “connected” population. 1 in 4 people across “lower- and middle- income countries” are unaware of the existence of mobile internet. * Looking ahead, Kemp says we should pass the 5 billion internet user milestone in 2022. We should expect to see regulation and taxation of cryptocurrencies. Wider reach of digitally distributed culture (i.e. shows, movies, etc from other places in the world). And finally, a convergence across digital activities and more immersive content experiences. The report ends with a sweet inclusion of which pets (dogs or cats) rule the internet. 4. [https://www.popularmechanics.com/technology/infrastructure/a29666802/future-of-the-internet/ Popular Mechanics: Future of the Internet] *This article was published September 30, 2021 by Matt Blitz. The article provides an assessment of what the future of the internet will be, based on professionals in the field. The article feels very futuristic but is grounded in conversations with folks engaging in this work which makes these futuristic claims more real.   *Within the next quarter century, our digital presence will be ingrained as part of the physical world. The web will be part of the real world. There are folks who state the internet will be “pre-loaded in our consciousness.” Verbal and auditory communications will be favored over typing messages. *As we discussed in class, privacy and surveillance will continue to be at the forefront as privacy continues to decrease for the sake of convenience. *While there is a lot of optimism around the future of the internet, there are conversations around the potential implications of expanded and ingrained internet and technology. How and who will regulate technology is up in the air - as governments are often behind on tech solutions and decisions will need to be made on a global scale. *It leaves the reader with lots to consider regarding potential futures of the internet and technology. plemdqaf3fbrtpc9tuqypf2pgrohehz 0 282299 2410941 2406218 2022-08-02T08:10:49Z 99.45.70.162 /* Units: $/stream or streams/$ */ Align decimals in table to make numbers easier to compare. wikitext text/x-wiki <!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DAF3EF; padding: 1em; overflow:hidden;"> <div style="overflow:hidden;"> <!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --> <div style="width:50%; margin-right:4em; float:left;"> {{Eventmath lesson plan overview <!--EDIT BELOW THIS LINE ONLY--> <!-- The summary below is displayed as a guide for visitors and contributors. You can publish without it, but it's required for removal from draft status.--> |assumed-knowledge = <!--Below, please briefly describe any mathematical skills or background knowledge students should have before completing this lesson.--> Students only need familiarity with basic arithmetic. |activities = <!--Below, please briefly summarize what students will do during this lesson and what they will learn.--> Students will consider a Tweet discussing streaming service pay rates for artists, and then they will create a graphical representation of the data, as well as alt-text for sharing that graph on social media. |class-time = <!--Below, please type one of the following, exactly as it appears: 00-15 minutes, 15-30 minutes, 30-45 minutes, 45-60 minutes, 60-90 minutes, more than 90 minutes. Your best guess is fine! It can be changed later.--> 15-30 minutes <!--Next, you'll provide all citation data for the source (article/social media post/video). Article example: | source-website = Huffington Post | source-title = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It’s Time To End The Electoral College. | source-date = 2016-11-10 | source-url = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-url = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-date = 2021-10-28 Post example: | source-website = Twitter | source-title = Remember: Germany is supposed to be a SUCCESS STORY. | source-date = 2020-12-22 | source-url = https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-url = https://web.archive.org/web/20201223064324/https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-date = 2021-02-16 Notes: 1. To ensure lesson plans can be sorted by date, please enter all dates in the format YYYY-MM-DD, including leading zeros: January is 01, not 1, for example. 2. If you cannot find a date, use n.d. as the date (no date). 3. To ensure that the article/post will remain accessible for educators and students... a. Archive it at https://archive.org/web/ (use the "Save Page Now" form to generate a permanent URL). b. Paste that URL here as the source-archive-url. c. Enter the date you archived the source as the source-archive date. 4. Linking to a website helps the site. If the lesson plan is debunking disinformation, you may avoid spreading disinformation by omitting the url of the original site entirely. In that case, use the URL of the archived version as the value of source-url, and omit values for archive-url and archive-date. --> | source-website = Twitter | source-title = Would you like another reason to switch to a music service that is NOT Spotify? | source-date = 2022-02-07 | source-url = https://twitter.com/zoecello/status/1490817801552027651 | source-archive-url = https://web.archive.org/web/20220305162020/https://twitter.com/zoecello/status/1490817801552027651 | source-archive-date = 2022-03-05 }} </div> <div style="float:left; width:20%;"> [[File:Eventmath_logo_bitmap_graphic.png|250px]] </div> </div> <div style="width:70%; float:left;"> <!--INSTRUCTIONS FOR THE BODY OF THE LESSON PLAN 1. Note that "==My heading==" is a level 2 section heading, "===My subheading===" is level 3, etc. Level 1 is reserved for the title. 2. After you fill in a section, please delete the ''italicized instructions.'' 3. Check out the "Advanced" and "Help" menus at the top of the editor if you don't know how to format lists, etc. --> ==Activities== The main idea is to help students make sense of very small/large numbers and compare them using appropriate units of measurement. ===Initial discussion=== Provide students with the text or a screenshot of the original Tweet and ask them to quickly answer some questions as if they were just seeing it in their timeline and trying to make sense of it. Consider using polling software to gather answers and display the results, or maybe give a "pop quiz" and then ask students to swap papers with their neighbor to compare ideas. For convenience, the text of the original Tweet is quoted below, along with some possible questions to ask about it: {{Quote box |quote = Would you like another reason to switch to a music service that is NOT Spotify? This is what services paid me on average per stream in 2021 Amazon - $0.0107463 Tidal - $0.007981 Youtube - $0.007795 Apple - $0.006040 Deezer - $0.0045861 Spotify - $0.003661 (I am my label fyi) |author = [https://en.wikipedia.org/wiki/Zo%C3%AB_Keating Zoë Keating] |source = https://twitter.com/zoecello/status/1490817801552027651 }} #Based on the information shown, how many streams would this artist need (on average) to earn $1 on the Amazon platform? What about for Apple? And for Spotify? #What would 1 million streams pay on Amazon? What about Apple? And Spotify? #How many Spotify streams are needed in 1 year to amount to the same pay as $15/hour for 40 hours/week? #How do the pay rates of Amazon and Apple compare? Write a sentence that expresses a numerical comparison between those two. Then, do the same for Apple and Spotify. (I had success with posing these questions to students on paper as a mock "quiz" for a couple of minutes, and then asking them to compare answers in groups of 2 or 3 to see if any questions arise before the main task.) ===Main task: create a graphical representation of the data=== The central activity of the lesson asks students to present the given information in a graphical form instead, and then write appropriate alt-text for sharing the graph as an image on social media. ====Units: $/stream or streams/$==== The original Tweet expresses the pay rates in the units of '''dollars per stream.''' While you discuss the solutions to the "quiz" questions, steer the discussion towards the idea that perhaps '''streams per dollar''' might be a more useful unit, for the sake of comparing the various platforms. From the point of view of an artist, it's probably much easier to compare the numerical values of "How many streams are needed to earn $1?", as opposed to comparing the tiny decimal numbers expressed in the original Tweet. Or, as hinted at by one of the possible questions above, students may want to express everything in the units of '''yearly streams needed to make annual minimum wage,''' which would essentially be a multiple of the '''streams per dollar''' unit. For your convenience, here is a table with the information from the original Tweet, as well as the corresponding value expressed in the '''Streams / $''' unit. (To obtain the '''yearly streams needed to make annual minimum wage''' value or something like it, simply multiply the '''Streams / $''' value by the annual wage you want to consider.) Aligning decimal points makes the sizes of the numbers easier to compare. {| class="wikitable sortable" |+ |- ! Platform !! $ / Stream !! Streams / $ |- | Amazon || $0.0107463 || style="text-align:right;" | 93.1 |- | Tidal || $0.007981 || style="text-align:right;" | 125.3 |- | YouTube || $0.007795 || style="text-align:right;" | 128.3 |- | Apple || $0.006040 || style="text-align:right;" | 165.6 |- | Deezer || $0.0045861 || style="text-align:right;" | 218.1 |- | Spotify || $0.003661 || style="text-align:right;" | 273.1 |} ====Graphical representation of the data==== You might also steer the discussion to considering whether the original Tweet is engaging or informative for a reader scrolling through their timeline. For example, you might ask students to reflect on what they would do if they encountered this Tweet in their feed or someone shared it with them. How would they process this information? What calculations, if any, would they do to make sense of these numbers? Through this discussion, students may realize that a visual data display, such as a bar graph, could be a more engaging way of presenting the underlying information. Ask students to resume discussion in their small groups and make a "mock up" of a graphical representation of this data. In other words, just ask them to think about the key features of their data display -- the labels and scales on the axes, the order of the categories, a title, etc. -- and make a quick sketch of what it might look like. They may also decide whether they want to use $/stream or streams/$ (or something else) as the unit of measurement. After a short while, reconvene the class-wide discussion so that students can learn from each other's ideas. Then, each group can create a more polished version of their graph during class time or as homework, depending on the classroom context. ====Writing alt-text==== Whether the students create a graph in class or for homework, they should also write appropriate alt-text for sharing that graph as an image on social media. Here is a prompt you might use on a worksheet or assignment description: <blockquote>Create ALT text for the graphic so that it's accessible to more people (e.g. people with visual or cognitive impairments, people who are searching with a search engine, or people for whom the image fails to load). [https://help.twitter.com/en/using-twitter/picture-descriptions According to the Twitter Help Center], "Good image descriptions are concise and descriptive, helping people understand what’s happening in an image." If the graphic conveys certain insights better than the numbers alone, an image description might convey those same insights verbally.</blockquote> ===Concluding discussion=== Finally, have the students share their creations with each other and discuss them! Perhaps you create a slideshow of each submission to view together during class time, or perhaps the students must share and comment on each other's submission on an online discussion forum, or... Whatever you do, it's important for students to see each other's work and share feedback. This will emphasize the point that there is no "right answer" here, and that there are many decisions to be made when sharing numerical information visually and in writing. Here are some questions you might use to spark some discussion. You might even use these as prompts for a follow-up survey or short writing assignment, too. * Look back at the original Tweet and then at your graphical representation of the data. How do they compare and contrast? If you encountered one of them for the first time in your social media feed, how might you react and process the information? * Which unit of measurement did you find most helpful when thinking about this topic: $/stream or streams/$? Or, did you find it helpful to go back and forth between the two? * Based on your work in this lesson, what advice do you have for yourself and your peers about how to process numerical information, especially when it involves small numbers (like the original Tweet)? * Reflect on the discussions you had with your group and the decisions you made regarding the alt-text for your image. Were there any challenges you faced in translating the data display into words? Did you gain any new insights into this topic or the underlying data by creating the graph and its alt-text? * ==Assignments== # '''Homework prompt:''' Find another example of an artist sharing their average pay rate for at least one streaming service. Express any information you find using both the '''Streams / $''' and '''$ / Stream''' units, and compare/contrast with the original source for this lesson. # '''Homework prompt:''' If you have an account on one of the streaming services considered in this activity, identify an artist you have listened to a lot. How many songs have you streamed from them in the last month? Year? (Perhaps you can find this information in your user profile or stats, or perhaps you'll need to do some estimation.) Using the rates provided in the original Tweet here, how much have you personally contributed to that artist's earnings on that platform this month/year? (But then, how certain can we be that your artist has the same pay rate per stream?) # '''Essay prompt:''' One issue raised by the original Tweet is that different artists likely have different pay rates per stream because of contracts with record labels. Do some research into this issue and share what you discover. Are the pay rates per stream shared in this lesson likely representative of "most artists" or not? # '''Essay prompt:''' Investigate how, for example, Spotify actually decides how much to pay an artist. It is not quite as simple as "(# of streams) X ($ rate per stream)." [https://www.headphonesty.com/2021/11/how-much-does-spotify-pay-per-stream/ Do some research] and share at least two new ideas you discover about what goes into deciding the amount an artist is paid. ==Resources== ===Background=== There is almost no prerequisite mathematical knowledge for this lesson plan. However, the activities and ensuing discussion may bring up issues of social justice and information/media literacy. In case it is helpful for you, here is some information about those ideas. ====Artists and labor==== * There is an [https://www.nytimes.com/2021/05/07/arts/music/streaming-music-payments.html ongoing conversation about pay rates of streaming services]. Some [https://www.nytimes.com/2022/01/26/arts/music/spotify-neil-young-joe-rogan.html prominent musicians recently removed their catalogs from Spotify]. Initially, this was done in protest of how the platform spreads misinformation (especially through podcasts), but some have also cited the poor compensation for working musicians, as well. There have even been protests by musicians and songwriters [https://www.latimes.com/entertainment-arts/music/story/2022-03-01/spotify-protest-songwriters-royalty-rates at various Spotify corporate offices], including the Union of Musicians and Allied Workers’s [https://www.unionofmusicians.org/justice-at-spotify “Justice at Spotify Day of Action.”] * Students may notice that the original Tweet begins and ends with some comments about the ''context'' for these numbers. Consider preempting the discussion by explicitly asking students to respond to these prompts: ** Why did the Tweet start with ''"Would you like another reason to switch to a music service that is NOT Spotify?"''? What framing/context does that provide for the rest of the information? How might you have interpreted this information differently if that sentence were not included? ** Why did the Tweet end with ''"(I am my label fyi)"'' at the end? What additional contextual information does that provide, and how might it change our interpretation of the quantitative information in the Tweet? (Answer: Artists typically split the profits from streaming services with their record label, according to whatever contract they've signed. In this case, Zoë Keating operating as her own label means that she does not have to split profits in that way. Accordingly, an artist who is ''not'' their own label would likely have noticeably ''smaller'' pay rates.) ** Does the Tweet specify where this information came from, or how the numbers were calculated? (Answer: Yes, she addresses this in [https://twitter.com/zoecello/status/1491081258629095430 a reply to her own Tweet] and implies that she took her annual earnings from each platform and divided by the number of streams she earned on that platform. This further emphasizes the idea that different artists may obtain different values from these calculations.) ====Alt-text for images==== Below are some guides about alt-text for accessibility. You may consider sharing some relevant portions with students as part of the main task of this lesson plan, or else ask students to read portions of these articles as part of their assignment. * [https://webaim.org/techniques/alttext/ WebAIM article on alt text] * [https://accessibility.umn.edu/what-you-can-do/start-7-core-skills/alternative-text University of Minnesota guide to alt text (with examples for charts and graphs)] * [https://accessibility.huit.harvard.edu/describe-content-images Advice from a Harvard University guide on digital accessibility] ===Explorations=== ''You're welcome to share references for additional learning and exploration, such as links to other articles, videos, spreadsheets, or computer code. When an open-access substitute is unavailable, links to paywalled sites are acceptable in this section.'' ==Feedback== <!--PLEASE DO NOT EDIT THIS SECTION DIRECTLY.--> Have you found this lesson plan helpful? Tell us about it! Just click ''Endorse'' below to open up an editor and type your comments. When you're ready, they'll appear at the bottom of this section to help other educators looking for good lesson plans. (Alternatively, if you see a way to improve this lesson plan, be bold and make an edit! You're also welcome to discuss the lesson plan or provide constructive feedback on its [[{{TALKPAGENAME}}|Discussion page]].) <div style="float:left;"> <inputbox> type=commenttitle hidden=yes page={{FULLPAGENAME}} buttonlabel=Endorse arialabel=Endorse the lesson plan editintro=Eventmath/Lesson plan endorsement instructions preload=Template:Eventmath lesson plan endorsement </inputbox> </div> </div> <div style="width:10%; padding:4em; float:left;"> __TOC__ </div> </div> <!--CATEGORY TAG INSTRUCTIONS Please follow these instructions so that people can find this lesson plan! 1. Select a primary event type from the following list: business, culture, economics, education, finance, government, health, politics, science, technology 2. Select a primary math type from the following list: algebra, arithmetic, calculus, combinatorics, geometry, logic, probability, statistics 3. Type your category tags directly underneath "[[Category: Eventmath lesson plans]]," at the bottom of this editor. For example, if you choose calculus and economics, you would have the following category tags on consecutive lines: [[Category: Eventmath lesson plans]] [[Category: Eventmath calculus]] [[Category: Eventmath economics]] NOTE: Be sure to use sentence casing for the category names (e.g. "Eventmath calculus" not "Eventmath Calculus"). FAQ Q: Can I apply multiple event or math tags? A: Yes! For example, you might want both a probability tag and a statistics tag. Q: Can I make a new category? A: Yes! Just type it in the same format. Example: [[Category: Eventmath number theory]] However, this category will not automatically have its own display box on the Lesson plans page. If you think there should be one, you can add one. If you don't know how, leave a comment on the Lesson plans "Discuss" tab. Alternatively, you might interpret an existing category more broadly. For example, if you have a lesson plan for differential equations, you might decide to include that under the calculus category. It's up to you. --> [[Category: Eventmath lesson plans]] [[Category: Eventmath arithmetic]] [[Category: Eventmath finance]] [[Category: Eventmath technology]] [[Category: Eventmath culture]] <div style="background-color: #DAF3EF; overflow:hidden;"> * I used an early version of this lesson plan with my class for liberal-arts students as an activity on numeracy in social media. The students were surprised to learn just how little artists are paid on these platforms and found the ensuing discussion helpful and engaging. I was also pleased to find that students actually ''wanted'' to do calculations with the numbers in the Tweet to better understand what the data is telling us. I have not yet tried the "create a graph and alt-text" task, but I'm looking forward to doing that next semester! -- [[User:Professorbrendan|Professorbrendan]] ([[User talk:Professorbrendan|discuss]] • [[Special:Contributions/Professorbrendan|contribs]]) 16:41, 13 July 2022 (UTC) </div> 73vwi31o2zp1ov4574coonvjddir62k 0 282659 2410843 2391171 2022-08-01T22:50:43Z Dave Braunschweig 426084 Dave Braunschweig moved page [[User:Jacsaw/sandbox/Social Work + Technology]] to [[Advocacy in Technology and Society/Social Work + Technology]] without leaving a redirect: Moving to main space wikitext text/x-wiki === '''Topic Summary''' === In this lecture we learned that we are all embedded in technological systems and technological systems are embedded in our social lives. There are people who are most at risk or most vulnerable to technological systems either due to their race, class, or privilege. Social workers use Data Justice to fight for fairer systems, both for ourselves and others. Social workers also play the role as an advocate by shaping the tech systems from within the system. We use technology tools to further our efforts, such as easy transfer of information and feedback loops for example. We discussed how social workers can insert ourselves to create a better, fairer, and just technology. Our framework was about how to be critical of something we depend on: technology. We then discussed Digital Social Work and that every social worker is a digital social worker. We must view internet as a site for everyone, including social workers to meet while we all go to different venues, or websites. Using digital tools can enhance many of our interactions. Though they may not be face to face like in person would be, we need service organizations to move to a digital model and move away from a more analog way of doing things, such as using paper charts. The issue here is that these service organizations do not necessarily have the bandwidth or skills to move everything over to digital. This becomes an issue when 90% of people are now online. We also discussed how to use our knowledge work, or mental models of social work, and digitizing them. How can we encode knowledge for those who are technology resistant? We discussed neoliberalism and how it has infiltrated social work and has caused it to shift from things like grassroots organizations to standardized social work. === '''Data Justice''' === The Data Justice Framework builds off of pursuits for fairness and goodness in the world. It uses OEDC's Fair Information Practice principles, which are collection, quality, purpose, use limitation, security, openness, individual participation, and accountability. This is a way to unify movements against tools that can harm people in the world. This is used as a global framework and not just in the United States. We discussed "Data For Good" which helps community based organizations that do not have the internal capacity to make sustainable technology platforms. The way to combat this issue is making the participation from other groups more sustainable. Social workers can help with these efforts by using our own knowledge as well. In order to sustain these organizations and move forward, we need feedback from stakeholders and who is impacted by them and skills that are needed. We need to also learn how to deal with Technology Superiority. The rise in automated products began in 2017, and this is when people believed that Data Science should also have a hippocratic oath, similar to social work. Data science borrowed and used the [https://www.socialworkers.org/About/Ethics/Code-of-Ethics/Code-of-Ethics-English NASW Code of Ethics] for their oath of data science. Their minimum standards include transparency, accountability, participatory, design/decision making. We also discussed using new tools to combat the issue of those who are vulnerable to being harmed online. We can create new tools using a PROP lens. [https://www.youtube.com/watch?v=RgtUxPtLS2o '''The Future of Data Justice Community Power and Data-Driven Systems'''] - A conversation with the Digital Equity Lab, based at the Milano School of Public Policy at The New School === '''Data Feminism: '''"What Gets Counted Counts" === This chapter discussed websites that require signing up or logging in.Most websites require you to choose your gender, but they only include male or female. It excludes non-binary individuals and the authors believe there's enough people to warrant changes. Joni Seager, a data scientist, says that what gets counted counts because gender is used as the basis for policymaking and resource allocation and being non-binary means becoming invisible in these issues. There is more global data on gender being collected than ever before but it still leaves people out, including non-binary people, lesbians, and older women. They are also asked very narrow questions about their lives like what contraception they use instead of their interests or likes. Evidently, this decision was made to make advertisers more easily market to one gender or the other. "This discrepancy leads right back to the issues of power we’ve been discussing since the start of this book: it’s corporations like Facebook, and not individuals like Maria Munir, who have the power to control the terms of data collection. This remains true even as it is people like Munir