The FactInt package provides a better method for the operation
Factors
for integer arguments, which supersedes the one included in
the GAP library:
Factors(
n ) M
This method returns a sorted list of the prime factors of n.
If it fails to compute the prime factorization of n, an error is signalled.
The returned factors pass the built-in probabilistic primality test of
GAP (IsProbablyPrimeInt
, see the GAP Reference Manual).
The same holds for all other factorization routines provided by this
package.
It follows a rough description how FactInt's method for Factors
works:
First of all it checks whether n = bk pm1 for some b, k and looks for factors corresponding to polynomial factors of xk pm1. Provided that b and k are sufficiently small, factors that do not correspond to polynomial factors are taken from Richard P. Brent's Factor Tables Brent04, which are available at
http://web.comlab.ox.ac.uk/oucl/work/richard.brent/factors.html.
The code for accessing these tables has been contributed by Frank Lübeck.
Then the method uses trial division and a number of cheap methods for special cases.
After the small and other ``easy'' factors have been found this way,
FactInt's method searches for ``medium-sized'' factors using
Pollard's Rho (by the library function FactorsRho
, see the GAP
Reference Manual), Pollard's p-1 (see FactorsPminus1),
Williams' p+1 (see FactorsPplus1) and the Elliptic Curves Method
(ECM, see FactorsECM) in this order.
If there is still an unfactored part remaining after that, it is factored using the MPQS (see FactorsMPQS).
The following options are interpreted:
"CFRAC"
and "MPQS"
(see FactorsCFRAC, FactorsMPQS). Default: "MPQS"
.
For the use of the GAP Options Stack, see Section Options Stack in the GAP Reference Manual.
Setting RhoSteps, Pminus1Limit1, Pplus1Residues, Pplus1Limit1, ECMCurves or ECMLimit1 equal to zero switches the respective method off. The method chooses defaults for all option values that are not explicitly set by the user. The option values are also interpreted by the routines for the particular factorization methods described in the next chapter.
gap> Factors( Factorial(44) + 1 ); [ 694763, 9245226412016162109253, 413852053257739876455072359 ] gap> Factors( 2^997 - 1 ); [ 167560816514084819488737767976263150405095191554732902607, 7993430605360222292860936960123884061988016846627213757686887976005930025638\ 602973712891518592878944687759622084106508783413855778177367022158878920741413\ 700868182301410439178049533828082651513160945607018874830040978453228378816647\ 358334681553 ]
The ``working horse'' of this package is
FactInt(
n ) F
This function computes the prime factorization of the integer n. The result is returned as a list of two lists. The first list contains the prime factors found, and the second list contains remaining unfactored parts of n, if there are any.
FactInt
interprets all options which are interpreted by the method
for Factors
described above. In addition, it interprets the options
cheap and FactIntPartial. If the option cheap is set, only usually
cheap factorization attempts are made. If the option FactIntPartial
is set, the factorization process is stopped before invoking the
(usually time-consuming) MPQS or CFRAC, if the number of digits of the
remaining unfactored part exceeds the bound passed as option value
MPQSLimit resp. CFRACLimit.
Factors( n )
is equivalent to
FactInt( n : cheap := false, FactIntPartial := false )[ 1 ]
.
gap> FactInt( Factorial(300) + 1 : cheap ); [ [ 461, 259856122109, 995121825812791, 3909669044842609, 4220826953750952739, 14841043839896940772689086214475144339 ], [ 10483128823176572317398383656043859405333629662907393256352061868792876450\ 580106888272460615410656311193456740818340859600641445970372439235869682208979\ 384309498719255615067943353399357029226058930732298505581697749539842674165663\ 346174704662364145104265524709331550541782037094517458717017420005463846144727\ 565841824785318809625948572758696907279733563594352516014206081210368516157890\ 709802912711149521530885498556124466779020824562030140449992853222252458594688\ 152833725706178959319799211283640357942345263781351 ] ]
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