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### 2 The General Factorization Routine

#### 2.1 The method for Factors

The FactInt package provides a better method for the operation Factors for integer arguments, which supersedes the one included in the GAP Library:

##### 2.1-1 Factors
 ‣ Factors( n ) ( method )

Returns: a sorted list of the prime factors of n.

The returned factors pass the built-in probabilistic primality test of GAP (IsProbablyPrimeInt, Baillie-PSW Primality Test; see the GAP Reference Manual). If the method fails to compute the prime factorization of n, an error is signalled. The same holds for all other factorization routines provided by this package. It follows a rough description how the factorization method works:

First of all, the method checks whether n = b^k ± 1 for some b, k and looks for factors corresponding to polynomial factors of x^k ± 1. Provided that b and k are not too large, the factors that do not correspond to polynomial factors are taken from Richard P. Brent's Factor Tables [Bre04]. 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 various common 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 (3.2-1)), Williams' p+1 (see FactorsPplus1 (3.3-1)) and the Elliptic Curves Method (ECM, see FactorsECM (3.4-1)) in this order.

If there is still an unfactored part remaining after that, it is factored using the Multiple Polynomial Quadratic Sieve (MPQS, see FactorsMPQS (3.6-1)).

The following options are interpreted:

TDHints

A list of additional trial divisors. This is useful only if certain primes p are expected to divide n with probability significantly larger than frac1p.

RhoSteps

The number of steps for Pollard's Rho.

RhoCluster

The number of steps between two gcd computations in Pollard's Rho.

Pminus1Limit1 / Pminus1Limit2

The first- / second stage limit for Pollard's p-1 (see FactorsPminus1 (3.2-1)).

Pplus1Residues

The number of residues to be tried by Williams' p+1 (see FactorsPplus1 (3.3-1)).

Pplus1Limit1 / Pplus1Limit2

The first- / second stage limit for Williams' p+1 (see FactorsPplus1 (3.3-1)).

ECMCurves

The number of elliptic curves to be tried by the Elliptic Curves Method (ECM) (see FactorsECM (3.4-1)). Also admissible: a function that takes the number n to be factored as an argument and returns the desired number of curves to be tried.

ECMLimit1 / ECMLimit2

The initial first- / second stage limit for ECM (see FactorsECM (3.4-1)).

ECMDelta

The increment per curve for the first stage limit in ECM. The second stage limit is adjusted appropriately (see FactorsECM (3.4-1)).

ECMDeterministic

If true, ECM chooses its curves deterministically, i.e. repeatable (see FactorsECM (3.4-1)).

FBMethod

Specifies which of the factor base methods should be used to do the "hard work". Currently implemented: "CFRAC" and "MPQS" (see FactorsCFRAC (3.5-1) and FactorsMPQS (3.6-1), respectively). Default: "MPQS".

For the use of the GAP Options Stack, see Chapter 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,
79934306053602222928609369601238840619880168466272137576868879760059\
3002563860297371289151859287894468775962208410650878341385577817736702\
2158878920741413700868182301410439178049533828082651513160945607018874\
830040978453228378816647358334681553 ]



The above method for Factors calls the following function, which is the actual "working horse" of this package:

##### 2.1-2 FactInt
 ‣ FactInt( n ) ( function )

Returns: a list of two lists, where the first list contains the determined prime factors of n and the second list contains the remaining unfactored parts of n, if there are any.

This function 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 or CFRACLimit, respectively.

Factors(n) is equivalent to FactInt(n:cheap:=false, FactIntPartial:=false).


gap> FactInt( Factorial(300) + 1 : cheap );
[ [ 461, 259856122109, 995121825812791, 3909669044842609,
4220826953750952739, 14841043839896940772689086214475144339 ],
[ 104831288231765723173983836560438594053336296629073932563520618687\
9287645058010688827246061541065631119345674081834085960064144597037243\
9235869682208979384309498719255615067943353399357029226058930732298505\
5816977495398426741656633461747046623641451042655247093315505417820370\
9451745871701742000546384614472756584182478531880962594857275869690727\
9733563594352516014206081210368516157890709802912711149521530885498556\
1244667790208245620301404499928532222524585946881528337257061789593197\
99211283640357942345263781351 ] ]



#### 2.2 Getting information about the factoring process

Optionally, the FactInt package prints information on the progress of the factorization process:

##### 2.2-1 InfoFactInt
 ‣ InfoFactInt ( info class )
 ‣ FactIntInfo( level ) ( function )

This Info class allows to monitor what happens during the factoring process.

If InfoLevel(InfoFactInt) = 1, then basic information about the factoring techniques used is displayed. If this InfoLevel has value 2, then additionally all "relevant" steps in the factoring algorithms are mentioned. If it is set equal to 3, then large amounts of details of the progress of the factoring process are shown.

Enter FactIntInfo(level) to set the InfoLevel of InfoFactInt to the positive integer level. The call FactIntInfo(level); is equivalent to SetInfoLevel(InfoFactInt,level);.

The informational output is usually not literally the same in each factorization attempt to a given integer with given parameters. For a description of the Info mechanism, see Section Info Functions in the GAP Reference Manual.

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