who have personally (and often painfully) run up against the limits of those classification systems—and who best know how they could be improved, remade, or in some cases, abolished altogether." (D’Ignazio, Catherine & Klein, Lauren,2020) The criteria used by which people are divided into the categories of man and woman is exactly a classification system. Feminist scholars tried to separate sex from gender, but could not because both gender and sex are social constructs. The American Medical Association now calls gender a “spectrum” rather than a binary, and as of 2018 it issued a firm statement that “sex and gender are more complex than previously assumed Early on, increasingly racist systems of classification began to emerge, along with pseudosciences like comparative anatomy and physiognomy. These allowed elite white men to provide a purportedly scientific basis for the differential treatment of people of color, women, disabled people, and gay people, among other groups. In each of these cases, as is true of any case of not fitting (or not wanting to fit) neatly into a box, it’s important to ask whether it’s the categories that are inadequate, or whether—and this is a key feminist move—it’s the system of classification itself. The matrix of domination describes how race, gender, and class (among other things) intersect to enhance opportunities for some people and constrain opportunities for others. Under the matrix of domination, normative bodies pass through scanners, borders, and bathrooms with ease; these systems have been designed by people like them, for people like them, with an aim—sometimes explicit—of keeping people not like them out. Being represented, though, also means being made visible, and being made visible to the matrix of domination—which continuously develops laws, practices, and cultural norms to police the gender binary. This poses significant risks to the health and safety of minorities. The book said that data must be classified in some way to be put to use because by the time that information becomes data, it’s already been classified in some way. “What distinguishes data from other forms of information is that it can be processed by a computer, or by computer-like operations,” as Lauren has written in an essay coauthored with information studies scholar Miriam Posner. And to enable those operations, which range from counting to sorting and from modeling to visualizing, the data must be placed into some kind of category—if not always into a conceptual category like gender, then at the least into a computational category like Boolean (a type of data with only two values, like true or false), integer (a type of  number with no decimal points, like 237 or −1), or string (a sequence of letters or words, like “this”). (D’Ignazio, Catherine & Klein, Lauren, 2020). The ethical complexity of whether to count gender, when to count gender, and how to count gender illuminates the complexity of acts of classification against the backdrop of structural oppression. By challenging the binary thinking that erases the experiences of certain groups while elevating others, we can work toward more just and equitable data practices and consequently toward a more just and equitable future. Binary categories fail to acknowledge what rests in between the two categories or outside them. The creators who fit in these binary groups rarely think about those who will not fit. For example, Facebook’s algorithms disproportionately flag Native American names for violation because those names often differ in structure and form from Anglo-Western names. The example of Facebook demonstrates the fundamental importance of obtaining consent when counting and of enabling individuals to refuse acts of counting and classification in light of potential harms. When data are collected about real people and their lives, risks ranging from exposure to violence are always present. But when deliberately considered, and when consent is obtained, counting can contribute to efforts to increase valuable and desired visibility and when a community is counting for itself, about itself, there is the potential that data collection can be not only be empowering but also healing. No person can fit their whole self into a form, regardless of how many blank text fields are provided. Counting and measuring do not always have to be tools of oppression. We can also use them to hold power accountable, to reclaim overlooked histories, and to build collectivity and solidarity. When we count within our own communities, with consideration and care, we can work to rebalance unequal distributions of power === Organizing in the Digital Age: Digital Macro Practice is Here…to Stay === These articles illustrate how digital technologies, including social media, have impacted the ways individuals, groups, and communities come together to advocate and effect social change. The authors bring awareness to the need for social workers in the technology industry. Their hope is to encourage social workers to take a more proactive role in integrating digital technologies in meaningful ways and advocating for digital justice. "The social work profession is at a critical crossroad where we can take a proactive role in influencing the ethical use of digital technologies to benefit social good and advance social change, rather than be reactive to the whims of technology companies and developers that thus far, have dictated the rules of digital engagement and participation"<ref>{{Cite journal|last=Rodriguez|first=Maria|last2=Storer|first2=Heather|last3=Shelton|first3=Jama|date=2021-07-03|title=Organizing in the digital age: digital macro practice is here…to stay|url=https://doi.org/10.1080/10705422.2021.1984178|journal=Journal of Community Practice|volume=29|issue=3|pages=199–202|doi=10.1080/10705422.2021.1984178|issn=1070-5422}}</ref> . The aspirational goals for the future of social work and technology are improving the quality and quantity of life for the most socially underrepresented and sidelined members of society. All technological advancements of the 21st century have primarily benefited the privileged. It has become crucial for social workers to apprehend and to protest advancing technological mechanisms of disenfranchisement. This responsibility also includes the necessary  step of being involved in producing, implementing, and evaluating technology’s myriad unintended impacts. === What is Data Justice? The Case for Connecting Digital Rights and Freedoms Globally === This paper address the concern that data-driven discrimination is advancing at a similar pace to data processing technologies. However, awareness and mechanisms for combating data-driven discrimination are not. There is a noticeable shift in policymaking worldwide from being data informed to being data driven. Data sources which allow researchers to gather people’s "movements, activities and behavior have ethical, political and practical implications for the way people are seen and treated by the state and by the private sector (and, importantly, by both acting in combination)"<ref name=":0" />. This distribution of privacy has significant social and political implications for lower-income communities, where authorities’ ability to gather accurate statistical data has previously been limited. The power data has to sort, categorize and intervene has not been connected to or addressed by any social justice agencies or authorities who collect, manage and use data. There is also a low level of awareness amongst architects of how new data technologies may not be unbiased in terms of access, use or impacts. Data justice, which is described as "fairness in the way people are made visible, represented and treated as a result of their production of digital data,<ref name=":0">{{Cite journal|last=Taylor|first=Linnet|date=2017-12|title=What is data justice? The case for connecting digital rights and freedoms globally|url=http://journals.sagepub.com/doi/10.1177/2053951717736335|journal=Big Data & Society|language=en|volume=4|issue=2|pages=205395171773633|doi=10.1177/2053951717736335|issn=2053-9517}}</ref>" is necessary to determine ethical paths through our data-driven world. <u>The article proposes three pillars for international data justice:</u> * (in)visibility * (dis)engagement with technology * and (anti)discrimination These pillars challenge the current data protection regulations and the growing assumption that being visible through the data we emit is part of the contemporary social contract. === Insights === In this topic, we discussed and identified the groups disproportionately at risk from the implementation of new technology. Social workers can add enormous value to the technology industry by sharing ethical standards of practice for fairer and more just technologies to ensure human rights on digital platforms. As social workers, we can offer our unique skillset and knowledge in human rights, social justice, and social and environmental ethics. These qualities and skills are widely lacking in the technology world, which has caused more social issues such as blatant discrimination in AI products and services, which have resulted in victimization and criminalization of marginalized communities. From now onwards, it will be crucial for social workers to use their social skills and knowledge in technologies for social justice. There is an emerging need for engaging social workers in technology development, deployment, and implementation to prevent further consequences in our society. === What does advocacy look like here? === '''How can advocates use technology and data-based practice to further their advocacy efforts?''' Social workers and advocates can use technology to bring further awareness to data injustice in many ways. As discussed in class, social workers are needed throughout all stages of the data and technology innovation. We must hold technology companies accountable when their data is mishandled or when data science disregards one or more of their minimum standards of transparency, accountability, participatory, and design/decision making. Additionally, technological advances are expanding advocates' ability to communicate with a wider audience and allow for the potential to make advocacy efforts more effective and efficient. Advocates can utilize technology to affect societal change by recruiting a larger audience, organizing efforts online, raising funding and communicating with major stakeholders. '''How are current and emergent technologies presenting challenges in society, and how can advocates work in pursuit of better, fairer, and more just technology?''' Breakthrough technological advances are being made across a range of fields. These technologies are highly disruptive and bring about major transformative shifts in how our society functions. These presenting challenges include: *Dislocation of labor markets and other market disruptions *Deepened inequalities *New risks to public and national security Social workers and data justice advocates are crucial to help mitigate the disruptive impact technology has on our society. There needs to be a stronger presence of social workers in the technology field to advocate for the more vulnerable groups at risk from the implementation of technology. And finally, social workers must advocate for our community members to have an opportunity to assist in developing, implementing, and evaluating emerging technologies. === Annotated References === * D’Ignazio, Catherine & Klein, Lauren (2020). Data Feminism, Chapter 4: ''What Gets Counted, Principle: Rethink Binaries and Hierarchies.'' * https://datajusticelab.org/ * ''Google Images: Where Digital Justice Can Begin | Angelica Poversky | TEDxUBC''. (n.d.). Www.youtube.com. <nowiki>https://www.youtube.com/watch?v=6FnWiXAV-aQ</nowiki> * Kavanagh, C. (2019, August 28). ''New Tech, new threats, and new governance challenges: An opportunity to craft smarter responses?'' Carnegie Endowment for International Peace. Retrieved April 25, 2022, from <nowiki>https://carnegieendowment.org/2019/08/28/new-tech-new-threats-and-new-governance-challenges-opportunity-to-craft-smarter-responses-pub-79736</nowiki> * Maria Rodriguez, Heather Storer & Jama Shelton (2021) Organizing in the digital age: digital macro practice is here…to stay, Journal of Community Practice, 29:3, 199-202, DOI: 10.1080/10705422.2021.1984178 * Rosemary Thackeray, Ph.D., MPH, MaryAnne Hunter, Empowering Youth: Use of Technology in Advocacy to Affect Social Change, ''Journal of Computer-Mediated Communication'', Volume 15, Issue 4, 1 July 2010, Pages 575–591, <nowiki>https://doi.org/10.1111/j.1083-6101.2009.01503.x</nowiki> * Taylor, L. (2017). What is data justice? The case for connecting digital rights and freedoms globally. Big Data & Society. <nowiki>https://doi.org/10.1177/2053951717736335</nowiki> kn71bzkdcoiu45u2ppvs3otyjzpw98f 0 282660 2410841 2391557 2022-08-01T22:50:43Z Dave Braunschweig 426084 Dave Braunschweig moved page [[User:Jacsaw/sandbox/Emerging Tech - Identifying the Stakeholders]] to [[Advocacy in Technology and Society/Emerging Tech - Identifying the Stakeholders]] without leaving a redirect: Moving to main space wikitext text/x-wiki <references /> === '''Topic Summary''' === Summarize what was covered during this topic through the lecture, discussion, and the readings assigned this week. '''''Stakeholder Analysis''''' Our class in Week 4 centered on how to identify the stakeholders of emerging technologies. To perform a stakeholder analysis, we must systematically narrow in on who is most impacted by emerging technologies, who has the control or decision-making to impact technology, who is likely to benefit from the technology, and what is the relationship between the impacted and impacting stakeholders. Systematically organizing information may include grouping people by different demographic, locations, or socioeconomic statuses. Once we can narrow in on the ‘who’ and ‘what’ of the situation, we can dive deeper into the changes likely to happen over time, and how each stakeholder will play a role in the direction of decision making. The goal of the stakeholder analysis is to identify risk to see who and what is missing from the current activity so that we organize to mitigate future negative impacts and have concrete criteria for identifying who should be held accountable if things do go wrong. {| class="wikitable" |+ ! colspan="2" |To prepare for this week's conversation, students read: |- |'''[https://data-feminism.mitpress.mit.edu/ Data Feminism by Catherine D'Ignazio and Lauren Klein]''' |In [https://data-feminism.mitpress.mit.edu/pub/h1w0nbqp/release/3 ''What Gets Counted Counts'', Chapter 4 of “Data Feminism”,] D’Ignazio and Klein discuss how, although classification systems are essential to any working infrastructure, they encompass and enforce binaries and hierarchies. This chapter prompts readers to gain an understanding of one of the principles of a data feminist perspective: rethinking binaries and hierarchies. In the process of assessing the classification systems that currently exist, Klein and D’Ignazio utilize the issue of the gender binary as an example that illustrates how acts of counting and classification must always find a balance between harming and benefitting marginalized communities. Chapter 4 further challenges readers to not only assess who is and is not being counted but to additionally recognize the power that classification systems hold in telling an individual’s or collective experience of oppression. Klein and D’Ignazio identify and discuss how counting and classification systems act as reflections of the social, cultural, or political values of a given community or time period in society. Thus, to provide an accurate reflection and depiction of marginalized communities, Klein and D’Ignazio assert that the role of the storyteller should be held by a community member. |- |'''[https://www.tandfonline.com/doi/full/10.1080/10705422.2021.1984354 Data Justice: Social Work and a More Just Future by Lauri Goldkind, Lea Wolf, and Walter LaMendola]''' |Goldkind et al. urge social workers to gain an understanding of the extent to which new data-driven technologies oppress the populations that they aim to serve across the micro, mezzo, and macro levels. The article highlights a selection of technologies whose data and algorithmic strategies have impacted the well-being of individuals, organizations, and communities. In discussing the ramifications of emerging data-driven technologies, Goldkind et al. reveal that so to have theories and frameworks that work to ameliorate the impact of said technologies. Ultimately, the article serves to promote the need for a data justice framework to be adopted by the social work profession in that this framework brings forth social work ethics up-to-date with the data revolution. |- | colspan="2" |'''Students were provided with a couple of optional readings:''' |- | rowspan="2" |'''Automating Inequality by Virginia Eubanks''' |In Chapter 1, “From Poor House to Database”, Virginia Eubanks provides a historical analysis of the poorhouse. Eubanks identifies how from its conception, although these institutions were developed to provide public economic relief for low-income folks, they instead functioned as systems that punished folks experiencing poverty. The 19th Century was a time period that heavily relied on poorhouses as a method of regulating poverty by acting as a classification system that differentiated the “deserving” from the “undeserving” poor. This binary of poorness was reinforced by the scientific charity movement that advocated for the need for caseworkers who primarily worked to filter out those who were not identified to be  “deserving” of public welfare benefits. Eubanks further discusses and identifies how periods of economic crisis, elected officials, and policies contributed to the shapeshifting of poorhouses over time. Although the physical infrastructure of poorhouses ceases to exist, the foundational values and beliefs that upheld those institutions for over a century persist and have transformed into automated decision-making systems that continue to profile, police, and punish folks experiencing poverty.  |- |In Chapter 3, “High-Tech Homelessness in the City of Angels”, Eubanks highlights the automated decision-making system that was created to streamline the process of connecting a person who is unhoused to a housing resource in Los Angeles: the Coordinated Entry System (CES). In discussing the increasing minimization of Skid Row – a 50-square-block area in downtown Los Angeles that is home to a community of people who are unhoused – Eubanks reveals that the CES does not account for the lack of affordable housing. This absence of affordable housing is identified to be attributed to the various instances where the city of Los Angeles, whether it be city officials or community members, continuously chose to prioritize profit and to reinforce the binary of “deserving” and “undeserving” poor. Eubanks asserts that this binary is ever-present in the CES algorithm that works to rank people who are unhoused according to their vulnerability level. In discussing the CES data entry process, Eubanks shines a light on the personhood of the people behind the CES data. Although CES data has acted to inform the public on the state of the homelessness crisis in Los Angeles, by providing a critical analysis of the CES privacy practices, or lack of, Eubanks reveals that this system is another system that surveils and criminalizes people experiencing poverty. |} '''''[https://ctb.ku.edu/en/table-of-contents/participation/encouraging-involvement/identify-stakeholders/main Stakeholders]''''' In order to systematically categorize information, people must be grouped into one of three levels levels of stakeholders: ''Primary Stakeholders:'' are the people most affected by the technology and most dependent on the resources. These are the people that are targeted during the development of emerging technology and have the most to gain or lose from its impacts.  For example, in class, we used an example of an app that sent automatic alerts if the user had been exposed to Covid-19.  In this example, the primary stakeholders, are the general public, or those who had the app and were either receiving or giving the Covid exposure information. ''Secondary Stakeholders:'' are the people who are indirectly affected or less affected by the technology than the primary stakeholders or less dependent on the resource. For example, apple who may house the app in their app store, but are not directly impacting the technology. ''Key Stakeholders:'' are the people who have significant influence over the technology or are the most impacted by the technology (may also be primary or secondary stakeholders). In the Covid exposure alert example, key stakeholders would be the technology builders, philanthropists, health advocacy groups, or policymakers.   '''Stakeholder Analyses''' Most stakeholder analyses are built on a grid, further made popular by the surge in use frequency of Microsoft Excel. ''[https://www.projectmanagement.com/wikis/368897/Stakeholder-Analysis--using-the-Power-Interest-Grid Power/Interest Grid]'': Eden & Ackerman 2014 This four-square grid analyzes how much power and control one has over the outcome (high or low) and how invested/interested/impacted they are in the success of the project. It can be important to maximize those in power who also have an interest in a positive outcome. ''[https://d1wqtxts1xzle7.cloudfront.net/60330089/Stakeholder_Analysis20190819-4314-t4c3ox-with-cover-page-v2.pdf?Expires=1651026055&Signature=dBRo4tw8wsWlonOGKbtKqL9aulUQvoCB~smlhZje5-hMw~Q-XboVPn5mxBSnN9QBYpybDZ-pI78dgPsqVcIxOyTk4XAe4dqEn9DUyE8VkACbwR3j~dBJBux~hggSHUE3a5A468wzMkxd9oUMUiH1-7gkMuP5~xvmQtHdF-8pSedw2ajA-a~hxBoHEUmpvfuzjvDWU4~9EZWf9N6wnocJb-3ojcr6rfqa9xc7lZUKbSMd16A0OJKFOBfKEPjx3WQ4TJ9hcENwN7Ncv8-ppNhpWTqWbEhLO9NSg~LQMRCg-mHEv2rvUZWG3t7wWnxQ~n9YBxIQ3cPUTSEPC2W41JWwyA__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA Power/Dynamism Grid] (Gardner et al., 1996):'' The four-square grid analyzes how power/influence shifts over time and how much the power can be influenced (Dynamism). The factors influencing the outcome must be aligned with the values of the mission, adding in the cultural aspects of changes such as policy and current advocacy efforts. ''[https://www.researchgate.net/publication/276859954_Three_dimensional_stakeholder_analysis_-_3dSA_Adding_the_risk_dimension_for_stakeholder_analysis 3dSA - Three Dimensional Stakeholder Analysis] :'' A three-dimensional grid that utilizes stakeholder power, interest, and attitudes. Attitude also adds in a layer of culture and context, looking at how everyday shifts impact the level of engagement of the stakeholder. This allows a more dynamic evaluation by incorporating size and degree of influence. ''[https://www.greycampus.com/blog/project-management/analyze-your-project-s-stakeholders-with-the-help-of-the-salience-model#:~:text=Stakeholder%20salience%20is%20the%20degree,This%20often%20leads%20to%20issues. Stakeholder Salience Method]:'' Three Venn Diagrams analyzing power, legitimacy, and urgency. Legitimacy represents the people and things that matter most, and by viewing the overlap with power and urgency, it identifies where there are aligned or misaligned goals. ''[https://project-management.com/understanding-responsibility-assignment-matrix-raci-matrix/ RACI (Responsible, Accountable, Consulted, and Informed) Matrix]:'' A chart representing an array of different primary stakeholders in a company based on their positionality, and their engagement with certain tasks. ''[https://orcaarisk.com/ Ethical Matrix (Cathy O’Neil) - Auditing Algorithms.]'' Looking at different levels of ecosystems: Community, Neighborhoods, Individual, Individuals belonging to certain groups. For example for Covid exposure app, in evaluation we may look at groups such as healthcare providers, google/apple, or those in high/low risk areas (ideally allowing users to identify and label themselves and thinking about structural oppression). Keeping in mind the layers of data protection - for example risks that may be imposed if a user identified as having pre-existing conditions or insurance coverage, and how that information could be used against the user in the future. Once the stakeholders are identified, it is compared based on Well-being, Autonomy, Fairness, Anti-oppressive, Anti-Racist & Inclusive. '''[https://www.crisistextline.org/ Crisis Text Line Case Study:]''' The Crisis Text Line is a tech based startup that responds to users who are reach out in a mental health emergency. The goal is to create care in a tech setting, however, the company is funded by Silicon Valley. This creates a conflict because the organization was invested in not for equity, but to receive coded user data. [https://www.politico.com/news/2022/01/31/crisis-text-line-ends-data-sharing-00004001 Until February 2022,] Crisis Tech Line shared their anonymized data with a company developing an Artificial Intelligence product. This inherently benefits the company, and while they don't receive the personal information, they still use conversations to further their AI learning. Additionally, organizations like Crisis Tech Line can take advantage of the gaps in the current mental health care system - users may benefit from the immediate help, but it may prevent them from receiving more extensive mental health service. However, these text lines may also be a very effective way for people who do not have access or resources to utilize mental health recourses. One addition consideration is that the Crisis responders are often volunteers, while the trainers are clinical based providers - while it is great that there is clinical oversight, the majority of the work of the organization is done by volunteers. === Insights === Go beyond your summary, synthesize what you learned on this topic, drawing on personal insights and that of your peers, the content, what you read in class and in the media Throughout this class it became clear that Social Workers have a vast role in the technology field. Considering who are the stakeholders and who has a genuine interest in the outcome leads me to believe that emerging tech companies would greatly benefit for the social work code of ethics, and the knowledge and expertise that social workers bring to the table. There are many conversations about whether it is more effective to work within systems or influence them externally, while these models note the influence of external key stakeholders such as advocacy organizations or philanthropists - it is important to recognize how important it is to have social workers ''within'' the companies themselves. From within, a social worker can take on the role of both an external advocate for the primary stakeholders, but retain the influence of an internal stakeholder. Furthermore, across all stakeholders, a wide array of perspectives and diverse