lucas_flt_factorization_method.sf

#!/usr/bin/ruby

# Author: Daniel "Trizen" Șuteu
# Date: 24 February 2021
# https://github.com/trizen

# A new factorization method, inspired by Fermat's Little Theorem (FLT), implemented
# using Lucas sequences, for numbers that have prime factors close to each other.

# The idea is to try to find a non-trivial factor of `n` by checking:
#     gcd(V_n(P, Q) - V_k(P, Q), n)
# for several small k >= 1, where V_n(P,Q) is the Lucas V sequence.

# See also:
#   https://en.wikipedia.org/wiki/Lucas_sequence

func lucas_FLT_factor(n, P = 3, Q = -1, tries = 1e4) {

    var (a, b, z) = (2, P, lucasVmod(P, Q, n, n))
    #var (a, b, z) = (0, 1, lucasUmod(P, Q, n, n))

    if (z == P) {    # can't factor Lucas pseudoprimes
        return 1
    }

    tries.times {

        var g = gcd(z - b, n)

        if (g.is_between(2, n-1)) {
            return g
        }

        (a, b) = (b, (P*b - Q*a) % n)
        (a, b) = (b, (P*b - Q*a) % n)
    }

    return 1
}

var p = 1e20.random_prime
var n = (p * p.next_prime * p.next_prime.next_prime)

say "Factoring: #{n}"
say ("Factor found: ", lucas_FLT_factor(n))

say ''

say lucas_FLT_factor(1169586052690021349455126348204184925097724507);                              #=> 166585508879747
say lucas_FLT_factor(61881629277526932459093227009982733523969186747);                             #=> 1233150073853267
say lucas_FLT_factor(57981220983721718930050466285761618141354457135475808219583649146881, 8, -7); #=> 213745738248483841
say lucas_FLT_factor(random_prime(1e30) * (2**128 + 1), 3, -4);                                    #=> 340282366920938463463374607431768211457

say ''

say lucas_FLT_factor(335603208601);
say lucas_FLT_factor(30459888232201);
say lucas_FLT_factor(162021627721801);
say lucas_FLT_factor(1372144392322327801);
say lucas_FLT_factor(7520940423059310542039581);
say lucas_FLT_factor(8325544586081174440728309072452661246289);
say lucas_FLT_factor(181490268975016506576033519670430436718066889008242598463521);
say lucas_FLT_factor(57981220983721718930050466285761618141354457135475808219583649146881, 8, -7);
say lucas_FLT_factor(131754870930495356465893439278330079857810087607720627102926770417203664110488210785830750894645370240615968198960237761, 8, -7);

say ''

say lucas_FLT_factor(16641689036184776955112478816668559);                                      #=> 140501852609
say lucas_FLT_factor(17350074279723825442829581112345759);                                      #=> 142467792809
say lucas_FLT_factor(61881629277526932459093227009982733523969186747);                          #=> 1233150073853267
say lucas_FLT_factor(173315617708997561998574166143524347111328490824959334367069087);          #=> 173823271649325368927
say lucas_FLT_factor(2425361208749736840354501506901183117777758034612345610725789878400467);   #=> 19760381716819645293083