`Factors`

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 ) | ( 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 \(n = b^k \pm 1\) for some \(b\), \(k\) and looks for factors corresponding to polynomial factors of \(x^k \pm 1\). 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 \(\frac{1}{p}\).

`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:

`‣ 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(`

is equivalent to `n`)`FactInt(`

.`n`:`cheap`:=false, `FactIntPartial`:=false)[1]

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

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

`‣ 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(`

to set the `level`)`InfoLevel`

of `InfoFactInt`

to the positive integer `level`. The call `FactIntInfo(`

is equivalent to `level`);`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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