group of people allows is imperative to conducting analysis effectively and throughout, as it increases the chances that all aspects of the evaluation are considered. The prevalence of systematic oppression across our systems and institutions, creates an increases level of risk for marginalized communities, particularly when there are not advocates in the room that share their identity. Creating more diverse stakeholders, in turn protects the most vulnerable, and thus protects the wider population as a whole, benefitting all primary stakeholders involved and mitigates potential risks. === What does advocacy look like here? === Considering your topic, respond to the following: 1. How can advocates use technology and data-based practice to further their advocacy efforts? And 2. How are current and emergent technologies presenting challenges in society, and how can advocates work in pursuit of better, fairer, and more just technology? '''''Equitable Participation of Key Stakeholders''''' When evaluating stakeholders, it is important to understand who is at the table and has power in the decision making, and who does not have influence but should. Stakeholders should represent a diverse range of perspectives and backgrounds, so that risks can be mitigated and primary stakeholders have advocates on their behalf. Considering the power/dynamism grid - the values of the key stakeholders should align with the goals of the company, keeping those in power more invested in the work they are doing. '''''[https://journals.sagepub.com/doi/full/10.1177/2053951717736335 Data Justice Framework (Linnet Taylor, 2017)]:''''' ''Visibility'': Analyzing the representation of the users of technology and the privacy protecting their personal data Engagement with Technology: How in practice, do we give people an option to opt-in or opt-out of tools that can be potential harmful id companies use their data in precarious or exploitive ways or sell sensitive data to other actors. ''Nondiscrimination:'' During development of technology, who are the people in decision making positions - what is the data being inputted into algorithms that could exasperate or challenge existing biases? How do we ensure that users will not be discriminated against based on bias data inputted into the system? '''Adding Internal and External Audits:''' Rather than placing the full onus on community members, adding in additional stakeholders representatives to evaluate technology. '''[https://www.latimes.com/california/story/2022-03-26/echo-park-lake-encampment-raises-questions-los-angeles-homeless-policy Echo Park Lake Residents Displaced]'''[[File:Echo Park Lake 3 2022.jpg|thumb|A photo of Echo Park Lake located in Los Angeles, California. Photo captures a lake, waterfall, and palm trees on a sunny Saturday in March of 2022.]]Social workers can incorporate a data justice framework when assessing the impact of automated-decision making technologies, like the Coordinated Entry System (CES) and the manner in which it was used to displace people who were unhoused and had established a home in the grassy parts of Echo Park Lake. Echo Park is one of the public parks owned and operated by Los Angeles City Parks. Over the course of the COVID-19 pandemic, people struggling with homelessness began to establish a commune-like encampment within the parameters of Echo Park. This community was short-lived in that people who were unhoused were uprooted from their homes and displaced on March 26, 2021. Nearly a year later, the park that was once open and bustling with community members and visitors, is restricted by a seemingly temporary fence that surrounds the park and is nearly empty most days. A Los Angeles Times column highlighted the experience of David Busch-Lilly, a former Echo Park Lake resident who was displaced by policing and city officials who promised that he, alongside with his neighbors, would be connected to housing. Unfortunately, David's case of being handed an empty promise was unfortunately not unique: {| class="wikitable" |+ !Of the 183 people who were unhoused and residing at Echo Park lake, 17 people received long-term housing, 48 are on a waiting list, 15 were unsuccessfully housed, 6 people died, and at least 15 went back to living on the streets - As reported and discussed by the UCLA Luskin Institute on Inequality and Democracy & Erika Smith, Columnist from the Los Angeles Times. |} In thinking about the experiences of these community members, social workers may advocate for their well-being by advocating for city and policing officials to adopt the view of CES as a tool that should them in their efforts to establish funding for long-term affordable housing. '''[https://automatingsociety.algorithmwatch.org/report2020/netherlands/netherlands-story/ Advocacy Success Efforts in Protecting Digital Rights for Low-Income Folks]''' As mentioned by Goldkind et al. in their discussion of data justice, Dutch civil rights actvitsts were successful in advocating for the digital rights and well-being of low-income folks to be prioritized over the Dutch government’s act of surveillance. Through these activists efforts, it was revealed that Netherland government agencies were utilizing a personal data system called SyRi, which stands for “system risk indication” to seek out “unlikely citizen profiles that warrant further investigation”. Civil rights activists challenged these acts of surveillance and by doing so, these efforts have prompted public organizations to reevaluate their process in collecting, managing and protecting data that is collected from the public. Social workers can learn from successful efforts like these to identify manners in which we can identify the stakeholders that need to be involved in the process of evoking similar change. === Annotated References === Cite what we covered in class and include 4 additional resources you find on the topic. Cite resources that are related to the topic, they can be in agreement with the topic, extending it, in disagreement or present conflict with the topic. D’Ignazio, Catherine & Klein, Lauren (2020). Data Feminism, Chapter 4: What Gets Counted, Principle: Rethink Binaries and Hierarchies. Data Justice, Social Work & a More Just Future (2021) access here: <nowiki>https://www.tandfonline.com/doi/full/10.1080/10705422.2021.1984354</nowiki> Eubanks, Virginia (2015). Automating Inequality, Chapter 1: From Poor House to Database & Chapter 3: High Tech in Homelessness in the City of Angels MindTools. (n.d.). Stakeholder analysis: Winning support for your projects. Project Management Skills From MindTools.com. Retrieved April 25, 2022, from <nowiki>https://www.mindtools.com/pages/article/newPPM_07.htm</nowiki> Smith, Erika D. “Column: He Had to Leave Echo Park Lake and He’s Still Homeless. Is This What L.A. Wants?” ''Los Angeles Times'', 26 Mar. 2022, <nowiki>https://www.latimes.com/california/story/2022-03-26/echo-park-lake-encampment-raises-questions-los-angeles-homeless-policy</nowiki>. Vervloesem, Koen. “How Dutch Activists Got an Invasive Fraud Detection Algorithm Banned.” ''Automating Society Report 2020'', Algorithm Watch, <nowiki>https://automatingsociety.algorithmwatch.org/report2020/netherlands/netherlands-story/</nowiki>. tio4uob03zervcf4ox71xb1ca337brb 0 282661 2410837 2391142 2022-08-01T22:50:42Z Dave Braunschweig 426084 Dave Braunschweig moved page [[User:Jacsaw/sandbox/Community Data and COVID-19]] to [[Advocacy in Technology and Society/Community Data and COVID-19]] without leaving a redirect: Moving to main space wikitext text/x-wiki === '''Topic Summary''' === COVID-19 revealed several realities about the strength of community organizing and the frailty in federal design. The virus had a disproportionate impact on working class communities and communities of color in the US and this was further reinforced in the mapping and presentation of how these communities were displayed. Much like the importance of representation in storytelling, when communities do not have access to the tools to voice their own lived realities, their experiences can be misrepresented and erased. Furthermore, inaccurate representations of a communities lived realities can be used to ensure that power and wealth are contained within the paradigms of whiteness (D’Ignazio & Klein, 2020). Just like storytelling, when communities are excluded from having access to tools to map and identify their lived realities, their experiences are threatened by erasure. During the peak of the pandemic, individuals took it upon themselves to organize and visualize the ways in which the in-real-time impact could be accessed, portrayed, and engaged with. Untapped New York is an example of building interactive and visual feeds through open source coding to build a collaborative environment to share information using found data. Untapped New York in particular captured data by pulling information from NYC Open Data and the Department of Health. However, how do we know that these outlets have accurately captured data? What filters have they used to represent a holistic view of the impact and losses in communities that were further exacerbated by things like the digital divide and educatio inequity regarding reporting COVID-19 cases? [https://untappedcities.com/2020/11/12/interactive-map-covid-nyc-hotspot-zip-codes/ Untapped NYC] shared the disclaimers in regards to how information was representing different communities: * The data in the map does not include probable deaths, which includes deaths with no known positive laboratory test for the virus that causes COVID-19, but the death certificate lists “COVID-19” or an equivalent as a cause of death. Probable deaths are being tracked by the Department of Health however, and over time, probable deaths may get recategorized as confirmed deaths. * The neighborhood names are assigned by the NYC Department of Health but they don’t always align with what residents term that area, so you may find yourself questioning some neighborhood designations (Young, 2020). While these maps provide valuable information to engage with the visualization of data that continues to be collected and updated, they raise questions about the “[https://www.ted.com/talks/chimamanda_ngozi_adichie_the_danger_of_a_single_story/transcript danger of a single story]” (Adichie, 2009). In her TedX Talk, Chimamanda Ngozi Adichie discusses the removal of unique details that occurs when a person is disempowered from the ability to share their own story.<blockquote>''Power is the ability not just to tell the story of another person, but to make it the definitive story of that person. The Palestinian poet Mourid Barghouti writes that if you want to dispossess a people, the simplest way to do it is to tell their story and to start with, "secondly." Start the story with the arrows of the Native Americans, and not with the arrival of the British, and you have an entirely different story. Start the story with the failure of the African state, and not with the colonial creation of the African state, and you have an entirely different story.''</blockquote>In class, we discussed the variations of data maps and the mobility of building accessible representations of data through community organized visualizations. The readings challenged the systems used to build technology and data-based practices. ==== <u>Design Justice</u> ==== ===== [https://data-feminism.mitpress.mit.edu/pub/ei7cogfn/release/4 <u>Chapter 2: Collect, Analyze, Imagine, Teach</u>] ===== In their book, Catherine D’Ignazio and Lauren Klein highlight the histories of data mapping that are not founded in people-centric approaches. They share the dangers of inaccurate data visualization by providing historical evidence of policies, such as redlining, that manipulated data to uphold white power and wealth. To challenge these systems, D’Ignazio and Klein propose alternatives to “envision equity” and “imagine co-liberation” by drawing from abolitionist and community centric lenses in order to “assert the notion of algorithmic fairness” (D’Ignazio & Klein, 2020). ===== [https://data-feminism.mitpress.mit.edu/pub/czq9dfs5/release/3 <u>Chapter 6: The Numbers Don’t Speak for Themselves</u>] ===== In this chapter, D’Ignazio and Klein discuss the power surrounding open data as “free access, use, modify, and share data for any purpose” (D’Ignazio & Klein, 2020). However even as policies have encouraged the open access and free use of government organized data, building transparency by uploading documents for access has not come with the addition of information to provide context for the data. The lack of appropriate explanation, then, aggravates ways in which data can be used to manipulate “racism, sexism, and other forms of oppression” (D’Ignazio & Klein, 2020). ===== [https://covidblack.medium.com/a-review-of-covid-19-intersectional-data-decision-making-a-call-for-black-feminist-data-analytics-da8e12bc4a6b <u>A Review of COVID-19 Intersectional Data Decision-Making: A Call for Black Feminist Data Analytics</u>] ===== In their article, Kim Gallon of COVID Black discusses the importance of centering Black Feminism in the collection of design of data frameworks. Gallon emphasizes the importance of applying intersectional frameworks to recognize the “multiple identities” that an individual carries and the threat to diversity, equity, and inclusion efforts without this focus. Gallon provides an example of a case study to highlight the benefits of applying intersectionality frameworks, and the losses of not challenging design filters enough. === '''Insights''' === During the Covid period, data collecting has been an effective way to safeguard public health. However, data and privacy are inextricably linked. In the first year of Covid, some states—including California and Texas— weren't sharing vaccine-related ethnicity and racial data with the CDC, citing patient privacy laws. Furthermore, incomplete demographic information associated with other COVID-19 metrics pointed to the presence of racial and ethnic disparities (Antonios, Chatterjee, Gee, Kravitz, &Senese, 2021). In this case, the idea of public health is questioned, since the data of certain communities are missing. Information could also do harm to people, and everyone should acknowledge information is a double-edged sword. There is a lot of information online and offline, but not everyone has adequate information literacy to process it. Accordingly, wrong information could undermine public responses, and disinformation can harm public health, which would disproportionately influence people as well. For example, the senior community, particularly seniors of color and LGBT older adults with a higher rate of poverty and illness are more vulnerable to disinformation, which may harm their physical health and cause a financial burden during the COVID. Besides, limited tech literacy further poses risks to people. Social media apps can be useful for sharing information about the outbreak, but there are fake apps claiming to track COVID cases, which is on the rise. In fact, they are often scams to infect and lock people’s devices and demand ransom (WebMD, 2020). In addition, Covid-19 data is not equally accessible to everyone, considering diversity in the New York City, and over 50% of New Yorkers can't speak English very well or not at all. This disproportionately impacts immigrants who are facing language and cultural barriers but need an accessible system to keep informed. === '''What does advocacy look like here?''' === As highlighted in [https://data-feminism.mitpress.mit.edu/ Data-Driven Approach to Addressing Racial Disparities in Health Care Outcomes] by Kim Gallon, approaching data collection through the contributions of Black Feminism provides an opportunity to acknowledge the intersectionality of our “multiple identities” that impact data representation (Gallon, 2020). In order to further understand how COVID-19 was impact particular communities in Boston, [https://hbr.org/2020/07/a-data-driven-approach-to-addressing-racial-disparities-in-health-care-outcomes a group or researchers, physicians, and health analysts at Brigham Health collected the following data from COVID-19 test spots: race, ethnicity, sex, language, insurance status, geographic location, and health-care worker status] (Gallon, 2020). The aim of this research was to apply an inclusive approach by specifically incorporating Kimberlé Crenshaw’s intersectionality framework. In doing so, the body of researchers were able to identify that “Hispanic non-English speaking patients were dying at higher rates than Hispanic English speaking patients.” However even with the intersectional framework that was applied to collect data, the approach failed to “consider the ways that the algorithms used to filter the data may have reflected a matrix of domination.” Without defining the filters that were created, questions are raised about the ways in which, for instance, gender is defined or how race is simplified. While technology and data-based practices provide opportunities to amplify the impact of issues that organizers are advocating for or against, this example is an important reminder of how systems are reflections of the values, principles, and beliefs of the team designing it. When individuals are not considering the restraints of who is and isn’t included in the process of collecting data, equity is threatened by norms and standards that weren’t created to be inclusive. Just as we challenge power narratives and structures in storytelling, it is important to challenge the power of mobilizing data by “pushing back against existing and unequal power structures and to work toward more just and equitable futures” (D’Ignazio & Klein, 2020). In Data Feminism, Catherine D’Ignazio and Lauren Klein provide alternatives to equitable and inclusive data mapping: # Collect: Compiling counter-data—in the face of missing data or institutional neglect. # Analyze: Challenging power often requires demonstrating inequitable outcomes across groups, and new computational methods are being developed to audit opaque algorithms and hold institutions accountable. # Imagine: We cannot only focus on inequitable outcomes, because then we will never get to the root cause of injustice. In order to truly dismantle power, we have to imagine our end point not as “fairness,” but as co-liberation. # Teach: The identities of data scientists matter, so how might we engage and empower newcomers to the field in order to shift the demographics and cultivate the next generation of data feminists? Director and Co-Founder of COVID Black, Kim Gallon, also captured the importance of centering Black feminism in data analysis frameworks to consider who we are centering and challenge how we are normalizing power dynamics. She highlights the following definitions to challenge the collection of data: * Data feminism: concentrates on “intersecting forces of privilege and oppression” that demonstrate that power is not shared equally in society; minoritized identities intersect (Gallon, 2020). * Data justice: any notion of algorithmic fairness must also acknowledge the systematic nature of the unfairness that has long been perpetrated by certain groups on others. * Design injustice: (design processes that intentionally and unintentionally engender individual and systematic inequity) becomes more difficult to identify without applying the mutually constitutive Black feminist frameworks, intersectionality, and matrix of domination, to COVID racial data analysis (Gallon 2020). * Intersectionality: “the paradigm of sex discrimination tends to be based on the experiences of white women; the model of race discrimination tends to be based on the experiences of the most privileged Blacks.” Race and sex discrimination are constructed to address a narrow set of circumstances that deny Black women’s lived experiences (Gallon, 2020). * Matrix of domination: a sociological paradigm that explains issues of oppression that deal with race, class, and gender, which, though recognized as different social classifications, are all interconnected.The matrix of domination is a way for people to acknowledge their privileges in society. How one is able to interact, what social groups one is in, and the networks one establishes is all based on different interconnected classifications (Wikipedia). === '''Annotated References''' === * Antonios, C., Chatterjee, M., Gee, G., Kravitz, D., & Senese, K. (2021, February). ''Analysis & Updates: Why some states won't share race and ethnicity data on vaccinations with the CDC-and why that's a problem''. The COVID Tracking Project. Retrieved April 26, 2022, from <nowiki>https://covidtracking.com/analysis-updates/why-some-states-wont-share-race-and-ethnicity-data-on-vaccinations-with-the-cdc-and-why-thats-a-problem</nowiki> This article talks about data-sharing issues at the state level. the Centers for Disease Control and Prevention asked every US state and territory to sign a contract, agreeing to share wide-ranging vaccine data and records with the federal government. Several states raised concerns, but every one signed. But it turns out that most states ended up withholding the names, addresses, ZIP codes, and dates of birth of those vaccinated, and at least seven states go a step further and redact race and ethnicity data from the federal government or don’t collect it in the first place. Besides, this article also explores the rationale for withholding data. * Centers for Disease Control and Prevention. (2022). ''CDC Covid Data tracker''. Centers for Disease Control and Prevention. Retrieved April 26, 2022, from <nowiki>https://covid.cdc.gov/covid-data-tracker/#demographicsovertime</nowiki> This resource provides data on COVID-19 weekly cases and deaths per 100,000 population by age, race/ethnicity, and sex. *''The Covid Tracking Project''. The COVID Tracking Project. (2021). Retrieved April 26, 2022, from <nowiki>https://covidtracking.com/</nowiki> This project tracked COVID-19 data since 2019, and provided a comprehensive, relatively standardized data system to support individuals. But this project stopped collecting data in 2021. * ''The Danger of a Single Story''. (n.d.). ''Chimamanda Ngozi Adichie: The danger of a single story | TED Talk''. Retrieved April 26, 2022, from <nowiki>https://www.ted.com/talks/chimamanda_ngozi_adichie_the_danger_of_a_single_story</nowiki>. In this video, author Chimamanda Ngozi Adichie emphasizes the importance of writers to share their own stories to represent their communities with accurate representations of lived experiences and histories. Adichie discusses the dangers and threats that communities face when those outside of them enter to tell their versions of stories. She stresses the importance of correcting histories through the act of storytelling. * D'Ignazio, C., & Klein, L. (2020, March 16). ''2. collect, analyze, imagine, teach · data feminism''. Data Feminism. Retrieved April 26, 2022, from <nowiki>https://data-feminism.mitpress.mit.edu/pub/ei7cogfn/release/4</nowiki> This chapter examines power applying an intersectional framework to reveal and acknowledge intersecting identities in data collection and data analysis. power and inequity are structured differently for different groups. Then, authors explore ways to challenge power, like collecting counter-data to quantify and visualize structural oppression, as Gwendolyn Warren and the DGEI did with their maps. This helps people who occupy positions of power understand the scope, scale, and character of the problems from which they are otherwise far removed. * D'Ignazio, C., & Klein, L. (2020, March 16). 6''. the number's don't speak for themselves · data feminism''. Data Feminism. Retrieved April 26, 2022, from <nowiki>https://data-feminism.mitpress.mit.edu/pub/ei7cogfn/release/4</nowiki> This chapter examines the phenomenon of open data and its relationship to the impact of availing raw data without additional information. The authors discuss ways in which raw data can be used to spread misinformation and manipulate forms of oppression. The authors encourage continuing to build access to open data, while also providing information in which the datas are seated in, to avoid the "reinforcement of unjust status quo." * Gallon, Kim. A Review of COVID-19 Intersectional Data Decision-Making: A Call for Black Feminist Data Analytics, Part I <nowiki>https://medium.com/@ktgallon/a-review-of-covid-19-intersectional-data-decision-making-a-call-for-black-feminist-data-analytics-da8e12bc4a6b</nowiki> (Links to an external site.) The author analyzes the intersectional approach to COVID-19 and talks about the significance of acknowledging people's multiple identities, which shape their vulnerability and experience with the coronavirus. Besides, this intersectional approach to data analysis is effective to develop strategies for ameliorating the impact of COVID-19 on Black and Brown people. However, the article also reveals the limits of this approach and discusses further algorithms of oppression, which is a specific manifestation of discriminatory design in which racist values and assumptions are built into technical systems. * Sivashanker, K., Duong, T., Ford, S., Clark, C., & Eappen, S. (2021, February 1). ''A data-driven approach to addressing racial disparities in health care outcomes''. Harvard Business Review. Retrieved April 26, 2022, from <nowiki>https://hbr.org/2020/07/a-data-driven-approach-to-addressing-racial-disparities-in-health-care-outcomes</nowiki> This research article shared the results of a case study taken place at Brigham Health in Boston to present COVID-19's disproportionate impact on Indigenous, Black, and other disadvantaged communities. The study illuminates an attempt made to center Black Feminist frameworks in a data collection model by using intersectionality to set the parameters for filters that would organize data. The study was used to reinforce the health inequities that are exacerbated when the collection of data is not organized to reflect on racial, ethnic, and class disparities. * WebMD. (2020, August 17). ''Mobile apps for Coronavirus (COVID-19): See the list''. WebMD. Retrieved April 26, 2022, from <nowiki>https://www.webmd.com/lung/coronavirus-apps#1</nowiki> This article introduces coronavirus mobile apps, like coronavirus symptoms apps, Covid contact tracing apps, health monitoring apps, social distancing apps, and Covid prevalence apps. Besides, it also talks about risks when people use these Covid apps. For example, the app can access personal information on one's device, but sometimes the rationale for this might not make sense for the app's purpose. * Wikimedia Foundation. (2022, February 9). ''Matrix of domination''. Wikipedia. Retrieved April 26, 2022, from <nowiki>https://en.wikipedia.org/wiki/Matrix_of_domination#cite_note-:0-3</nowiki> This source was used to help define the contributions of Patricia Hill Collins to emphasize another framework to center Black Feminism in data. * Young, M. (2021, August 17). ''Interactive map of COVID cases shows NYC's hotspot ZIP codes''. Untapped New York. Retrieved April 26, 2022, from <nowiki>https://untappedcities.com/2020/11/12/interactive-map-covid-nyc-hotspot-zip-codes/</nowiki> This article profiles an interactive data visualization map that was published in 2020 to track hotspot COVID cases using NYC open data sources. The interactive map uses color gradients to visualize COVID cases in different zip codes across NYC captured by testing sites, and other city collected data. While the visualization provides a valuable resource for city residents to engage with health and safety concerns in their neighborhoods, the profile also underscores gaps in data that prevent the map from accurately shaping certain communities. f52ip4t516gnwp68ksmu2zhooq6fr2b 0 282662 2410835 2391193 2022-08-01T22:50:41Z Dave Braunschweig 426084 Dave Braunschweig moved page [[User:Jacsaw/sandbox/Building Informed Critique]] to [[Advocacy in Technology and Society/Building Informed Critique]] without leaving a redirect: Moving to main space wikitext text/x-wiki === '''Topic Summary''' === Summarize what was covered during this topic through the lecture, discussion, and the readings assigned this week. The overarching theme for this week revolves around accountability. Whether there is misinformation being spread or a lack of fact checking with news sources and social media posts, there is also a lack of accountability and transparency occurring when it comes to the algorithms being applied for use in the public sector. [https://www.pewresearch.org/journalism/2021/09/20/news-consumption-across-social-media-in-2021/pj_2021-09-20_news-social-media_0-01/ '''Rise in Usage of Social Media as a News Platform'''] According to a PEW Research Center study, about half of Americans get news on social media at least sometimes, down slightly from 2020 (Walker & Matsa, 2021). With the increasing prevalence of social media in people's lives, the question of whether social media serves as a reliable news platform continues to be raised. Considering how a third of Americans choose Facebook as their primary news source, we explore how their fact-checking procedures affect such an audience that is largely White, about 60% of the site's regular news consumers. '''[https://www.aljazeera.com/news/2022/2/25/russia-limiting-access-to-facebook-over-fact-checking-row Accountability in Media, Misinformation, and International Relations]''' With the recent invasion into Ukraine from Russia, Facebook has been conducting independent fact checking and labeling of content to which the Russian government has requested come to a halt (Al Jazeera, 2022). As Facebook refused, Russia has since announced that it will place restrictions for people using Facebook in the country. Facebook felt compelled to continue their fact checking and labeling work to provide an outlet for ordinary Russians who feel anti-war and want to make their voices heard. '''[https://www.facebook.com/journalismproject/programs/third-party-fact-checking/how-it-works Facebook Factchecks]''' Facebook partnered with third-party fact-checkers from the International Fact-Checking Network (IFCN) to achieve two goals: 1. Empower users to access reliable information, and 2. Fight the spread of misinformation. Facebook and IFCN work through a multi-step process to achieve these goals, first by reviewing content shared on the Facebook platform, identifying the site where it first took off, the associated keywords, and tip-offs from the community. Then, fact-checkers will review the material's verity and either certify the information as acceptably accurate or tag it with a warning. One unique point about this partnership is fact-checkers will not remove content, but will only "apply strong warning labels" and stem future dissemination to other sources. This approach aims to strike a balance amongst free speech, "safety, authenticity, privacy, and dignity" for contributors and users to communicate in a shared space with confidence. '''[https://www.cnn.com/2022/02/26/politics/fake-ukraine-videos-fact-check/index.html Media & Misinformation]''' An example discussed in class referred to a fake video where conflict appeared to be occurring in Ukraine when it in fact was not, as a result of stitching a video game clip with actual clips from the Russian invasion on Ukraine (Subramaniam, 2022). Other instances exist as well where images are mirrored and even broadcasted to depict war in other regions of the world. Videos from war occurring in Palestine were taken and said to be part of the war in Ukraine without ever raising awareness to the war in Palestine. '''[https://www.opengovpartnership.org/documents/+algorithmic-accountability-public-sector/ Algorithmic Accountability in The Public Sector]''' A global study was conducted by the Ada Lovelace Institute (Ada), AI Now Institute (AI Now), and Open Government Partnership (OGP) to analyze the first wave of algorithmic accountability policy in the public sector. According to the study, an '''algorithm''' is described as a series of steps through which particular inputs can be turned into outputs (Ada Lovelace Institute, AI Now Institute and Open Government Partnership, 2021). An '''algorithmic system''' consists of one or more algorithms that are often used to produce outputs used for making decisions to replace certain decisions that would be made by humans. This report calls for algorithmic accountability policies to make sure that those who build, obtain, and use these algorithms are held responsible in the public sector. The report is constructed to be comprehendible for policymakers, researchers, and civil society members. It is important to note that the public sector is comprised of government agencies and institutions. Additionally, it is important to note the contexts and systems that these policies are designed and implemented into. The report discusses 8 different policy mechanisms and aims that have been previously used in attempts to achieve algorithmic accountability in the public sector. # Principles and Guidelines # Prohibitions and Moratoria # Public Transparency # Impact Assessments # Audits and Regulatory Inspection # External/Independent Oversight Bodies # Rights to Hearings and Appeal # Procurement Conditions 6 Lessons Learned from the Implementation of Effective Algorithmic Accountability Policies # Clear institutional incentives and binding legal frameworks can support consistent and effective implementation of accountability mechanisms, supported by reputational pressure from media coverage and civil society activism. # Algorithmic accountability policies need to clearly define the objects of governance as well as establish shared terminologies across government departments. # Setting the appropriate scope of policy application supports their adoption. existing approaches for determining scope such as risk-based tiering will need to evolve to prevent under-and over-inclusive application. # Policy mechanisms that focus on transparency must be detailed and audience appropriate to underpin accountability. # Public participation supports policies that meet the needs of affected communities. Policies should prioritize public participation as a core policy goal, supported by appropriate resources and formal public engagement strategies. # Policies benefit from institutional coordination across sectors and levels of governance to create consistency in application and leverage diverse expertise. === Insights === Go beyond your summary, synthesize what you learned on this topic, drawing on personal insights and that of your peers, the content, what you read in class and in the media Overall, the main takeaway from the topic is that accountability and transparency are becoming increasingly more important. With an increase in social media being used as a news platform, the onus is on the public to decipher what is or is not "fake news". While this may be difficult enough to do on a daily basis, this issue becomes even more difficult for individuals in the United States if English is not their first language. With the "death of the expert," no one is trusting of news and even facts appear as subjective. Talk show host John Oliver critiques the state of digital misinformation and poignantly discusses: “There needs to be public pressure on platforms to do more about misinformation whether they are in English or not because until they do, if you are a member of one if these diaspora communities you may have to prepare yourself with more difficult conversations with your least favourite uncles" (Guardian News and Media, 2021). Classmates have shared that the passive act of mindless scrolling and liking on social media are actually active in feeding into the algorithms embedded within the technology and it is also important to note that certain media platforms continue to perpetuate divisive content as they profit off of doing so yet they are not held accountable for their actions. Algorithms on social media platforms amplify divisive content and the key point is this: "This is not about free speech and what individuals post on these platforms. It is about what the platforms choose to do with that content, which voices they decide to amplify, which groups are allowed to thrive and even grow at the hand of the platforms’ own algorithmic help" (Harvard Business Review, 2021). Social media evolves ever so quickly that it's difficult to anticipate when the next new app or trend will be. TikTok, Instagram, and Youtube are a few examples of "fast media" where information is passed from one source to another, more often than not with insufficient citation. However, it's critical to consider data "laterally" and observing what related sources are reporting. It's interesting to consider how social work blends with other professions like law and journalism that make for a multi-faceted way to digest and disseminate data. While journalism revolves around the newsroom and widely sharing "factual information," social workers abide by the ethical code of confidentiality, transparency, and informed consent. In this intersection with social work and technology, advocates can consider how to develop language around "informed critique" as well to empower all communities by meeting their needs as they are. Oftentimes this can mean communities without internet access or a personal device; however, this should not undermine their contribution to discussions on accountability and misinformation in technology. === What does advocacy look like here? === Considering your topic, respond to the following: 1. How can advocates use technology and data-based practice to further their advocacy efforts? And 2. How are current and emergent technologies presenting challenges in society, and how can advocates work in pursuit of better, fairer, and more just technology? Social workers in technology are critical to expounding data dissemination, misinformation, and informed critique by bringing in voices and experiences from marginalized and overlooked communities. They have the unique blend as a community organizer, case manager, and policy advocate along with other hats to serve as a liaison between the public and Big Tech. Advocates can use social media and technology to their advantage by holding space for communities who've been hurt by misinformation or misrepresentations and bridging the gap through public hearings, offering data analysis workshops, or building a technology wellness toolkit with guidelines on how to recover if they saw a triggering news headline. Social workers bring a myriad of skills to any workspace but even more so in technology and data-based practice because they center the user as a person, as a human being, whereas Big Tech see users as numbers, digits, and followers. It's important for social workers to humanize the technology experience and bear witness to the harms and benefits that technology offers. '''Current and Emergent Technologies''' Deepfake is one practice in artificial intelligence where an individual in an existing picture or video is scanned, matched, and replaced with another individual of a similar likeness. This AI practice threatens the ethical framework of technology where users and developers should agree to disseminating accurate information. With technology becoming increasingly advanced, it's become more and more difficult to pinpoint misinformation and hold the responsible parties accountable, presenting a challenge on credibility and accountability in society. Social work advocates in pursuit of fair tech that works ''for'' the people are encouraged to consider where the technology and information gap is between current and emerging technologies and where marginalized populations stand in the midst of it. Questions to consider can include "Who is the target population, and why? How is this technology informative, empowering, or transparent? Does the technology overlook or isolate a particular population, and how?" === Annotated References === Cite what we covered in class and include 4 additional resources you find on the topic. Cite resources that are related to the topic, they can be in agreement with the topic, extending it, in disagreement or present conflict with the topic. Ada Lovelace Institute, AI Now Institute and Open Government Partnership. (2021). ''Algorithmic Accountability for the Public Sector''. Available at: [https://www.opengovpartnership.org/documents/&#x20;algorithmic-accountability-public-sector/ https:// www.opengovpartnership.org/documents/ algorithmic-accountability-public-sector/] Al Jazeera. (2022, February 25). ''Russia limiting access to facebook over fact-checking row''. Russia-Ukraine war News | Al Jazeera. Retrieved April 26, 2022, from https://www.aljazeera.com/news/2022/2/25/russia-limiting-access-to-facebook-over-fact-checking-row Guardian News and Media. (2021, October 11). ''John Oliver on digital misinformation: 'there needs to be more public pressure on platforms'''. The Guardian. Retrieved April 26, 2022, from https://www.theguardian.com/tv-and-radio/2021/oct/11/john-oliver-last-week-tonight-digital-misinformation How Facebook’s third-party fact-checking program works. (2021, June 1). Retrieved April 26, 2022, from https://www.facebook.com/journalismproject/programs/third-party-fact-checking/how-it-works ''How to hold social media accountable for undermining democracy''. Harvard Business Review. (2021, January 12). Retrieved April 26, 2022, from https://hbr.org/2021/01/how-to-hold-social-media-accountable-for-undermining-democracy Myers, S. L., & Kang, C. (2022, April 21). ''Barack Obama Takes On a New Role: Fighting Disinformation''. The New York Times. https://www.nytimes.com/2022/04/20/technology/barack-obama-disinformation.html Subramaniam, T. (2022, February 26). ''Fact-checking fake videos of Ukraine conflict''. CNN. Retrieved April 26, 2022, from https://www.cnn.com/2022/02/26/politics/fake-ukraine-videos-fact-check/index.html The Associated Press. (2022, April 23). ''EU law targets Big Tech over hate speech, disinformation''. NPR. https://choice.npr.org/index.html?origin=https://www.npr.org/2022/04/23/1094485542/eu-law-big-tech-hate-speech-disinformation Walker, M., & Matsa, K. E. (2021, September 20). ''News consumption across social media in 2021''. Pew Research Center's Journalism Project. Retrieved April 26, 2022, from https://www.pewresearch.org/journalism/2021/09/20/news-consumption-across-social-media-in-2021/ s1v9k0w6s9gxv8w2v474mg6khvlaf0n 0 282663 2410839 2391151 2022-08-01T22:50:42Z Dave Braunschweig 426084 Dave Braunschweig moved page [[User:Jacsaw/sandbox/Design Justice + Creative Media]] to [[Advocacy in Technology and Society/Design Justice + Creative Media]] without leaving a redirect: Moving to main space wikitext text/x-wiki === '''Topic Summary''' === For the topic of "Design Justice & Creative Media" students engaged with a presentation provided by [https://renatagaui.com/ Renata Gaui] and [http://www.franrodriguezsawaya.com/ Francesca Rodruiguez Sawaya] on their experiences as women in the digital/design world. To prepare for this week's conversation, students read ''Design Justice, A.I., an Escape from the Matrix of Domination'' by Samantha Costanza-Chock. ''[https://youtu.be/4eccsPy3Wfs CLICK HERE] to watch Design Justice 101 with Samantha Costanza-Chock'' === [https://jods.mitpress.mit.edu/pub/costanza-chock/release/4&#x20;HERD Principles of Design Justice] === 1.    “We use design to '''sustain, heal, and empower''' our communities, as well as to seek liberation from exploitative and oppressive systems.” 2.    “We '''center the voices of those who are directly impacted''' by the outcomes of the design process.” 3.    “We '''prioritize design’s impact on the community''' over the intentions of the designer.” 4.    “We view '''change as emergent from an accountable, accessible, and collaborative process''', rather than as a point at the end of a process.” 5.    “We see the role of the '''designer as a facilitator rather than an expert.”''' 6.    “We believe that '''everyone is an expert based on their own lived experience''', and that we all have unique and brilliant contributions to bring to a design process.” 7.   “We '''share design knowledge and tools''' with our communities.” 8.    “We work towards '''sustainable, community-led and -controlled''' outcomes.” 9.    “We work towards '''non-exploitative solutions''' that reconnect us to the earth and to each other” 10.    “Before seeking new design solutions, '''we look for what is already working''' at the community level. We honor and uplift traditional, indigenous, and local knowledge and practices.” (Costanza-Chock, 2018) === [http://www.warpingthefuture.online/ Warping the Future] === ==== Lecture:''"How craft led to the digital world we know"'' ==== Renata Gaui and Francesca Rodriguez Sawaya explored the connection between weaving, craft, and computational technologies in their presentation, "How Craft led to the Digital World we Know". Parallels between these two histories, namely the extreme perishability and/or the intentional erasure present in both, served as primary topics of the discussion. Further, "[https://www.warpingthefuture.online Warping the Future]", an interactive exhibit co-curated by Gaui and Sawaya was summarized. The exhibit, inspired by the co-curators research around craft and technology, placed emphasis on recognizing women's distinct role throughout the histories of both domains of design and technology. === [https://design-justice.pubpub.org/pub/ap8rgw5e/release/1 Artifacts Have Politics] === ''Notes on: Design Justice, A.I., an Escape from the Matrix of Domination'' by Samantha Costanza-Chock. In design justice there is acknowledgment for how our larger society's norms, values and assumptions become encoded and reproduced through the design of socio-technical data driven systems. This complex idea can be shown in concrete examples of individuals lived experiences. Imagine, for example, you are heading to the airport. Maybe you are taking a needed vacation or out on a business trip. The stress of making it through security and getting to your terminal on time are enough to make anyone sweat a little. However if you identify as cisgender and white the experiences afforded to you through TSA may be completely different to individuals of color, transgender, or non binary experience. ''"TSA says it does not discriminate on the basis of gender identity, and that travelers will be treated with respect as members of whatever gender they present themselves as. But in fact, the "Automated Target Recognition" software used by body scanners is set up to police both binary sex and cisgender bodily expectations. Bodies that vary from these expectations set off alarms and are treated as potential terrorist threats."'' Here, an individual further details their experience as a non binary person (assigned female at birth) going through the TSA scanner: ''"Since my gender presentation is nonbinary femme, usually the operator selects ‘female.’ However, the three dimensional contours of my body, at millimeter resolution, differ from the statistical norm of ‘female bodies’ as understood by the dataset and risk algorithm designed by the manufacturer of the millimeter wave scanner (and its subcontractors), and as trained by a small army of click workers tasked with labelling and classification (as scholars Lilly Irani and Nick Dyer-Witheford, among others, remind undefined). If the agent selects ‘male,’ my breasts are large enough, statistically speaking, in comparison to the normative ‘male’ body-shape construct in the database, to trigger an anomalous warning and a highlight around my chest area. If they select ‘female,’ my groin area deviates enough from the statistical ‘female’ norm to trigger the risk alert, and bright yellow pixels highlight my groin, as visible on the flat panel display. In other words, I can’t win."'' Individuals of color face similar discrimination and barriers going through TSA from things such as hair deviating from the societal "norm", therefore being flagged for additional security. These are few examples in one specific area of our life that culture values, norms, and assumptions get encoded into data sets to marginalize minority groups. However these values and assumptions are encoded into data sets way beyond TSA and effect the lives of many marginalized communities. === Community Led Practices to Build the Worlds We Need === This article seeks to explore the complex and ever changing relationship between design, power, and social justice. By encouraging people of marginalized communities to have involvement in community-based social justice organizations on the global scale, to spearhead the movement of creating equity in the products and services created and distributed in the technology fields. This is a communal effort that will ultimately strengthen the relationship between technology and social justice, effectively making the argument that in order to create a world of ethical and sound technology, communities must unite in their advocacy to diversify fields and build a better world that works towards liberation and sustainability. By centering the voices of marginalized individuals in the conception and maintenance of ethical and emerging technology, the hope is to continuously challenge structural and institutional inequalities that permeate into the world of technology and design professions. Having marginalized voices at the forefront of community efforts can ensure the collective liberation of all peoples living under the matrix of domination, which includes white supremacist hetero-patriarchy, ableism, capitalism, and settler colonialism". With the goal to improve the material conditions of all modern peoples under the existing systems of oppression, collective action steps must be taken to combat the existing inequity within the technology and design fields. === Advocacy in Design Justice & Creative Media === This presentation, and topic in general, prompts one to consider the exciting potential that rests among the crossover between the humanities and technology. While generally understood as separate domains, efforts such as those put forth by Renata Gaui and Francesca Rodriguez Sawaya prove there is amble possibilities for effective advocacy to manifest through efforts that emphasize both social and technological experiences and it's impact on culture(s). Moreover, this discussion illuminates the fruitful outcomes of technological efforts informed by human experience. This weeks readings illustrate the need for a social work lens to be present in the innovation and execution of socio-technical data driven systems. Ultimately, socio-technical data driven systems, such as TSA scanners, must be understood as technology informed by social schemas, as they are created by individuals living in an organized society. Thus, social workers, who can be understood as investigators of bias and stigma, could be effective resources in the effort to ensure such systems are widely accessible and maintain as minimal bias as possible. Current and emerging technologies present challenges in society that reflect the current sociopolitical state of society. Issues of power, race, oppression, and privilege are reflected in the product design of the technologies available and accessible to the population. To combat a more ethical connection to technology, it is crucial to have individuals on the design team that are proportional to the users they aim to serve. Consulting with users, receiving and incorporating feedback, and employing a diverse advocacy team will call for a consistently improving relationship with technology. == '''Resources for Design Justice:''' == '''Podcast:''' Data & Society: https://datasociety.net/library/design-justice/ '''General resources:''' https://designjustice.org/resources-overview '''Book:''' Designs for the Pluriverse, Arturo Escobar '''Book:''' Design Justice, Sasha Costanza-Chock '''Book:''' Emergent Strategy, Adrienne Maree Brown '''Book:''' Race after technology: Abolitionist tools for the New Jim code, Benjamin Ruha '''Book:''' Artificial unintelligence: how computers misunderstand the world, Meredith Broussard '''Book:''' Design justice: community-led practices to build the worlds we need, Sasha Costanza-Chock '''Book:''' Automating inequality: How high-tech tools profile, police, and punish the poor, Virginia Eubanks '''Book:''' Value sensitive design: Shaping technology with moral imagination, Batya Friedman '''Book:''' Mismatch: How Inclusion Shapes Design, Kat Holmes '''Book:''' Algorithms of oppression: How search engines reinforce racism, Safiya Umoja Noble '''Book:''' Participatory Design: Principles and Practices, Douglas Schuler === Annotated References === 1. Costanza-Chock, S. (2018,July 16). ''Design Justice, A.I., and escape from the Matrix of Domination. Journal of Design and Science''. Journal of Design and Science. Retrieved April 25, 2022, from <nowiki>https://jods.mitpress.mit.edu/pub/costanza-chock/release/4</nowiki> 2. Costanza-Chock, S. (2020). ''Design Justice : Community-Led Practices to Build the Worlds We Need''. MIT Press. 3.Costello, C. G. (1970, January 1). ''Traveling while trans: The false promise of better treatment''. TransFusion. Retrieved April 25, 2022, from <nowiki>https://trans-fusion.blogspot.com/2016/01/traveling-while-trans-false-promise-of.html</nowiki>. 4. The Gottesman Libraries at Teachers College. (n.d.). ''Warping the future''. Warping The Future.Retrieved April 25, 2022, from <nowiki>http://www.warpingthefuture.online/</nowiki> etplmd0l35um1luery2wmf7hsquja52 0 282664 2410836 2391190 2022-08-01T22:50:41Z Dave Braunschweig 426084 Dave Braunschweig moved page [[User:Jacsaw/sandbox/Civic and GovTech, Regulation in the time of Big Tech]] to [[Advocacy in Technology and Society/Civic and GovTech, Regulation in the time of Big Tech]] without leaving a redirect: Moving to main space wikitext text/x-wiki === '''Civic and GovTech: Regulation in The Time of Big Tech''' === This week we discussed the ethics around surveillance technology, machine learning, and black-box algorithms, and how we may be automating discrimination due to data bias. We discussed the various use and misuse of facial recognition tech and we did a case study of an apartment complex in Brooklyn where a landlord was using facial technology to monitor the tenants. The tenants felt violated by the tech, organized together, and successfully got the biometric technology out of their apartment complex. '''Machine Learning''' Machine learning is a subfield of Artificial Intelligence that uses advanced programming to predict an algorithm. For example, inputting primary data and processed data into a machine learning program that processes that into and outputs a formula for how the data was processed. The following is a brief description of how machine learning works using the pizza analogy. * Traditional Programming: In traditional programming, we have "ingredients" (input), which are used in a "recipe"(construction) to create a "pizza"(product) ** ''Ingredients (input: what do you have available)?'' ** ''Recipe (logic: putting everything together)'' * Machine Programming: In machine programming, we have "ingredients" and "pizza", allowing us to use logic to discover "recipe". ** ''Pizza (target/what is made)'' ** ''Ingredients (input/whats available)'' '''Surveillance technology''' Surveillance technology can be a device or system that collects and stores data to track and monitor an individual or community. This could include biometric information, location tracking, or dietary preferences. The combination of data and A.I. can be helpful to track and identify unauthorized credit card purchases but can also be used against marginalized communities and contribute to gentrification. '''Black box algorithms''' Machine learning generates algorithms based on primary data. These algorithms (AKA predictive algorithms) are often complicated and a mystery to even the designers of the program. The designers don’t always know what connections the program has made and what predictions it will continue making. The possibilities are endless and potential good or harm is unknown but it doesn’t prevent designers and corporations to push them out into the real world as commercial products. '''Case study: Atlantic Towers Facial Technology''' <ref name=":5" /> While facial recognition technology is normalized in personal devices such as unlocking your phone or logging into your banking app. Its use in the public sphere to track and surveil citizens has long been criticized. In 2019, an apartment complex in Brooklyn came under public scrutiny when they installed facial recognition tech at the building to collect data on tenants and use the tech to gate keep everyone who enters and leaves the building. The tenants felt violated and came together to take action. Under the leadership of a long-term resident, Tranae Moran<ref name=":6" /> organized the neighbors and filed a complaint with the state, and won against the landlord. The group has raised this concern on a city level and has led to the “No Barriers to Housing Act.” '''Recent Legislations''' San Francisco becomes the first city in the US to ban facial recognition technology.<ref name=":4" /> "Stop Secret Surveillance" bill will prevent local government agencies from using this technology. Businesses and private individuals will be able to continue using it, however, it prevents tech giants like Amazon and Microsoft from selling to US law enforcement.         Sparked by the Atlantic Towers incident, Brooklyn legislator introduced the “No Biometric Barriers Housing Act of 2019"<ref name=":7" />. If the bill becomes law, it would ban public housing agencies like HUD from using facial and biometric recognition technology in most federally funded housing. The bill would further require HUD to report on how this emerging tech plays out in public housing. '''To prepare for this week's discussion, students read the following texts:''' ''[https://www.theguardian.com/cities/2019/may/29/new-york-facial-recognition-cameras-apartment-complex New York tenants fight as landlords embrace facial recognition cameras], by Erin Durkin'' ''[https://logicmag.io/beacons/organizing-as-joy-an-ocean-hill-brownsville-story-with-tranae-moran-and/ Organizing as Joy: An Ocean-Hill Brownsville Story, with Tranae Moran and Fabian Rogers]'' ''[https://www.natlawreview.com/article/get-ready-new-york-city-s-new-biometric-identifier-information-law Get Ready for New York City’s New Biometric Identifier Information Law]'' === Insights === The common theme among student presentations of the day and class material was surveillance and regulations. Arguments were presented both for and against surveillance. A student shared how her mom was driving cross-state solo for 8 hours and having the tech access her location anytime was reassuring. Other shared, parents and in-laws can be overbearing with access to location and battery power. The class concluded that the tech isn’t inherently harmful but depends on who has access to it and how they use this information. A suggestion was to give more agency to the user by notifying them when someone checks in on their location. We agreed there needs to be a better regulation to prevent misuse. This led to a conversation on accountability. '''Accountability concerns''' With machine learning and automation technology becoming a new standard the largest concern was who would be held accountable when harm is done by a faceless machine and who is responsible for mitigating the danger and preventing harmful designs. For so long, tech giants have encouraged an experimental space with no restriction in design or policy. This allowed the space to create and accumulate innovation, talent, money, and historic power. However, with unchecked power came ethically questionable products that have led to automation of racism, sexism, ablism and more. For example, the growing concern of social media addiction among youth and the high rates of mental health issues related to use of social media. Silicon Valley’s ”Do First Apologize Later” motto reflects poorly on the dominant group for being careless of products and algorithms that burden marginalized communities. '''Running themes''' ''Surveillance isn't inherently bad but depends on whose has access and if they are weaponizing it.'' Surveillance is regularly seen as a negative for the public, often thought of as a way for the government to watch over and control their people, but maybe there is more to that. As displayed in a variety of presentations this week, surveillance is much more nuanced than some may make it out to be. It seems that the good and ugly of surveillance tech relies more so on the user and the purpose for which it is being used. For example, one student discussed how using tracking apps like Life360 have really benefitted her and her family as they all tend to be spread across the country and her father works a potentially dangerous job, thus having the app saves all of them from worrying too much about whether the family is safe. Another student had a very different reaction to the same app and felt she wasn’t comfortable sharing her location with others. In response, those individuals felt so strongly about her using the app to the point where she felt she was being nonconsensually tracked. Other examples from the day included the idea of ancestry DNA testing being uploaded to websites that aid the police in arrests, but can also help solve Doe cases and geotags being great for finding lost items but being a threat when unsuspectingly used for nefarious purposes. It seems most surveillance tech operates as a double-edged sword. ''Biases in machine learning occur due to the data being input.'' Is there such a thing as neutral data? This was a question asked in class this week as bias in machine learning was discussed. Due to the racial implications of the society we live in, even neutral data points may encode race where race was not originally part of the data set, something that tends to happen with machine learning. The example used in class was referencing loan allotment based on neutral data sets like mileage on car, but while a machine may see a high mileage car of less worthy of a loan, it isn’t taking into account that it may be primarily BIPOC or lower-income individuals wanting to purchase higher mileage vehicles. This algorithm has now developed a tendency to up the repayment fees for people of color or people experiences poverty looking to take out car loans. This is largely because algorithms and technology tend to only be as good as the initial programmers and continue to carry the bias of the coder, transforming it as the technology evolves. ''Machine Learning and Artificial Intelligence is only as good as the initial programmers.'' So long as humans have a hand in technology, it will fall victim to the same societal archetypes its creators do. This is why it is imperative that programmers are educated in social justice issues, consider history in their design and employ multidisciplinary approaches to tech development that incorporates a large swath of viewpoints and experiences to mitigate the potential of harm and stop the further marginalization of already marginalized groups. Examples of biases built on neutral data further marginalizing already marginalized individuals: * Costanza-Chock (2020)<ref name=":3">{{Cite book|url=http://dx.doi.org/10.7551/mitpress/12255.001.0001|title=Design Justice|last=Costanza-Chock|first=Sasha|date=2020|publisher=The MIT Press|isbn=978-0-262-35686-2}}</ref> ** In an example from Costanza-Chock (2020)<ref name=":3" />, the idea of neutral data points driving marginalization feels especially potent. In the opening pages of the book the author details their anxiety over traveling as a trans individual through airports– in specific, security. They’ve come to learn that the UI of millimeter security scanning does not take into account bodies outside the “standard” binary, meaning those who may present outwardly feminine but not have the female genitalia to match tend to be flagged for further inspection. This then becomes quickly embarrassing as TSA agents respond to the machine’s cries and debate over who will pat down the author– a duty normally assigned based on binary gender/sex. No real way to bypass this procedure, it’s become easier for the author to simply not travel by flight and spare themselves the embarrassment that comes with the procedures, limiting their access to much of what those within the binary neglect to think about. * Class example: ** Similarly, one student in class shared their own experience of an interaction with tech that may have been using neutral data points but still clearly targeted them due to their race/natural hair. This student used an example of how every time they go to an airport they tend to be searched for drugs. Though never actually caught with drugs, this continues to happen to the student. At one point they inquired with airport security over what about them kept getting them flagged as a potential risk, in which the security officer could only point to their hair. The student regularly wears their hair in dreads and the officer noted that it seems the computer is more likely to flag people with natural hairstyles like dreads as risks for drug trafficking. So while this algorithm was hopefully never meant to raise alarms on people based on their appearance, clearly it was designed with enough bias that seemingly neutral data points have lead to a program keen on stereotyping passengers. === What does advocacy look like here? === '''Disruptive Tech''' Advocacy in technology may take on many forms, but by far the most reported on form is that of disruption. Disruptive technologies can look drastically different from one another, but they all serve the same goal: to alter the way consumers interact with a technology. Examples: * Gender PayBot<ref name=":0" /> ** Within this week, there was a presentation that discussed the Gender PayBot, a disruptive Twitter bot that took to the app on International Women’s Day 2022 to call out UK organizations for gender pay disparities. Targeting any organization that tweeted in favor of the international holiday, the bot made waves throughout the Twitterverse for pulling already published data on pay gaps within each company and publishing it, making it overwhelmingly accessible to the masses. * Google takes down apps that surveils Muslim people<ref name=":1" /> ** Google recently took down dozens of apps from its play store that were found to have spyware secretly tracking users, a lot of which targeted Muslim users. This comes after a recent VICE expose on Muslim Pro<ref name=":8">{{Cite web|url=https://www.vice.com/en/article/jgqm5x/us-military-location-data-xmode-locate-x|title=How the U.S. Military Buys Location Data from Ordinary Apps|website=www.vice.com|language=en|access-date=2022-04-27}}</ref>, a app popular among Muslims across the world to check prayer times and read the Quran. The VICE expose found MuslimPro to be selling user data to multiple third parties including the US military. The American Muslim community was outraged given the history of surveillance and illegal spying in the community. Google taking initiative to monitor the app store and banning harmful apps, is a step in the right direction in upholding users rights to privacy and prevent further harm caused by the US police state on marginalized communities. * Hyphen-Labs facial-recognition scarves<ref name=":2" /> ** Hypen-Labs, a femme BIPOC collective, released multiple disruptive accessories at the Tribeca Film Festival in 2017, making waves across the event. At the time of the debut, Hyphen-Labs was approaching the commercial release of silk scarf designed with a carefully plotted pattern that threw off facial recognition software, rendering the surveillance tactic useless. The design had the potential to overload software by creating thousands of faux-faces, taking up the space where human faces would normally be analyzed. Tested on apps like Snapchat that utilize filters, the scarf proved effective at throwing off facial recognition and completely penetrated the algorithms normally successful at recognizing features. Disruptive technology like this is especially helpful at times where facial recognition may put someone in danger. === Annotated References === <ref name=":5">{{Cite web|url=http://www.theguardian.com/cities/2019/may/29/new-york-facial-recognition-cameras-apartment-complex|title=New York tenants fight as landlords embrace facial recognition cameras|date=2019-05-30|website=the Guardian|language=en|access-date=2022-04-27}}</ref><ref name=":2">{{Cite web|url=https://www.inverse.com/article/30589-facial-recognition-scrambling-scarf-surveillance-earrings|title=Facial Recognition Scrambling Scarf From Hyphen-Labs on Sale Soon|last=Zakarin|first=Jordan|website=Inverse|language=en|access-date=2022-04-27}}</ref><ref name=":0">{{Cite news|url=https://www.nytimes.com/2022/03/09/business/pay-gap-international-womens-day-twitter.html|title=Twitter Bot Highlights Gender Pay Gap One Company at a Time|last=Holpuch|first=Amanda|date=2022-03-09|work=The New York Times|access-date=2022-04-27|language=en-US|issn=0362-4331}}</ref><ref name=":6">{{Cite web|url=https://logicmag.io/beacons/organizing-as-joy-an-ocean-hill-brownsville-story-with-tranae-moran-and/|title=Organizing as Joy: An Ocean-Hill Brownsville Story, with Tranae Moran and Fabian Rogers|website=Logic Magazine|access-date=2022-04-27}}</ref><ref>{{Cite web|url=https://www.natlawreview.com/article/get-ready-new-york-city-s-new-biometric-identifier-information-law|title=Get Ready for New York City’s New Biometric Identifier Information Law|website=The National Law Review|language=en|access-date=2022-04-27}}</ref><ref name=":1">{{Cite web|url=https://www.forbes.com/sites/zacharysmith/2022/04/06/google-reportedly-bans-dozens-of-apps-containing-spyware/|title=Google Reportedly Bans Dozens Of Apps Containing Spyware|last=Smith|first=Zachary Snowdon|website=Forbes|language=en|access-date=2022-04-27}}</ref><ref name=":4">{{Cite news|url=https://www.nytimes.com/2019/05/14/us/facial-recognition-ban-san-francisco.html|title=San Francisco Bans Facial Recognition Technology|last=Conger|first=Kate|date=2019-05-14|work=The New York Times|access-date=2022-04-27|last2=Fausset|first2=Richard|language=en-US|issn=0362-4331|last3=Kovaleski|first3=Serge F.}}</ref><ref name=":3" /><ref name=":7">{{Cite web|url=https://clarke.house.gov/nobiometricsbarriers/|title=Reps. Clarke, Pressley & Tlaib Announce Bill to Ban Public Housing Usage of Facial Recognition & Biometric Identification Technology|date=2019-07-25|website=Congresswoman Yvette Clarke|language=en-US|access-date=2022-04-27}}</ref> Cite what we covered in class and include 4 additional resources you find on the topic. Cite resources that are related to the topic, they can be in agreement with the topic, extending it, in disagreement or present conflict with the topic. 9eoj7cwz9gqyi6cu48zdngxf80r5147 0 282665 2410842 2384892 2022-08-01T22:50:43Z Dave Braunschweig 426084 Dave Braunschweig moved page [[User:Jacsaw/sandbox/Social Media, VR + the metaverse]] to [[Advocacy in Technology and Society/Social Media, VR + the metaverse]] without leaving a redirect: Moving to main space wikitext text/x-wiki === '''Topic Summary''' === Summarize what was covered during this topic through the lecture, discussion, and the readings assigned this week. === Insights === Go beyond your summary, synthesize what you learned on this topic, drawing on personal insights and that of your peers, the content, what you read in class and in the media === What does advocacy look like here? === Considering your topic, respond to the following: 1. How can advocates use technology and data-based practice to further their advocacy efforts? And 2. How are current and emergent technologies presenting challenges in society, and how can advocates work in pursuit of better, fairer, and more just technology? === Annotated References === Cite what we covered in class and include 4 additional resources you find on the topic. Cite resources that are related to the topic, they can be in agreement with the topic, extending it, in disagreement or present conflict with the topic. dlxgi4pgjyaxiy31t2sgmc4iw6fse28 0 282668 2410840 2384882 2022-08-01T22:50:42Z Dave Braunschweig 426084 Dave Braunschweig moved page [[User:Jacsaw/sandbox/Digital Landscape + Open Access]] to [[Advocacy in Technology and Society/Digital Landscape + Open Access]] without leaving a redirect: Moving to main space wikitext text/x-wiki === '''Topic Summary''' === Summarize what was covered during this topic through the lecture, discussion, and the readings assigned this week. === Insights === Go beyond your summary, synthesize what you learned on this topic, drawing on personal insights and that of your peers, the content, what you read in class and in the media === What does advocacy look like here? === Considering your topic, respond to the following: 1. How can advocates use technology and data-based practice to further their advocacy efforts? And 2. How are current and emergent technologies presenting challenges in society, and how can advocates work in pursuit of better, fairer, and more just technology? === Annotated References === Cite what we covered in class and include 4 additional resources you find on the topic. Cite resources that are related to the topic, they can be in agreement with the topic, extending it, in disagreement or present conflict with the topic. dlxgi4pgjyaxiy31t2sgmc4iw6fse28 0 282774 2410844 2391106 2022-08-01T22:50:44Z Dave Braunschweig 426084 Dave Braunschweig moved page [[User:Jacsaw/sandbox/Social Work Futures]] to [[Advocacy in Technology and Society/Social Work Futures]] without leaving a redirect: Moving to main space wikitext text/x-wiki This module focuses on the intersections of social work and technology in an innovative manner. === Topic Summary === In this lecture, we watched presentations provided by Lauri Goldkind, PhD and Laura, Nissen, PhD. Dr. Lauri Goldkind's lecture focused on topics of Data Justice, Data protection, Racial justice and technology, and Design Justice. The social work future includes bringing technology into human services. She brought up a question: How can social workers leverage different technological tools for making human services? She used an article, [https://www.tandfonline.com/doi/full/10.1080/1369118X.2018.1486870 The biggest lie on the Internet: ignoring the privacy policies and terms of service policies of social networking services] , to argue that many online platforms which promotes therapy is not taking responsibility for the therapy. Social workers need to make clients understand the importance of End User Agreement (EULA) and the Terms of Service (TOS) and make them understand what they are signing on. In addition, Dr. Lauri Goldkind introduced concept of data justice, created by Lynette Taylor, including visibility, engagement with technology, and non-discrimination. Through Dr. Nissen's lecture, she encouraged students to use social work think in post normal time. She also brought up ideas on the future of issues, such as social change, work, organizational life, cities, technology, climate, etc. She led an interesting question of, "Who will decide the future of social work would look like?", and introduced the website of [https://socialworkfutures.com/background/ Social Work Futures]. Students are encouraged to learn organizations that challenge big tech and promote ethical tech. '''Here are some learning materials recommended by Dr. Goldkind and Dr. Nissen:''' ==== [https://socialworkfutures.com/background/ Social Work Futures] ==== This website explores how social workers can increase their impact through futures frameworks. It covers contents on About Laura, Background, Recent Presentations and Talks, Some Favorite Future Voices, and What is Futures Thinking. ==== [https://mitpress.mit.edu/books/distance-cure The Distance Cure: A History of Teletherapy] <ref>{{Cite book|url=https://mitpress.mit.edu/books/distance-cure|title=The Distance Cure: A History of Teletherapy|last=Zeavin|first=Hannah|date=2021-08-17|publisher=MIT Press|isbn=978-0-262-04592-6|location=Cambridge, MA, USA|language=en}}</ref> ==== In ''The Distance Cur''e, Hannah Zeavin proposes a reconfiguration of the traditional therapeutic dyad of therapist and patient as a triad: therapist, patient, and communication technology. Zeavin tracks the history of teletherapy (therapeutic interaction over distance) and its metamorphosis from a model of cure to one of contingent help. She describes its initial use in ongoing care, its role in crisis intervention and symptom management, and our pandemic-mandated reliance on regular Zoom sessions. Her account of the “distanced intimacy” of the therapeutic relationship offers a powerful rejoinder to the notion that contact across distance (or screens) is always less useful, or useless, to the person seeking therapeutic treatment or connection. At the same time, these modes of care can quickly become a backdoor for surveillance and disrupt ethical standards important to the therapeutic relationship. The history of the conventional therapeutic scenario cannot be told in isolation from its shadow form, teletherapy. Therapy, Zeavin tells us, was never just a “talking cure”; it has always been a ''communication'' cure. '''Here is a recommended reading material:'''[https://www.wiley.com/en-us/Race+After+Technology%3A+Abolitionist+Tools+for+the+New+Jim+Code-p-9781509526437 Race After Technology: Abolitionist Tools for the New Jim Code] This book is written by Ruja Benjamin<ref>{{Cite web|url=https://aas.princeton.edu/people/ruha-benjamin|title=Ruha Benjamin|website=Department of African American Studies|language=en|access-date=2022-04-13}}</ref>, a professor in the Department of African American studies at Princeton University. It addresses how emerging technologies can reinforce White supremacy and deepen social inequity. Ruja Benjamin argues that automation has the potential to "hide, speed up, and deepen discrimination while appearing neutral and even benevolent when compared to the racism of a previous era." Presenting the concept of the “New Jim Code,” Ruja Benjamin shows how discriminatory designs encode inequity by explicitly amplifying racial hierarchies; by ignoring but thereby replicating social divisions; or by aiming to fix racial bias but ultimately doing quite the opposite. Moreover, she makes a compelling case for race itself as a kind of technology, designed to stratify and sanctify social injustice in the architecture of everyday life.<ref>{{Cite web|url=https://www.wiley.com/en-us/Race+After+Technology%3A+Abolitionist+Tools+for+the+New+Jim+Code-p-9781509526437|title=Race After Technology: Abolitionist Tools for the New Jim Code {{!}} Wiley|website=Wiley.com|language=en-us|access-date=2022-04-27}}</ref> === Insights === <u>Insights on Lauri Goldkind, PhD and Laura, Nissen, PhD.</u> The presenter argued that many online platforms which promote therapy are not taking responsibility for the therapy. The reason people seek therapy is because they are in discomfort, which means they are already suffering, and to add additional digital burdens may be counter-productive. The therapist should make sure and make it clear what the clients are getting into by agreeing to their terms and conditions. Clients should be made aware of what will happen to the data, and what is the risk associated with sharing it on an online platform. We understand that being transparent with the clients goes beyond being ethical. The therapist would not want to see that more harm was done to the clients by seeking the therapy. It should be the client's self- determination to be comfortable with the technology, but it should not be used primarily because the therapist would have easy access to reach out to the clients. <u>Insights on Social Work Futures Website</u> When we think about Social Work in a traditional setting, it is common for the general public to think about Child Protective Services removing children from parents. The dynamics of social workers have changed significantly. Now, social workers can be found working for the government, corporate, INGOs and NGOs, in disaster relief missions, at domestic violence shelters, with the police departments, in the military and the list goes on. So what does the future hold for social workers? We believe that this website has provided a glimpse of where and in what roles social workers can find themselves in the future. We understand that future of social work is an important prospect. For instance, if someone is interested in technology and also cares for social impact, this website will provide that person on how technology and social work can go hand in hand. The website has the potential to inspire people to be involved in the changing dynamics of social work. We cherished the idea that the website acknowledges that future cannot be predictable however, it has highlighted that as social workers are planners, they can plan for a more equitable society. <u>Insights on [https://mitpress.mit.edu/books/distance-cure The Distance Cure: A History of Teletherapy]</u> This book highlights an important notion that is often swept under the rug. We have been praising on all fronts that remote access to services has been handy. Recently we have been seeing an increase in advertisements prompting us to seek therapy online, but are they effective as in-person therapy? We do not want to convey the message that remote services are not productive. In some technical fields, they might be effective, however in professions where human emotions and interactions are a major part of interventions, services performed through remote means might be futile. We admit that virtual sessions get the job done easily, in dire scenarios, technology might even help avert crisis situations, but just for the sake of easiness, we should not take away the essence of human connection from therapies by being virtual. <u>Insights on [https://www.wiley.com/en-us/Race+After+Technology%3A+Abolitionist+Tools+for+the+New+Jim+Code-p-9781509526437 Race After Technology: Abolitionist Tools for the New Jim Code]</u> We have been trained in such a way that anything related to race and racialization seems more of a political matter. We tend to believe that technology and science are exclusive matters regarding race and racialization. We admired the idea that this book helps to explain how the technologies that we have been using are perpetuating White supremacy. One basic example is the use of facial recognition in surveillance of Black Lives Matter protestors by the FBI. The use of technology to stop a race from exercising their First Amendment Right might be the tip of the iceberg. This practice can far go beyond surveillance but also on the allocation of resources which will divide already strained inequality in the society. Not only does this technology target marginalized folks, but it also has the potential to errors that will result in the wrongful criminalization of innocent civilians, which we have already witnessed in our cities. The positive light that we have seen is that grassroots levels organizations are advocating for restricting the use of facial recognization technologies. === What does advocacy look like? === Social workers need to help the most vulnerable populations, who have limited access with services delivered on digital platforms. For examples, social workers can serve these these vulnerable populations by providing informational sessions to help them understand the terms of service agreement and user agreement. Additionally, social workers need to challenge companies, or large organizations to advocate for more clear practices on data usage, collection, terms of service agreement, user agreement. Social workers should also challenge the encoded bias and racism in technological innovations. This can help to advocate and promote ethical tech. In addition, Technology can be an important tool to protect the vulnerable population in the future as it will be important to protect vulnerable populations from the technology itself. Technology can be a boom, it might help a financially struggling student to finance his/her study by enabling a part-time job through Uber or in a similar manner, or it could doom a mass who are in the realm of surveillance because of utilizing the right to free speech by the authority. Social Workers and advocates should act as vigilantes to deter technology from attempting to harm vulnerable and marginalized people in many ways. Social workers should advocate at every level of government to protect these populations against digital harm. It is in the social worker's competencies to be engaged in practice-informed research and research-informed practice. Future social workers should invest in enriching their competencies by research and education on every changing dynamic of the society that has the potential to benefit the society as well as may harm the society. The emerging issues need to be researched and understood by the social work domain to be able to prevent any harm by being the representation of the marginalized population and being the voice of the voiceless. === Annotated References === [https://www.youtube.com/watch?v=_8RrX4hjCr0&ab_channel=TEDxTalks From park bench to lab bench. What kind of future are we designing? (2015)]<ref>{{Citation|title=From park bench to lab bench - What kind of future are we designing? {{!}} Ruha Benjamin {{!}} TEDxBaltimore|url=https://www.youtube.com/watch?v=_8RrX4hjCr0|accessdate=2022-04-27|language=en}}</ref> Ruja Benjamin shared three stories of discriminatory designs. She suggests that the interests and the concerns of those people with power and privilege should not govern the decisions that affect everyone.She urges for more inclusive design and inspires the audience to think, "whose voices are missing?", and "who we expect to be at the table and not just on the table." The futures will be shaped by who we are consulting, who are participating, who is thought important to get us there. [https://www.youtube.com/watch?v=7XrbzlR9QmI&ab_channel=TEDxTalks Imagining the Future: The Transformation of Humanity]<ref>{{Citation|title=Imagining the Future: The Transformation of Humanity {{!}} Peter Diamandis {{!}} TEDxLA|url=https://www.youtube.com/watch?v=7XrbzlR9QmI|accessdate=2022-04-27|language=en}}</ref> Peter Diamandis shares how society is heading towards a human-scale transformation. A future where people are highly connected, sharing ideas, knowledge and actions. Peter highlights 4 driving forces as well as 4 steps that transform humanity. <u>[https://www.tandfonline.com/doi/full/10.1080/10437797.2021.1912676 Teaching Note—Preparing Social Workers for the Digital Future of Social Work Practice]</u> The purpose of this teaching note is to highlight the need to prepare social work students for practice using ICT during and beyond COVID-19. The COVID-19 pandemic prompted an abrupt shift in the use of information and communication technology (ICT) in practice. The pre-pandemic use of informal ICT was already ubiquitous in social work practice. The purpose of this teaching note is to highlight that social work education must prepare students for practice using ICT during COVID-19 and beyond. Incorporated are findings from an international study on informal ICT use before COVID-19, which confirmed its ubiquity in social work, underscoring the need to educate and prepare social workers to use ICTs. Six knowledge areas important for social work education related to ICT use are discussed: integrating theoretical foundations; ethical considerations; confidentially and documentation; access and equity; risk assessment and management; and supervision and policy. COVID-19 was a major event that required an abrupt shift in the use of information and communication technology (ICT) in many professions including social work. The purpose of this article is to highlight the importance of information and communication technology. https://www-tandfonline-com.ezproxy.cul.columbia.edu/doi/full/10.1080/10437797.2021.1912676 Annotated References: Faye Mishna, Jane E. Sanders, Karen M. Sewell & Elizabeth Milne (2021) Teaching Note—Preparing Social Workers for the Digital Future of Social Work Practice, Journal of Social Work Education, 57:sup1, 19-26, DOI: 10.1080/10437797.2021.1912676 <references /> 1m1g0bj5ntqgc5xq1expdrhvybdh3rl 0 285225 2410852 2406676 2022-08-01T23:57:34Z Guy vandegrift 813252 /* table */ wikitext text/x-wiki TO DO #Finish this page and post wiki sisterlink at [[w:List_of_pitch_intervals#List]] #Record sister link but also mention need to change to ogg files of saw versus sine #Go back to the audacity study of the perfect fifth (original files lost) using python to generate the signals. I believe it was a 4 second signal with a drum to illustrate the beats. It should also include the tritone, I think. __TOC__ [[Draft:Sing_free/The_physics_of_music#new]] | [[w:List_of_pitch_intervals#List]] {| class="wikitable" style="text-align:right; vertical-align:bottom;" |- style="text-align:left;" ! cents ! name |- | 0 | style="text-align:left;" | P1 (unison) |- | 100 | style="text-align:left;" | S (semitone) |- | 200 | style="text-align:left;" | T (tone) |- | 300 | style="text-align:left;" | m3 (minor third) |- | 400 | style="text-align:left;" | M3 (major third) |- | 500 | style="text-align:left;" | P4 (fourth) |- | 600 | style="text-align:left;" | TT (tritone) |- | 700 | style="text-align:left;" | P5 (fifth) |- | 800 | style="text-align:left;" | m6 (minor sixth) |- | 900 | style="text-align:left;" | M6 (major sixth) |- | 1000 | style="text-align:left;" | m7 (minor seventh) |- | 1100 | style="text-align:left;" | M7 (major seventh) |- | 1200 | style="text-align:left;" | P8 (octave) |} ==table== {| class="wikitable" style="font-size:90%; float:right; clear:right; margin: 0 0 1em 1em;" |- ! Column || Legend |- | TET || bgcolor="#ffff78" | '''12'''-tone equal temperament (12-tet, etc.). |- | rowspan="8" | Limit || bgcolor="#EFCFCF" | '''3'''-limit intonation, or ''Pythagorean''. |- | bgcolor="#CFEFCF" | '''5'''-limit "just" intonation, or ''just''. |- | bgcolor="#CFEFEF" | '''7'''-limit intonation, or ''septimal''. |} {| class="wikitable" style="font-size:100%;text-align:center;" |+style="background: black; color:white"| Important musical intervals ! Cents || Note || Ratio ||Factors || Name || Play ! |- | {{right|294.13 }} || style = "background: blue; color:white"; rowspan=4 | '''m3''' | 32 : 27 || 2⁵ : 3³ || Pythagorean minor third || {{audio|help=no|Pythagorean minor third on C.mid|play}} |&minus;5.87 |- bgcolor="#FFFF78"; | '''300''' | '''2^3/12''' || '''2^1/4''' || '''Equal-tempered minor third''' || {{audio|help=no|Minor third on C.mid|play}} |'''0''' |- | {{right|301.85 }} | 25 : 21|| 5² : 3×7 ||Quasi-equal-tempered minor third||{{audio|help=no|Second septimal minor third on C.mid|play}} |1.85 |- | {{right|315.64 }} | 6 : 5 || 2×3 : 5 || Just minor third || {{audio|help=no|Just minor third on C.mid|play}} |15.64 |- | {{right|386.31 }} || style="background: red; color:white"; rowspan = 5 | '''TBA''' | 5 : 4 || 5 : 2²|| Just major third || {{audio|help=no|Just major third on C.mid|play}} |&minus;13.69 |- bgcolor="#FFFF78"; | '''400''' | 2^4/12 || 2^1/3 || Equal-tempered major third || {{audio|help=no|Major third on C.mid|play}} |'''0''' |- | {{right|407.82 }} | 81 : 64 || 3⁴ : 2⁶ || Pythagorean major third || {{audio|help=no|Pythagorean major third on C.mid|play}} |7.82 |- | {{right|427.37 }} | 32 : 25 || 2⁵ : 5² || Just diminished fourth || {{audio|help=no|Just diminished fourth on C.mid|play}} |27.37 |- | {{right|435.08 }} | 9 : 7 || 3² : 7 ||Septimal major third || {{audio|help=no|Septimal major third on C.mid|play}} |35.08 |- | {{right|498.04 }} || style="background: blue; color:white"; rowspan = 3 | '''TBA''' | 4 : 3 || 2² : 3 || Pythagorean perfect fourth || {{audio|help=no|Just perfect fourth on C.mid|play}} |&minus;1.96 |- bgcolor="#FFFF78"; | '''500''' | 2^5/12 || &nbsp; || Equal-tempered fourth || {{audio|help=no|Perfect fourth on C.mid|play}} |'''0''' |- | {{right|519.55 }} | 27 : 20 || 3³ : 2²×5 || 5-limit wolf fourth || {{audio|help=no|Wolf fourth on C.mid|play}} |19.55 |- | {{right|568.72 }} || style="background: red; color:white"; rowspan = 7 | '''TBA''' | 25 : 18 || 5² : 2×3² || Just augmented fourth||{{audio|help=no|Classic augmented fourth on C.mid|play}} |&minus;31.28 |- | {{right|582.51 }} | 7 : 5 || 7 : 5 || Lesser septimal tritone || {{audio|help=no|Lesser septimal tritone on C.mid|play}} |&minus;17.49 |- | {{right|590.22 }} | 45 : 32 || 3²×5 : 2⁵|| Just augmented fourth||{{audio|help=no|Just augmented fourth on C.mid|play}} |&minus;9.78 |- bgcolor="#FFFF78"; | '''600''' | 2^6/12 || 2^1/2={{radic|2}} || Equal-tempered tritone || {{audio|help=no|Tritone on C.mid|play}} |'''0''' |- | {{right|609.78 }} | 64 : 45 || 2⁶ : 3²×5 || Just tritone||{{audio|help=no|Just tritone on C.mid|play}} |9.78 |- | {{right|617.49 }} | 10 : 7 || 2×5 : 7 || Greater septimal tritone || {{audio|help=no|Greater septimal tritone on C.mid|play}} |17.49 |- | {{right|631.28 }} | 36 : 25 || 2²×3² : 5² || Just diminished fifth || {{audio|help=no|Just diminished fifth on C.mid|play}} |31.28 |- | {{right|680.45 }} || style="background: blue; color:white"; rowspan = 3 | '''TBA''' | 40 : 27 || 2³×5 : 3³|| 5-limit wolf fifth || {{audio|help=no|Wolf fifth on C.mid|play}} |&minus;19.55 |- bgcolor="#FFFF78"; | '''700''' | 2^7/12 || 2^7/12 || Equal-tempered perfect fifth || {{audio|help=no|Perfect fifth on C.mid|play}} |'''0''' |- | {{right|701.96 }} | 3 : 2 || 3 : 2 || Perfect fifth || {{audio|help=no|Just perfect fifth on C.mid|play}} |1.96 |- | {{right|764.92 }} || style="background: red; color:white"; rowspan = 5 | '''TBA''' | 14 : 9 || 2×7 : 3²|| Septimal minor sixth || {{audio|help=no|Septimal minor sixth on C.mid|play}} |&minus;35.08; |- | {{right|772.63 }} | 25 : 16 || 5² : 2⁴ || Just augmented fifth || {{audio|help=no|Just augmented fifth on C.mid|play}} }} |&minus;27.37 |- bgcolor="#FFFF78"; | '''800''' | 2^8/12 || 2^2/3 || Equal-tempered minor sixth || {{audio|help=no|Minor sixth on C.mid|play}} |'''0''' |- | {{right|813.69 }} | 8 : 5 || 2³ : 5 || Just minor sixth || {{audio|help=no|Just minor sixth on C.mid|play}} |13.69 |- | {{right|835.19 }} | 81 : 50 || 3⁴ : 2×5² || Acute minor sixth||{{audio|help=no|Acute minor sixth on C.mid|play}} |35.19 |- | {{right|884.36 }} || style="background: blue; color:white"; rowspan = 4 | '''TBA''' | 5 : 3 || 5 : 3 || Just major sixth || {{audio|help=no|Just major sixth on C.mid|play}} |&minus;15.64 |- bgcolor="#FFFF78"; | '''900''' | 2^9/12 || 2^3/4 || Equal-tempered major sixth || {{audio|help=no|Dim seventh on C.mid|play}} |'''0''' |- | {{right|905.87 }} | 27 : 16 || 3³ : 2⁴ || Pythagorean major sixth || {{audio|help=no|Pythagorean major sixth on C.mid|play}} |5.87 |- | {{right|933.13 }} | 12 : 7 || 2²×3 : 7 || Septimal major sixth || {{audio|help=no|Septimal major sixth on C.mid|play}} |33.13 |- | {{right|968.83 }} || style="background: red; color:white"; rowspan = 4 | '''TBA''' | 7 : 4 || 7 : 2²|| Septimal minor seventh || {{audio|help=no|Harmonic seventh on C.mid|play}} |&minus;31.17 |- | {{right|996.09}} | 16 : 9 || 2⁴ : 3² || Pythagorean minor seventh || {{audio|help=no|Lesser just minor seventh on C.mid|play}} |&minus;3.91 |- | {{right|1000.00}} | 2^10/12 || 2^5/6 || Equal-tempered minor seventh || {{audio|help=no|Minor seventh on C.mid|play}} |'''0''' |- | {{right|1017.60}} | 9 : 5 || 3² : 5 || Greater just minor seventh || {{audio|help=no|Greater just minor seventh on C.mid|play}} |17.60 |} ==b== {| class="wikitable" style="font-size:90%; float:right; clear:right; margin: 0 0 1em 1em;" |- ! Column || Legend |- | TET || bgcolor="#ffff78" | '''12'''-tone equal temperament (12-tet, etc.). |- | rowspan="8" | Limit || bgcolor="#EFCFCF" | '''3'''-limit intonation, or ''Pythagorean''. |- | bgcolor="#CFEFCF" | '''5'''-limit "just" intonation, or ''just''. |- | bgcolor="#CFEFEF" | '''7'''-limit intonation, or ''septimal''. |} {| class="wikitable" style="font-size:100%;text-align:center;" |+style="background: black; color:white"| Important musical intervals ! Cents || Note || Ratio ||Factors || Name || Play |- | {{right|294.13 }} || style = "background: blue; color:white"; rowspan=4 | '''m3''' | 32 : 27 || 2⁵ : 3³ || Pythagorean minor third || {{audio|help=no|Pythagorean minor third on C.mid|play}} |- bgcolor="#FFFF78"; | '''300''' | '''2^3/12''' || '''2^1/4''' || '''Equal-tempered minor third''' || {{audio|help=no|Minor third on C.mid|play}} |- | {{right|301.85 }} | 25 : 21|| 5² : 3×7 ||Quasi-equal-tempered minor third||{{audio|help=no|Second septimal minor third on C.mid|play}} |- | {{right|315.64 }} | 6 : 5 || 2×3 : 5 || Just minor third || {{audio|help=no|Just minor third on C.mid|play}} |- | {{right|386.31 }} || style="background: red; color:white"; rowspan = 5 | '''TBA''' | 5 : 4 || 5 : 2²|| Just major third || {{audio|help=no|Just major third on C.mid|play}} |- bgcolor="#FFFF78"; | '''400''' | 2^4/12 || 2^1/3 || Equal-tempered major third || {{audio|help=no|Major third on C.mid|play}} |- | {{right|407.82 }} | 81 : 64 || 3⁴ : 2⁶ || Pythagorean major third || {{audio|help=no|Pythagorean major third on C.mid|play}} |- | {{right|427.37 }} | 32 : 25 || 2⁵ : 5² || Just diminished fourth || {{audio|help=no|Just diminished fourth on C.mid|play}} |- | {{right|435.08 }} | 9 : 7 || 3² : 7 ||Septimal major third || {{audio|help=no|Septimal major third on C.mid|play}} |- | {{right|498.04 }} || style="background: blue; color:white"; rowspan = 3 | '''TBA''' | 4 : 3 || 2² : 3 || Pythagorean perfect fourth || {{audio|help=no|Just perfect fourth on C.mid|play}} |- bgcolor="#FFFF78"; | '''500''' | 2^5/12 || &nbsp; || Equal-tempered fourth || {{audio|help=no|Perfect fourth on C.mid|play}} |- | {{right|519.55 }} | 27 : 20 || 3³ : 2²×5 || 5-limit wolf fourth || {{audio|help=no|Wolf fourth on C.mid|play}} |- | {{right|568.72 }} || style="background: red; color:white"; rowspan = 7 | '''TBA''' | 25 : 18 || 5² : 2×3² || Just augmented fourth||{{audio|help=no|Classic augmented fourth on C.mid|play}} |- | {{right|582.51 }} | 7 : 5 || 7 : 5 || Lesser septimal tritone || {{audio|help=no|Lesser septimal tritone on C.mid|play}} |- | {{right|590.22 }} | 45 : 32 || 3²×5 : 2⁵|| Just augmented fourth||{{audio|help=no|Just augmented fourth on C.mid|play}} |- bgcolor="#FFFF78"; | '''600''' | 2^6/12 || 2^1/2={{radic|2}} || Equal-tempered tritone || {{audio|help=no|Tritone on C.mid|play}} |- | {{right|609.78 }} | 64 : 45 || 2⁶ : 3²×5 || Just tritone||{{audio|help=no|Just tritone on C.mid|play}} |- | {{right|617.49 }} | 10 : 7 || 2×5 : 7 || Greater septimal tritone || {{audio|help=no|Greater septimal tritone on C.mid|play}} |- | {{right|631.28 }} | 36 : 25 || 2²×3² : 5² || Just diminished fifth || {{audio|help=no|Just diminished fifth on C.mid|play}} |- | {{right|680.45 }} || style="background: blue; color:white"; rowspan = 3 | '''TBA''' | 40 : 27 || 2³×5 : 3³|| 5-limit wolf fifth || {{audio|help=no|Wolf fifth on C.mid|play}} |- bgcolor="#FFFF78"; | '''700''' | 2^7/12 || 2^7/12 || Equal-tempered perfect fifth || {{audio|help=no|Perfect fifth on C.mid|play}} |- | {{right|701.96 }} | 3 : 2 || 3 : 2 || Perfect fifth || {{audio|help=no|Just perfect fifth on C.mid|play}} |- | {{right|764.92 }} || style="background: red; color:white"; rowspan = 5 | '''TBA''' | 14 : 9 || 2×7 : 3²|| Septimal minor sixth || {{audio|help=no|Septimal minor sixth on C.mid|play}} |- | {{right|772.63 }} | 25 : 16 || 5² : 2⁴ || Just augmented fifth || {{audio|help=no|Just augmented fifth on C.mid|play}} }} |- bgcolor="#FFFF78"; | '''800''' | 2^8/12 || 2^2/3 || Equal-tempered minor sixth || {{audio|help=no|Minor sixth on C.mid|play}} |- | {{right|813.69 }} | 8 : 5 || 2³ : 5 || Just minor sixth || {{audio|help=no|Just minor sixth on C.mid|play}} |- | {{right|835.19 }} | 81 : 50 || 3⁴ : 2×5² || Acute minor sixth||{{audio|help=no|Acute minor sixth on C.mid|play}} |- | {{right|884.36 }} || style="background: blue; color:white"; rowspan = 4 | '''TBA''' | 5 : 3 || 5 : 3 || Just major sixth || {{audio|help=no|Just major sixth on C.mid|play}} |- bgcolor="#FFFF78"; | '''900''' | 2^9/12 || 2^3/4 || Equal-tempered major sixth || {{audio|help=no|Dim seventh on C.mid|play}} |- | {{right|905.87 }} | 27 : 16 || 3³ : 2⁴ || Pythagorean major sixth || {{audio|help=no|Pythagorean major sixth on C.mid|play}} |- | {{right|933.13 }} | 12 : 7 || 2²×3 : 7 || Septimal major sixth || {{audio|help=no|Septimal major sixth on C.mid|play}} |- | {{right|968.83 }} || style="background: red; color:white"; rowspan = 4 | '''TBA''' | 7 : 4 || 7 : 2²|| Septimal minor seventh || {{audio|help=no|Harmonic seventh on C.mid|play}} |- | {{right|996.09}} | 16 : 9 || 2⁴ : 3² || Pythagorean minor seventh || {{audio|help=no|Lesser just minor seventh on C.mid|play}} |- | {{right|1000.00}} | 2^10/12 || 2^5/6 || Equal-tempered minor seventh || {{audio|help=no|Minor seventh on C.mid|play}} |- | {{right|1017.60}} | 9 : 5 || 3² : 5 || Greater just minor seventh || {{audio|help=no|Greater just minor seventh on C.mid|play}} |} ==must add NO== {|class="wikitable" | |- | |- | {{right|90.22 }} || 256 : 243 || 2⁸ : 3⁵ ||Pythagorean [[minor second]] || {{audio|help=no|Pythagorean minor semitone on C.mid|play}} |- | {{right|100.00 }} || 2^1/12 || 2^1/12 || Equal-tempered [[minor second]] ||{{audio|help=no|Minor second on C.mid|play}} |- | {{right|111.73}} ||16 : 15 || 2⁴ : 3×5 || [[Just minor second]] || {{audio|help=no|Just diatonic semitone on C.mid|play}} |- | {{right|119.44 }} || 15 : 14 || 3×5 : 2×7 || [[Septimal diatonic semitone]] ||{{audio|help=no|Septimal diatonic semitone on C.mid|play}} |- | |- | {{right|500.00 }} || 2^5/12 || 2^5/12 || Equal-tempered [[perfect fourth]]||{{audio|help=no|Perfect fourth on C.mid|play}} |- | |- |} ==original== {| class="wikitable" style="font-size:90%; float:right; clear:right; margin: 0 0 1em 1em;" |- ! Column || Legend |- | TET || bgcolor="#EFCFCF" | '''X'''-tone equal temperament (12-tet, etc.). |- | rowspan="8" | Limit || bgcolor="#EFEFCF" | '''3'''-limit intonation, or ''Pythagorean''. |- | bgcolor="#CFEFCF" | '''5'''-limit "just" intonation, or ''just''. |- | bgcolor="#CFEFEF" | '''7'''-limit intonation, or ''septimal''. |- | bgcolor="#CFCFEF" | '''11'''-limit intonation, or ''undecimal''. |- | bgcolor="#DFCFEF" | '''13'''-limit intonation, or ''tridecimal''. |- | bgcolor="#DFDFDF" | '''17'''-limit intonation, or ''septendecimal''. |- | bgcolor="#CFCFCF" | '''19'''-limit intonation, or ''novendecimal''. |- | bgcolor="#BC8F8F" | '''H'''igher limits<!--23 and greater-->. |- | M || bgcolor="#CFFFAF" | '''M'''eantone temperament or tuning. |- | S || '''S'''uperparticular ratio (no separate color code). |} b5z71l7mpl0tqugu5xw63wh8yjewcr9 2 285229 2410832 2405799 2022-08-01T21:52:48Z U3216256 2942574 /* About Me */ wikitext text/x-wiki == About me == Hello everyone, My name is Ebony and I am a second year Bachelor of Science in Psychology student at the ''University of Canberra''. I am excited to contribute to the 2022 [[Motivation and emotion/Book/2022|Motivation and Emotion]] Book. As a first-generation University student, I am passionate about education and bringing awareness to issues that first-generation students face. == Hobbies == University Related * [https://clubs.canberra.edu.au/Clubs/CLSS CLSS Treasurer] * UCPS President (2021) Other Hobbies * Knitting/Crocheting * Baking * Reading == The book chapter I am working on == == Social contributions == lncj3h8htuuxfw9337jvebw0ki9ype4 3 285256 2410802 2406038 2022-08-01T16:33:19Z MdsShakil 2913655 MdsShakil moved page [[User talk:Baggaet]] to [[User talk:Ratekreel]]: Automatically moved page while renaming the user "[[Special:CentralAuth/Baggaet|Baggaet]]" to "[[Special:CentralAuth/Ratekreel|Ratekreel]]" wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Baggaet!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:54, 13 July 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} bj1mnijzkpazbduxpdnkglgyl8acqb9 0 285380 2410782 2410396 2022-08-01T13:26:40Z Young1lim 21186 /* Handling Series of Data */ wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Arrays ([[Media:C04.Series1.Array.1.A.20220801.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Pointers ([[Media:C04.Series2.Pointer.1.A.20180726.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.Series3.ArrayPointer.1.A.20220801.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20220418.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.11.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Applications ==== * Applications of Arrays ([[Media:C04.Series1App.Array.1.A.20220801.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.Series7.AppPoint.1.A.20200424.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.Series3App.ArrayPointer.1.A.2022024.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming]] </br> tfl0xl8f3c5ahgaqbdl3voa1xp4c39k 2 285382 2410897 2408372 2022-08-02T04:24:55Z Jtwsaddress42 234843 /* Lovelock, James E. (1919 - ) */ wikitext text/x-wiki {{RoundBoxTop|theme=3}} === [[w:James Lovelock|Lovelock, James E. (1919 - 2022)]] === <hr /> [[File:James Lovelock in 2005.jpg|thumb|James_Lovelock]] [[File:James Lovelocks Electron capture detector for a gas chromatograph, 1960. (9660569973).jpg|thumb|James Lovelocks Electron capture detector for a gas chromatograph, 1960.]] '''Notable Accomplishments''' * Electron Capture Detector * Gaia Hypothesis <br /><hr /> '''Publications''' {{User:Jtwsaddress42/Bibliography/Lovelock, James E.}} {{RoundBoxBottom}} <hr /> qnu6pk5paep9dtnnrn3julawc1uhcjj 0 285384 2410800 2410374 2022-08-01T16:26:14Z Young1lim 21186 /* Workings of the GNU Compiler */ wikitext text/x-wiki === Workings of the GNU Compiler === * Overview ([[Media:Overview.20200211.pdf |pdf]]) * Access ([[Media:Access.20200409.pdf |pdf]]) * Operators ([[Media:Operator.20200427.pdf |pdf]]) * Conditions ([[Media:Condition.20220801.pdf |pdf]]) * Control ([[Media:Control.20220616.pdf |pdf]]) * Procedure ([[Media:Procedure.20220412.pdf |pdf]]) * Recursion ([[Media:Recursion.20210824-2.pdf |pdf]]) * Arrays ([[Media:Array.20211018.pdf |pdf]]) * Structures ([[Media:Structure.20220101.pdf |pdf]]) * Alignment ([[Media:Alignment.20201117.pdf |pdf]]) * Pointers ([[Media:Pointer.20201106.pdf |pdf]]) </br> === Workings of the GNU Linker === ==== Overview ==== * Static Linking Overview ([[Media:Link.1.StaticOverview.20181120.pdf |pdf]]) * Dynamic Linking Overview ([[Media:Link.2.DynamicOverview.20181120.pdf |pdf]]) ==== Linking Process ==== * Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]]) * Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]]) * Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]]) * Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]]) * Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]]) * Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]]) * Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]]) ==== Example I ==== * Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]]) * Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]]) * Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]]) ==== Examples II ==== * analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]]) * analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]]) * analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming]] sp76nquo8ay4f1o8lu7prcrpk1dalru 2410861 2410800 2022-08-02T01:59:00Z Young1lim 21186 /* Workings of the GNU Compiler */ wikitext text/x-wiki === Workings of the GNU Compiler === * Overview ([[Media:Overview.20200211.pdf |pdf]]) * Access ([[Media:Access.20200409.pdf |pdf]]) * Operators ([[Media:Operator.20200427.pdf |pdf]]) * Conditions ([[Media:Condition.20220802.pdf |pdf]]) * Control ([[Media:Control.20220616.pdf |pdf]]) * Procedure ([[Media:Procedure.20220412.pdf |pdf]]) * Recursion ([[Media:Recursion.20210824-2.pdf |pdf]]) * Arrays ([[Media:Array.20211018.pdf |pdf]]) * Structures ([[Media:Structure.20220101.pdf |pdf]]) * Alignment ([[Media:Alignment.20201117.pdf |pdf]]) * Pointers ([[Media:Pointer.20201106.pdf |pdf]]) </br> === Workings of the GNU Linker === ==== Overview ==== * Static Linking Overview ([[Media:Link.1.StaticOverview.20181120.pdf |pdf]]) * Dynamic Linking Overview ([[Media:Link.2.DynamicOverview.20181120.pdf |pdf]]) ==== Linking Process ==== * Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]]) * Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]]) * Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]]) * Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]]) * Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]]) * Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]]) * Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]]) ==== Example I ==== * Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]]) * Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]]) * Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]]) ==== Examples II ==== * analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]]) * analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]]) * analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming]] gafloh2vaopsvncmt168s4ic1ob2799 2 285475 2410898 2408374 2022-08-02T04:37:47Z Jtwsaddress42 234843 /* Lefkowitz, Robert J. (1943 - ) */ wikitext text/x-wiki {{RoundBoxTop|theme=3}} === [[w:Robert J. Lefkowitz|Lefkowitz, Robert J. (1943 - )]] === <hr /> [[File:Robert Lefkowitz 2 2012.jpg|thumb|Robert Lefkowitz]] '''Notable Accomplishments''' * [https://www.nobelprize.org/prizes/chemistry/2012/lefkowitz/facts/ The Nobel Prize in Chemistry 2012] - shared with Brian K. Kobilka “for studies of G-protein-coupled receptors.” * G-Protein Coupled Receptors <br /><hr /> {| align= center | width= 640px |'''GPCR in Membrane''' [[File:GPCR in membrane.png|640px|GPCR in membrane]] <br /> |} <hr /> '''Publications''' {{User:Jtwsaddress42/Bibliography/Lefkowitz, Robert J.}} {{RoundBoxBottom}} <hr /> dbtdws7sm9r2lrssi2kwy8mocqbayoz 4 285491 2410937 2409404 2022-08-02T07:22:01Z Alexis Jazz 791434 Updating gadget usage statistics from [[Special:GadgetUsage]] ([[phab:T121049]]) wikitext text/x-wiki {{#ifexist:Project:GUS2Wiki/top|{{/top}}|This page provides a historical record of [[Special:GadgetUsage]] through its page history. To get the data in CSV format, see wikitext. To customize this message or add categories, create [[/top]].}} The following data is cached, and was last updated 2022-07-31T20:58:59Z. A maximum of {{PLURAL:5000|one result is|5000 results are}} available in the cache. {| class="sortable wikitable" ! Gadget !! data-sort-type="number" | Number of users !! data-sort-type="number" | Active users |- |CleanDeletions || 75 || 2 |- |EnhancedTalk || 1304 || 3 |- |HideFundraisingNotice || 749 || 7 |- |HotCat || 815 || 10 |- |LintHint || 70 || 1 |- |ReferenceTooltips || data-sort-value="Infinity" | Default || data-sort-value="Infinity" | Default |- |Round Corners || 1108 || 5 |- |contribsrange || 346 || 3 |- |dark-mode || 13 || 0 |- |dark-mode-toggle || 27 || 2 |- |edittop || 456 || 5 |- |popups || 789 || 3 |- |purge || 663 || 8 |- |sidebartranslate || 497 || 1 |- |usurper-count || 74 || 2 |} * [[Special:GadgetUsage]] * [[m:Meta:GUS2Wiki/Script|GUS2Wiki]] <!-- data in CSV format: CleanDeletions,75,2 EnhancedTalk,1304,3 HideFundraisingNotice,749,7 HotCat,815,10 LintHint,70,1 ReferenceTooltips,default,default Round Corners,1108,5 contribsrange,346,3 dark-mode,13,0 dark-mode-toggle,27,2 edittop,456,5 popups,789,3 purge,663,8 sidebartranslate,497,1 usurper-count,74,2 --> 2hd2ubj69qd2g42uhy0nu8uke64rbhl 0 285546 2410798 2410621 2022-08-01T15:54:16Z Ncharamut 2824970 /* Grant funds spent */ updated finances wikitext text/x-wiki == Goals == ''Did you meet your goals? Are you happy with how the project went?''<br>We did meet our goals and are very happy with how the project went. We were able to create a template for conferences to use seen [[Template:Conferences|here]]. And updated all of the previous Future Directions Forum pages seen [[Journal of Clinical Child and Adolescent Psychology Future Directions Forum (JCCAP FDF)|here]]. Additionally, we were able to create Wikipedia pages for some of the previous presenters of the Forum with a focus on women and other minority groups. By starting this, we have decided to try and continue this project moving forward. We have also recruited several new Wiki editors and trained those individuals how to use our template and how to edit Wiki pages. Many of the current editors working on the project have increased their skills on Wiki platforms and have attended workshops on transclusion and creating templates in order to increase our ability to effectively update pages and create the template for conferences to use. == Outcome == ''Please report on your original project targets. Please be sure to '''[[Grants:Metrics#Three_shared_metrics|review and provide metrics required for Rapid Grants]]''''' {| class="wikitable" |+ ! !Target outcome !Achived outcome !Explanation |- |1 |Host 2 edit-a-thons with HGAPS team members from both UNC and UMD |Hosted 2 edit-a-thons |Both edit-a-thons were successful. We had HGAPS members come from both UNC and UMD as well as UCLA. We also had individuals attend the edit-a-thons who are not a part of HGAPS. |- |2 |15+ contributors and attendees per edit-a-thon |10 contributors per edit-a-thon |Although we were not able to get 15+ attendees at our edit-a-thons, we were able to get 10 at both events. |- |3 |5 new editors |10 new editors |We had more than 5 new editors and ended with about 10 new editors who have not edited Wiki before. |- |4 |1 new article created, 20+ articles improved |5 new pages created and 25 pages updated |We were successful on this target and created 5 new pages related to keynote speakers and improved many existing articles with around 25 total. |- |5 |increase views within 6 months of launch |Views have increased |We have had an increase in views on FDF related pages since updating the pages and creating new ones. |} == Learning == ''Projects do not always go according to plan. Sharing what you learned can help you and others plan similar projects in the future. Help the movement learn from your experience by answering the following questions:'' *What worked well? #We made page templates using both Wiki and Canva for every page we created or improved. This was very helpful as it guided new editors and created a professional, organized, and visually continuous layout for every page we edited, even though most of our work was done collaboratively. This practice also helped to ensure that necessary and beneficial information was present on every page we edited. #Our team met regularly (at least once a week) via video call throughout the entirety of this project. Between meetings, we used a professional instant messaging platform to communicate thoughts, questions, ideas, and reminders both within our team and to other members of our larger organization. Having a clearly defined and hierarchical communication system that was still adaptable enough to meet the needs of our team (working internationally) allowed us to collaborate efficiently, solve problems as they arose, avoid miscommunication, and make the most of our time. #We did the majority of our planning work in rough drafts, instead of working/brainstorming directly on the wiki pages themselves. This allowed us to ensure that everything we put on Wiki was accurate, well thought out, and organized. The use of collaborative documents for brainstorming outside of Wiki also served to streamline the teamwork process, especially since a significant number of our editors did not have prior experience editing Wiki. #We created documents with lists of web links for use on specific pages in advance of each Edit-A-Thon. Since these link lists were created by experienced team members, they allowed us to ensure that the information editors used to build pages was accurate and extensive. Additionally, this process allowed editors to use their time most efficiently during the Edit-A-Thon, as they could focus on brainstorming and editing, instead of having to search for and vet links. #Including prizes, gift cards to virtually by lunch for editors, and designated time at the end of the Edit-A-Thon to collaboratively review pages helped not only to gather interest in the events we hosted, but also to incentivize editors to stay focused and productive over the two hour time frame. #Establishing a detailed time frame and plan for each Edit-A-Thon helped us avoid administrative errors during events and allowed us to plan how much we should expect participants to accomplish. This practice also made sure we used our time provided efficiently, and made sure that we were able to fulfill our goals at every Edit-A-Thon. #Providing new editors with Wiki training links in advance of the Edit-A-Thon, as well as with editing workshops during the first few minutes of every event helped us to ensure that new editors learned from this experience and were able to make valuable contributions to their teams. More advanced editing workshops were also provided to individuals with more experience, allowing people of all levels of Wiki editing skills to benefit from this experience and feel encouraged to continue to build their Wiki editing skills in the future. #With the help of the HGAPS Social Media team, we created Google Forms that participants were able to fill out to sign up for Edit-A-Thons. This way, our team was able to know how many editors to expect at each event, as well as to know how many new versus experienced editors planned to show up. These forms were created and advertised during HGAPS meetings and on social media well in advance of each event, in order to maximize our potential audience and gague interest. #Additionally, we used Google Forms to employ a page rating system that allowed teams to both give and receive feedback on their work, further refining editing skills and serving as an additional motivator for focused work. *What did not work so well? # *What would you do differently next time? # == Finances == ===Grant funds spent=== ''Please describe how much grant money you spent for approved expenses, and tell us what you spent it on.'' * '''Wages for template/tool-kit creators''': $2,040 ($20 per hour x 3 creators x 34 hours) ** The remaining $2589.73 was spent here for our creators. * '''Wages for trainers''': $720 ($20 per hour x 3 trainers x 12 hours) ** We spent the anticipated $720 here on trainers. * '''Gift cards to virtually buy lunch for edit-a-thon participants''': $600 ($20 for 15 participants x 2 events) ** We spent $260 on gift cards to buy lunch for edit-a-thon participants. Not everyone that attended wanted one, which led to us being under budget here. We have reallocated these remaining funds ($340) to "wages for template/tool-kit creators". * '''Incentives for sustained editing''': $500 (1st prize $100, 2nd prize $50, and 3rd prize $25 at 1 and 3 month follow-ups after last edit-a-thon = $175 x 4 "contests") ** Upon organizing the edit-a-thons, we decided to change our method for prizes. Since we organized into teams we decided to have participants evaluate each teams work at the end of each edit-a-thon. We used these evaluations to come up with a first place and second place team and these teams were given prizes. We spent $350 in this category and reallocated these remaining funds ($150) to "wages for template/tool-kit creators". * '''HGAPS merch for milestones completed in editor training''': $300 ** We spent $248.27 on HGAPS merch for editors. The remaining funds ($51.73) were reallocated to "wages for template/tool-kit creators". * '''Fiscal sponsor administrative fees (including access to Google Suites for Nonprofits platform and analytics)''': $832 (20% final budget) ** We spent the anticipated $832 for fiscal sponsor administrative fees. '''Total''': USD $4,992 ===Remaining funds=== ''Do you have any remaining grant funds?'' No, we don't have any remaining grant funds left after the completion of the project. <!--Let us know if you would like to use the remaining funds on a similar or new project. Remember, a grants officer must approve this request before you spend the money.--> ==Anything else== ''Anything else you want to share about your project?'' [[Category:Project/Rapid/Report]] tppcsseebtoqiui3ox0lnolq0gv63vy 2410799 2410798 2022-08-01T16:12:39Z Parodda 2936296 Made a series of brief edits to the "what went well" section for clarity wikitext text/x-wiki == Goals == ''Did you meet your goals? Are you happy with how the project went?''<br>We did meet our goals and are very happy with how the project went. We were able to create a template for conferences to use seen [[Template:Conferences|here]]. And updated all of the previous Future Directions Forum pages seen [[Journal of Clinical Child and Adolescent Psychology Future Directions Forum (JCCAP FDF)|here]]. Additionally, we were able to create Wikipedia pages for some of the previous presenters of the Forum with a focus on women and other minority groups. By starting this, we have decided to try and continue this project moving forward. We have also recruited several new Wiki editors and trained those individuals how to use our template and how to edit Wiki pages. Many of the current editors working on the project have increased their skills on Wiki platforms and have attended workshops on transclusion and creating templates in order to increase our ability to effectively update pages and create the template for conferences to use. == Outcome == ''Please report on your original project targets. Please be sure to '''[[Grants:Metrics#Three_shared_metrics|review and provide metrics required for Rapid Grants]]''''' {| class="wikitable" |+ ! !Target outcome !Achived outcome !Explanation |- |1 |Host 2 edit-a-thons with HGAPS team members from both UNC and UMD |Hosted 2 edit-a-thons |Both edit-a-thons were successful. We had HGAPS members come from both UNC and UMD as well as UCLA. We also had individuals attend the edit-a-thons who are not a part of HGAPS. |- |2 |15+ contributors and attendees per edit-a-thon |10 contributors per edit-a-thon |Although we were not able to get 15+ attendees at our edit-a-thons, we were able to get 10 at both events. |- |3 |5 new editors |10 new editors |We had more than 5 new editors and ended with about 10 new editors who have not edited Wiki before. |- |4 |1 new article created, 20+ articles improved |5 new pages created and 25 pages updated |We were successful on this target and created 5 new pages related to keynote speakers and improved many existing articles with around 25 total. |- |5 |increase views within 6 months of launch |Views have increased |We have had an increase in views on FDF related pages since updating the pages and creating new ones. |} == Learning == ''Projects do not always go according to plan. Sharing what you learned can help you and others plan similar projects in the future. Help the movement learn from your experience by answering the following questions:'' *What worked well? #We made page templates on Wiki using transclusion for every page we created or improved. This was very helpful as it guided new editors and created a professional, organized, and visually continuous layout for every page we edited, even though most of our work was done collaboratively. This practice also helped to ensure that necessary and beneficial information was present on every page we edited. #We also made a checklist on Canva, summarizing key aspects to include living biography pages. This checklist provided newer Wiki editors with a framework off of which they could build their pages, and ensured that necessary elements were not left off. The checklist was presented in conjunction with an example living biography page ([[wikipedia:Steven_Pinker|Steven Pinker]]) during the introduction of each Edit-A-Thon and was linked on collaborative editing documents, which served to further demonstrate our expectations to newer editors and ensure that quality pages were created. #Our team met regularly (at least once a week) via video call throughout the entirety of this project. Between meetings, we used a professional instant messaging platform to communicate thoughts, questions, ideas, and reminders both within our team and to other members of our larger organization. Having a clearly defined and hierarchical communication system that was still adaptable enough to meet the needs of our team (working internationally) allowed us to collaborate efficiently, solve problems as they arose, avoid miscommunication, and make the most of our time. #We did the majority of our planning work in wiki sandboxes and separate rough draft documents, instead of working/brainstorming directly on the publicly visible wiki pages themselves. This allowed us to ensure that everything we put on Wiki was accurate, well thought out, and organized. The use of collaborative documents for brainstorming outside of Wiki also served to streamline the teamwork process, especially since a significant number of our editors did not have prior experience editing Wiki. #We created documents with lists of web links for use on specific pages in advance of each Edit-A-Thon. Since these link lists were created by experienced team members, they allowed us to ensure that the information editors used to build pages was accurate and extensive. Additionally, this process allowed editors to use their time most efficiently during the Edit-A-Thon, as they could focus on brainstorming and editing, instead of having to search for and vet links. #Including prizes, gift cards to virtually by lunch for editors, and designated time at the end of the Edit-A-Thon to collaboratively review pages helped not only to gather interest in the events we hosted, but also to incentivize editors to stay focused and productive over the two hour time frame. #Establishing a detailed time frame and plan for each Edit-A-Thon helped us avoid administrative errors during events and allowed us to plan how much we should expect participants to accomplish. This practice also made sure we used our time provided efficiently, and made sure that we were able to fulfill our goals at every Edit-A-Thon. #Providing new editors with Wiki training links in advance of the Edit-A-Thon, as well as with editing workshops during the first few minutes of every event helped us to ensure that new editors learned from this experience and were able to make valuable contributions to their teams. More advanced editing workshops were also provided to individuals with more experience, allowing people of all levels of Wiki editing skills to benefit from this experience and feel encouraged to continue to build their Wiki editing skills in the future. #With the help of the HGAPS Social Media team, we created Google Forms that participants were able to fill out to sign up for Edit-A-Thons. This way, our team was able to know how many editors to expect at each event, as well as to know how many new versus experienced editors planned to show up. These forms were created and advertised during HGAPS meetings and on social media well in advance of each event, in order to maximize our potential audience and gague interest. #Additionally, we used Google Forms to employ a page rating system that allowed teams to both give and receive feedback on their work, further refining editing skills and serving as an additional motivator for focused work. *What did not work so well? # *What would you do differently next time? # == Finances == ===Grant funds spent=== ''Please describe how much grant money you spent for approved expenses, and tell us what you spent it on.'' * '''Wages for template/tool-kit creators''': $2,040 ($20 per hour x 3 creators x 34 hours) ** The remaining $2589.73 was spent here for our creators. * '''Wages for trainers''': $720 ($20 per hour x 3 trainers x 12 hours) ** We spent the anticipated $720 here on trainers. * '''Gift cards to virtually buy lunch for edit-a-thon participants''': $600 ($20 for 15 participants x 2 events) ** We spent $260 on gift cards to buy lunch for edit-a-thon participants. Not everyone that attended wanted one, which led to us being under budget here. We have reallocated these remaining funds ($340) to "wages for template/tool-kit creators". * '''Incentives for sustained editing''': $500 (1st prize $100, 2nd prize $50, and 3rd prize $25 at 1 and 3 month follow-ups after last edit-a-thon = $175 x 4 "contests") ** Upon organizing the edit-a-thons, we decided to change our method for prizes. Since we organized into teams we decided to have participants evaluate each teams work at the end of each edit-a-thon. We used these evaluations to come up with a first place and second place team and these teams were given prizes. We spent $350 in this category and reallocated these remaining funds ($150) to "wages for template/tool-kit creators". * '''HGAPS merch for milestones completed in editor training''': $300 ** We spent $248.27 on HGAPS merch for editors. The remaining funds ($51.73) were reallocated to "wages for template/tool-kit creators". * '''Fiscal sponsor administrative fees (including access to Google Suites for Nonprofits platform and analytics)''': $832 (20% final budget) ** We spent the anticipated $832 for fiscal sponsor administrative fees. '''Total''': USD $4,992 ===Remaining funds=== ''Do you have any remaining grant funds?'' No, we don't have any remaining grant funds left after the completion of the project. <!--Let us know if you would like to use the remaining funds on a similar or new project. Remember, a grants officer must approve this request before you spend the money.--> ==Anything else== ''Anything else you want to share about your project?'' [[Category:Project/Rapid/Report]] o4i51jt7hxvwuf7meibikjzwmkbhtwj 2410804 2410799 2022-08-01T16:43:06Z Parodda 2936296 Added a rough draft of the "What did not work so well?" and "What would you do differently next time?" sections of "Learning" wikitext text/x-wiki == Goals == ''Did you meet your goals? Are you happy with how the project went?''<br>We did meet our goals and are very happy with how the project went. We were able to create a template for conferences to use seen [[Template:Conferences|here]]. And updated all of the previous Future Directions Forum pages seen [[Journal of Clinical Child and Adolescent Psychology Future Directions Forum (JCCAP FDF)|here]]. Additionally, we were able to create Wikipedia pages for some of the previous presenters of the Forum with a focus on women and other minority groups. By starting this, we have decided to try and continue this project moving forward. We have also recruited several new Wiki editors and trained those individuals how to use our template and how to edit Wiki pages. Many of the current editors working on the project have increased their skills on Wiki platforms and have attended workshops on transclusion and creating templates in order to increase our ability to effectively update pages and create the template for conferences to use. == Outcome == ''Please report on your original project targets. Please be sure to '''[[Grants:Metrics#Three_shared_metrics|review and provide metrics required for Rapid Grants]]''''' {| class="wikitable" |+ ! !Target outcome !Achived outcome !Explanation |- |1 |Host 2 edit-a-thons with HGAPS team members from both UNC and UMD |Hosted 2 edit-a-thons |Both edit-a-thons were successful. We had HGAPS members come from both UNC and UMD as well as UCLA. We also had individuals attend the edit-a-thons who are not a part of HGAPS. |- |2 |15+ contributors and attendees per edit-a-thon |10 contributors per edit-a-thon |Although we were not able to get 15+ attendees at our edit-a-thons, we were able to get 10 at both events. |- |3 |5 new editors |10 new editors |We had more than 5 new editors and ended with about 10 new editors who have not edited Wiki before. |- |4 |1 new article created, 20+ articles improved |5 new pages created and 25 pages updated |We were successful on this target and created 5 new pages related to keynote speakers and improved many existing articles with around 25 total. |- |5 |increase views within 6 months of launch |Views have increased |We have had an increase in views on FDF related pages since updating the pages and creating new ones. |} == Learning == ''Projects do not always go according to plan. Sharing what you learned can help you and others plan similar projects in the future. Help the movement learn from your experience by answering the following questions:'' *What worked well? #We made page templates on Wiki using transclusion for every page we created or improved. This was very helpful as it guided new editors and created a professional, organized, and visually continuous layout for every page we edited, even though most of our work was done collaboratively. This practice also helped to ensure that necessary and beneficial information was present on every page we edited. #We also made a checklist on Canva, summarizing key aspects to include living biography pages. This checklist provided newer Wiki editors with a framework off of which they could build their pages, and ensured that necessary elements were not left off. The checklist was presented in conjunction with an example living biography page ([[wikipedia:Steven_Pinker|Steven Pinker]]) during the introduction of each Edit-A-Thon and was linked on collaborative editing documents, which served to further demonstrate our expectations to newer editors and ensure that quality pages were created. #Our team met regularly (at least once a week) via video call throughout the entirety of this project. Between meetings, we used a professional instant messaging platform to communicate thoughts, questions, ideas, and reminders both within our team and to other members of our larger organization. Having a clearly defined and hierarchical communication system that was still adaptable enough to meet the needs of our team (working internationally) allowed us to collaborate efficiently, solve problems as they arose, avoid miscommunication, and make the most of our time. #We did the majority of our planning work in wiki sandboxes and separate rough draft documents, instead of working/brainstorming directly on the publicly visible wiki pages themselves. This allowed us to ensure that everything we put on Wiki was accurate, well thought out, and organized. The use of collaborative documents for brainstorming outside of Wiki also served to streamline the teamwork process, especially since a significant number of our editors did not have prior experience editing Wiki. #We created documents with lists of web links for use on specific pages in advance of each Edit-A-Thon. Since these link lists were created by experienced team members, they allowed us to ensure that the information editors used to build pages was accurate and extensive. Additionally, this process allowed editors to use their time most efficiently during the Edit-A-Thon, as they could focus on brainstorming and editing, instead of having to search for and vet links. #Including prizes, gift cards to virtually by lunch for editors, and designated time at the end of the Edit-A-Thon to collaboratively review pages helped not only to gather interest in the events we hosted, but also to incentivize editors to stay focused and productive over the two hour time frame. #Establishing a detailed time frame and plan for each Edit-A-Thon helped us avoid administrative errors during events and allowed us to plan how much we should expect participants to accomplish. This practice also made sure we used our time provided efficiently, and made sure that we were able to fulfill our goals at every Edit-A-Thon. #Providing new editors with Wiki training links in advance of the Edit-A-Thon, as well as with editing workshops during the first few minutes of every event helped us to ensure that new editors learned from this experience and were able to make valuable contributions to their teams. More advanced editing workshops were also provided to individuals with more experience, allowing people of all levels of Wiki editing skills to benefit from this experience and feel encouraged to continue to build their Wiki editing skills in the future. #With the help of the HGAPS Social Media team, we created Google Forms that participants were able to fill out to sign up for Edit-A-Thons. This way, our team was able to know how many editors to expect at each event, as well as to know how many new versus experienced editors planned to show up. These forms were created and advertised during HGAPS meetings and on social media well in advance of each event, in order to maximize our potential audience and gauge interest. #Additionally, we used Google Forms to employ a page rating system that allowed teams to both give and receive feedback on their work, further refining editing skills and serving as an additional motivator for focused work. *What did not work so well? #While we still were able to recruit a significant number of both new and experienced editors to participate in our Edit-A-Thons, we did not manage to achieve the participant turnout rate we initially hoped for. This led to minor complications during incentive allocation, wherein we had more incentive money to award than editors to award it to. #Our team ended up with more clean-up work at the end of each Edit-A-Thon than we initially anticipated. This additional work was partly due to the lower than target turnout rate at Edit-A-Thons, but was able to be accounted for when our finances were re-examined. #We had a good turnout in terms of RSVP response rate to the sign-up and informational forms we advertised, but we struggled with getting everyone who RSVP'd to attend. This was largely anticipated, but does provide room for improvement if/when we complete similar projects in the future. *What would you do differently next time? #Due to the aforementioned challenges, we have discussed strengthening our advertising effort for Edit-A-Thons in the future. While we did post informational infographics in advance of the Edit-A-Thons on social media, and dedicated time in our HGAPS weekly meetings, we realized that it would likely be beneficial to start the process of advertising the Edit-A-Thons earlier than we did, and post subsequent reminders as the events got closer (as opposed to just having 1 or 2 social media posts). #We also considered that it might be a good idea to include more information about the specific goals of future HGAPS Edit-A-Thons in our social media advertisements. This practice would help garner interest in specific topics, drawing in an increased number of dedicated editors from outside HGAPS UNC. A specific idea we had for implementing this change is to advertise the Edit-A-Thons by way of infographic, as opposed to simply listing the title and time of the event. #As a significant amount of time ended up being spent coordinating bringing in outside trainers, we discussed having experienced Wiki editors prerecorded training sessions. This would allow us to better plan for the time that training would take, and decrease the time and resources spent on recruiting trainers that had to present live talks. If we used prerecorded training sessions we would still recruit experienced editors to be present at the events to answer any questions that may arise. #Overall, our team was able to learn a lot during the organization and implementation of this project, and we hope to use what we have learned to scale up this project in the future. Next time, we strive to further perfect our process in order to recruit more new editors, involve an even greater audience from outside of HGAPS, and create/update more pages in order to continue to disseminate critical psychological research by way of sustained Wiki editing. == Finances == ===Grant funds spent=== ''Please describe how much grant money you spent for approved expenses, and tell us what you spent it on.'' * '''Wages for template/tool-kit creators''': $2,040 ($20 per hour x 3 creators x 34 hours) ** The remaining $2589.73 was spent here for our creators. * '''Wages for trainers''': $720 ($20 per hour x 3 trainers x 12 hours) ** We spent the anticipated $720 here on trainers. * '''Gift cards to virtually buy lunch for edit-a-thon participants''': $600 ($20 for 15 participants x 2 events) ** We spent $260 on gift cards to buy lunch for edit-a-thon participants. Not everyone that attended wanted one, which led to us being under budget here. We have reallocated these remaining funds ($340) to "wages for template/tool-kit creators". * '''Incentives for sustained editing''': $500 (1st prize $100, 2nd prize $50, and 3rd prize $25 at 1 and 3 month follow-ups after last edit-a-thon = $175 x 4 "contests") ** Upon organizing the edit-a-thons, we decided to change our method for prizes. Since we organized into teams we decided to have participants evaluate each teams work at the end of each edit-a-thon. We used these evaluations to come up with a first place and second place team and these teams were given prizes. We spent $350 in this category and reallocated these remaining funds ($150) to "wages for template/tool-kit creators". * '''HGAPS merch for milestones completed in editor training''': $300 ** We spent $248.27 on HGAPS merch for editors. The remaining funds ($51.73) were reallocated to "wages for template/tool-kit creators". * '''Fiscal sponsor administrative fees (including access to Google Suites for Nonprofits platform and analytics)''': $832 (20% final budget) ** We spent the anticipated $832 for fiscal sponsor administrative fees. '''Total''': USD $4,992 ===Remaining funds=== ''Do you have any remaining grant funds?'' No, we don't have any remaining grant funds left after the completion of the project. <!--Let us know if you would like to use the remaining funds on a similar or new project. Remember, a grants officer must approve this request before you spend the money.--> ==Anything else== ''Anything else you want to share about your project?'' [[Category:Project/Rapid/Report]] qslkqjnvco79xt1t43lo6caj0e0xngl 2 285603 2410899 2408674 2022-08-02T04:49:17Z Jtwsaddress42 234843 wikitext text/x-wiki {| align= center | style="font-size:85%" |{{Gallery | title = ''Lars Onsager'' | width = 100 | height = 100 | align= center | File:Onsager-medal adverse.jpg | alt1= Onsager Medal | Onsager Medal | File:Onsager-medal-reverse.jpg | alt2=Onsager Medal - back | Onsager Medal - back | File:Egyszerűsített Onsager relációs táblázat.jpg | alt3= Egyszerűsített Onsager relációs táblázat | Egyszerűsített Onsager relációs táblázat | File:Onsager reziprozitaet.png | alt4= Onsager reziprozitaet | Onsager reziprozitaet }} |} bs7h3rhxnb13eu72qvfia8t4n4b07pl 0 285800 2410954 2410370 2022-08-02T11:50:31Z Dave Braunschweig 426084 Dave Braunschweig moved page [[Topic:Physical security]] to [[Physical security]]: Moving to main space wikitext text/x-wiki = Physical Security = === Introduction === Physical Security is the implementation of security measures for the protection of assets, information, or people. It aims to prevent as much unauthorized or malicious access as possible, while still allowing those authorized or permitted access to what they need. This topic aims to outline and define aspects of physical security, alongside providing necessary resources for further research. Terminologies are defined in as standard a way as possible considering the multi-resource nature of the topic, but have a preference to the terminology of other Wikiversity topic or lecture pages. Physical Security measures rely on a combination of physical and psychological methodologies for preventing unwanted access to assets. <h1>Physical Security Terminology</h1> === General Terminologies === * '''Security System''' - a linked or related set of physical security implementations. May apply to: an individual object, such as a lock; A complex object or set of objects, such as a car, computer, or engine; Or some other large scale system, such as a facility and its grounds, a series of complex objects, or a complex of buildings or rooms. * '''CIA Triad''' - A concept from information security that applies well to physical security. The CIA triad is a triangular representation of compromises between the Confidentiality, Integrity, and Availability of an asset (in the typical case, information). When considering Access Controls, particularly to things such as devices for Security Testing, Surveillance, or any other component of the security system, the CIA Triad is of particular relevance. The Triad is not an absolute representation, since some systems can contain high levels of multiple categories, but an important conceptual representation. === The Five Components of Physical Security === The core ideas of physical security can be organized into five components: The three key components - Access Control, Surveillance, and Security testing - alongside Deterrence and Detection. * '''Access Control''' - Access Control is the concept of managing and restricting the accessibility and use of certain assets and areas, either to a whitelist of individuals, or from a blacklist of individuals. Those who need a resource, such as a ledger, account number, or stored paperwork, should have access to it, and those who don't should not. * '''Surveillance''' - Surveillance is the concept of watching, monitoring, observing, or recording an area and/or asset, alongside the entities that access it. For an area or asset to be considered surveilled in some form, authorized individuals should be capable of retrieving information about recent accesses and/or interactions with the surveilled entity. * '''Security Testing''' - Security Testing refers to the act and practice of testing the implementation of physical security methods, the practices of agents involved in the enforcement of physical security methods, and the grading or assessment of the capability of a system of security implementations. Security Testing is an active, repeated practice, and should often result in changes, reimplementation, and enhancements alongside evolving information and a changing threat landscape. * '''Deterrence''' - Deterrence as a concept refers to methods to prevent an intrusion attempt in the first place, and is typically the first layer of defense against intruders. Deterrence can take a variety of forms, and may not be a physically implemented system so much as an implied, warned or suggested concept or response. * '''Detection''' - Detection is the concept of identifying a physical intrusion, and possibly the intruders involved. For a method to classify as detection it must be capable of, often 'autonomously' (agents such as human guards, dogs, or Computer-Vision systems classify here), indicating a breach or attempt against a security implementation, site, or asset. None of the above methods are, nor are expected to be, full-proof, which applies further to the definitions. Methods to evade Detection, Access-Control, Surveillance, and Deterrence do exist and will likely be discussed further under Security Testing. The above components experience overlap/ Access Control may serve as Deterrence, Surveillance may perform a Detection role, Detection may be required for Access Control, and so on; These components are conceptual, and not hard limitations. Beyond the components of security, a security system must be applied to their own areas of a physical system or environment, defined below. === The Three Levels of Physical Security === * '''Outer Perimeter Security''' - Refers to the marginal outer layer of a security system such as: The physical property boundary lines of a site; The entrance buildings or hubs to a large site such as Disney World or other multi-building site; The room or area around a device such as a server box, car, or cash-register. * '''Inner Perimeter Security''' - Refers to the direct outer layer of a security system such as: The walls, doors, windows or entry points of a building; The accessible areas of a device such as a laptop or car in standard use (not reverse engineered, disassembled, etc;). * '''Interior Security''' - Refers to the internal operations layer of a security system such as: inner spaces, cubbies, and cabinets of an office, building, or room; Internal mechanics of a device such as a lock, engine, or computer; Not every system of security will be capable of realistically or permissively implementing protections on every level listed above, however, every level should still be held in consideration as a vector of intrusion, or consideration for secondary protections. Additionally, some texts and references may include grades of security (often 5 grades) for physical security implementation. These grades are omitted simply due to the variance in definition across sources, combined with their lack of necessity for the understanding of the content. <h1>Physical Security as a Practice</h1> Various concepts of Physical Security will be broken down, linked to, or explained below. === Planning a System of Security === Before creating a system of security, one must create a plan or set of plans, with an underlying goal. This is particularly the case in regards to large organizations or security teams, which must standardize and communicate ideas clearly and quickly for an efficient and correct implementation. * '''[[Physical Threat Landscaping]]''' - The first aspect of planning a system of security, identifying threats and considerations that a physical security system must defend against. <h1>Research Links</h1> * Example of the results of poor physical security: https://www.youtube.com/watch?v=R5RE0mVbJ3s * Reference building codes in your area or government for further information. Often a security section concerning exact rules and techniques for preventing intrusion is contained within, ie: https://www.kcmo.gov/home/showpublisheddocument/6420/637509790534730000 oy18pnlvlp0w8t9p1sg819ui7ta7nvo 2 285818 2410901 2410509 2022-08-02T05:09:22Z Jtwsaddress42 234843 /* Olah, George A. (1927 - 2017) */ wikitext text/x-wiki {{RoundBoxTop|theme=3}} === [[w:George Olah|Olah, George A. (1927 - 2017)]] === <hr /> [[File:Oláh György előadása 8299.jpg|thumb|George Olah (1927 - 2017)]] [[File:Onium-ioonid lahustes.png|thumb|Onium-ioonid lahustes]] [[File:Nitrogen pentahydride possible structures.svg|thumb|Nitrogen pentahydride possible structures]] [[File:Metüülliitium.aspx.gif|thumb|Metüülliitium.aspx]] '''Notable Accomplishments''' * [https://www.nobelprize.org/prizes/chemistry/1994/olah/facts/ The Nobel Prize in Chemistry 1994] - “for his contribution to carbocation chemistry.” * Superacids & Carbocations <br /><hr /> {| align= center | witdth= 640px |'''Carbo-Cation Stabilities''' [[File:CarboCationStabilities.svg|640px|CarboCationStabilities]] <br> |}<hr /> '''Publications''' {{User:Jtwsaddress42/Bibliography/Olah, George A.}} {{RoundBoxBottom}} <hr /> c651r0ltpsi5pyu5y5jtcevv42rpqo0 2410902 2410901 2022-08-02T05:14:12Z Jtwsaddress42 234843 /* Olah, George A. (1927 - 2017) */ wikitext text/x-wiki {{RoundBoxTop|theme=3}} === [[w:George Olah|Olah, George A. (1927 - 2017)]] === <hr /> [[File:Oláh György előadása 8299.jpg|thumb|George Olah (1927 - 2017)]] [[File:Onium-ioonid lahustes.png|thumb|Onium-ioonid lahustes]] [[File:Nitrogen pentahydride possible structures.svg|thumb|Nitrogen pentahydride possible structures]] [[File:Metüülliitium.aspx.gif|thumb|Metüülliitium.aspx]] '''Notable Accomplishments''' * [https://www.nobelprize.org/prizes/chemistry/1994/olah/facts/ The Nobel Prize in Chemistry 1994] - “for his contribution to carbocation chemistry.” * Superacids & Carbocations <br /><hr /> {| align= center | witdth= 640px |'''Carbo-Cation Stabilities''' [[File:CarboCationStabilities.svg|640px|CarboCationStabilities]] <br> |}<hr /> '''Publications''' {{User:Jtwsaddress42/Bibliography/Olah, George A.}} <br /><hr /> {| align= center | witdth= 640px |'''Classical and Non-classical Interpretations of Bonding in the 2-Norbornyl Cation''' [[File:Classical and Non-classical Interpretations of Bonding in the 2-Norbornyl Cation.jpg|Classical and Non-classical Interpretations of Bonding in the 2-Norbornyl Cation]] <br> |}{{RoundBoxBottom}} <hr /> 7nyg2jir40gieb6xkondv00yeodutd9 2 285830 2410865 2410557 2022-08-02T02:18:03Z U3183521 2947387 /* Book chapter I am working on */ wikitext text/x-wiki == About me == Psychology undergraduate == Book chapter I am working on == [[Motivation and emotion/Book/2022/Kama muta|Kama muta]] for [[w:Emotion|emotion]] t2xbsqxi77rzrdgfikdekevn3eiqsdg 0 285832 2410958 2410583 2022-08-02T11:52:46Z Dave Braunschweig 426084 Advise wikitext text/x-wiki {{Advise}} b5s86rnq2qoshcmza44axrg6scq1qoz 2 285834 2410879 2410600 2022-08-02T03:13:24Z Jtwsaddress42 234843 /* Axelrod, Julius (1912 - 2004) */ wikitext text/x-wiki {{RoundBoxTop|theme=3}} === [[w:Julius Axelrod|Axelrod, Julius (1912 - 2004)]] === <hr/> [[File:Axelrod01.jpg|thumb|Julius Axelrod (1912 - 2004)]] [[File:Julius Axelrod 1970.jpg|thumb|Julius Axelrod 1970]] [[File:Julius Axelrod.jpg|thumb|]] '''Notable Accomplishments''' * [https://www.nobelprize.org/prizes/medicine/1970/axelrod/facts/ The Nobel Prize in Physiology or Medicine 1970] -shared with Sir Bernard Katz and Ulf von Euler “for their discoveries concerning the humoral transmittors in the nerve terminals and the mechanism for their storage, release and inactivation.” * Catecholamine Neurotransmitters & their Metabolism <br /><hr /> {| align= center | width= 640px |'''Biosynthesis of the Catecholamine Neurotransmitters''' [[File:Biosynthese catecholaminen.png|640px|Biosynthese catecholaminen]] <br /> |}<hr /> '''Publications''' {{User:Jtwsaddress42/Bibliography/Axelrod, Julius}} <br /><hr /> {| align= center | witdh= 640px |'''Julius Axelrod at the blackboard working on the structure of catecholamines''' [[File:Axelrod.jpg|640px|Julius Axelrod working at the blackboard on the structure of catecholamines]] <br /> |} {{RoundBoxBottom}} <hr /> 13vcu1zodifuw04qf3g87u8ug7t3v9q 6 285856 2410784 2022-08-01T13:27:49Z Young1lim 21186 {{Information |Description=Laurent.5: Applications 6A (20220801 - 20220730) |Source={{own|Young1lim}} |Date=2022-08-01 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Applications 6A (20220801 - 20220730) |Source={{own|Young1lim}} |Date=2022-08-01 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 2hob8iphvusdgu8nakjkjczfnp8t0b9 6 285857 2410785 2022-08-01T13:28:37Z Young1lim 21186 {{Information |Description=C04.Series.1: Arrays 1A (20220801 - 20220730) |Source={{own|Young1lim}} |Date=2022-00-01 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=C04.Series.1: Arrays 1A (20220801 - 20220730) |Source={{own|Young1lim}} |Date=2022-00-01 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} afde6h5gb3do127er07x7o6711fexcq 2410789 2410785 2022-08-01T13:32:26Z Young1lim 21186 /* Summary */ wikitext text/x-wiki == Summary == {{Information |Description=C04.Series.1: Arrays 1A (20220801 - 20220730) |Source={{own|Young1lim}} |Date=2022-08-01 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} muyssceukcfxb9p5aihelb7tgnps0zu 6 285858 2410786 2022-08-01T13:29:31Z Young1lim 21186 {{Information |Description=C04.Series.3: Array Pointers 1A (20220801 - 20220730) |Source={{own|Young1lim}} |Date=2022-08-01 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=C04.Series.3: Array Pointers 1A (20220801 - 20220730) |Source={{own|Young1lim}} |Date=2022-08-01 |Author=Young W. 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Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=VLSI.Arith.1.A: Variable Block Adders (20220801 - 20220730) |Source={{own|Young1lim}} |Date=2022-08-01 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 3luowl02yriev0h8f1lib5bkrwrap8s 2 285861 2410791 2022-08-01T15:18:36Z Isivile Nkungu Khenkebe 2947451 The Actor wikitext text/x-wiki Isivile Nkungu Khenkebe pizkuu7blncj1b4nzidb6sxfbfrf566 6 285862 2410801 2022-08-01T16:27:28Z Young1lim 21186 {{Information |Description=Condition (20220801 - 20220730) |Source={{own|Young1lim}} |Date=2022-08-01 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} wikitext text/x-wiki == Summary == {{Information |Description=Condition (20220801 - 20220730) |Source={{own|Young1lim}} |Date=2022-08-01 |Author=Young W. 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I majored in Psychology (B.A.). t2357jfmtbppgv2v5ducx8hari4irqk 2410819 2410818 2022-08-01T20:15:22Z Ardenguo 2944162 wikitext text/x-wiki Hi my name is Arden and I graduated in 2022 from the [[wikipedia:University_of_California,_Los_Angeles|University of California, Los Angeles]]. I majored in Psychology (B.A.). s90ovx0m4by42heokk2l10ekjjgzs33 2 285865 2410854 2022-08-02T00:08:40Z U3083720 2947468 added my user info wikitext text/x-wiki Hi my name is Andrew, I am a third year undergraduate student studying psychology at the University of Canberra. 79dar0vn864899g7aykwz6oelbd7ici 6 285866 2410862 2022-08-02T01:59:34Z Young1lim 21186 {{Information |Description=Condition (20220802 - 20220801) |Source={{own|Young1lim}} |Date=2022-08-02 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} wikitext text/x-wiki == Summary == {{Information |Description=Condition (20220802 - 20220801) |Source={{own|Young1lim}} |Date=2022-08-02 |Author=Young W. 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You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:57, 2 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} kkiq85rha12t08lc3u7dtjx5gfou5n8 3 285872 2410919 2022-08-02T05:58:09Z Jtneill 10242 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Lana E. W. Jones!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Jtneill|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:58, 2 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} pygf1w8nvf8qxct3hk5j9hweuh4bgil 3 285873 2410921 2022-08-02T05:58:55Z Jtneill 10242 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] KingMob221!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Jtneill|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:58, 2 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} py0adveencvj12ue8caz42zwfd6dkyz 3 285874 2410922 2022-08-02T05:59:13Z Jtneill 10242 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Sebastian Armstrong!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Jtneill|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:59, 2 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} i3acdjo767m911e9atie4jaab9ms9ps 3 285875 2410923 2022-08-02T05:59:24Z Jtneill 10242 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3218292!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Jtneill|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:59, 2 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} 0uuwza4d6pvj2ji9km9lodt7vlp4uko 3 285876 2410924 2022-08-02T05:59:47Z Jtneill 10242 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Brewerjr!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Jtneill|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:59, 2 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} ooxrwwi5icr6t0vt0yngz1boorwhmol 3 285877 2410925 2022-08-02T05:59:59Z Jtneill 10242 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3141987!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Jtneill|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:59, 2 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} davk8rrfh46479obuhx9ke8sgxjpzmu 3 285878 2410953 2022-08-02T11:49:26Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Abigail Richie!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 11:49, 2 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} ofkl963wfpzivzh05pjsbrd1tr7u5j3 104 285879 2410955 2022-08-02T11:50:31Z Dave Braunschweig 426084 Dave Braunschweig moved page [[Topic:Physical security]] to [[Physical security]]: Moving to main space wikitext text/x-wiki #REDIRECT [[Physical security]] n2mdfcsqespdq8ntuucptgsqry9lsl6 3 285880 2410957 2022-08-02T11:51:29Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] IncarnationOfTarnation!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 11:51, 2 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} 9kxo609qh7cmi6fkxhheqh1myx31vuj 3 285881 2410959 2022-08-02T11:53:07Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Rudrasiasbpl!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 11:53, 2 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} 32k5t798kw4vj6xd2bkbjifznu1ljtl 2410960 2410959 2022-08-02T11:54:07Z Dave Braunschweig 426084 /* Wikiversity and Wikipedia */ new section wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Rudrasiasbpl!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 11:53, 2 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} == Wikiversity and Wikipedia == Wikiversity welcomes most types of [[Portal:Learning Materials|learning materials]] that are not [[w:Wikipedia:Contributing to Wikipedia|encyclopedia]] articles, [[b:Help:How to start a book|textbooks]], or [[s:Wikisource:What is Wikisource?|source texts]]. Please note that [[Wikiversity:What Wikiversity is not|Wikiversity is not]] a place for duplication of other Wikimedia projects. [[Wikiversity:Colloquium|Let us know]] if you have any questions. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 11:54, 2 August 2022 (UTC) rcla3m5srwoy2e2buxonldmtp23mjv6 3 285882 2410962 2022-08-02T11:56:42Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Isivile Nkungu Khenkebe!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 11:56, 2 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} b4gmlfidb9j0js4uja4ic2yoe150kqc 3 285883 2410963 2022-08-02T11:58:02Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3191574 (PHP)!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 11:58, 2 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} 0ceu1ms4c4ti4kuq49mcheteal2kaok