carmichael_factorization_method_generalized.sf

#!/usr/bin/ruby

# Author: Daniel "Trizen" Șuteu
# Date: 04 May 2019
# https://github.com/trizen

# A simple factorization method, using the binary search algorithm, for numbers of the form:
#
#   n = x * Prod_{k=1..r} ((x±1)*a_k ± 1)
#
# for `r` relatively small.

# Many Carmichael numbers and Lucas pseudoprimes are of this form and can be factorized relatively fast by this method.

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

func carmichael_factorization(n, k=3, l=2, h=6) {

    var blocks = [
        {|x,params|
            params.map {|k|
                ((x-1)*k + 1)
            }
        },

        {|x,params|
            params.map {|k|
                ((x+1)*k - 1)
            }
        },
    ]

    @(l..h).combinations(k-1, {|*params|

            for block in blocks {

                var r = bsearch_le(n.iroot(k), {|x| block(x, params).prod*x <=> n })
                var g = gcd(r, n)

                if (g > 1) {
                    var f = [r, block(r, params)...].grep { .divides(n) }
                    return (f ? f : g)
                }
            }
    })

    return 1
}

say carmichael_factorization(7520940423059310542039581, k:3)                                         #=> 79443853
say carmichael_factorization(570115866940668362539466801338334994649, k:3)                           #=> 4563211789627
say carmichael_factorization(8325544586081174440728309072452661246289, k:3)                          #=> 11153738721817

say carmichael_factorization(1169586052690021349455126348204184925097724507, k:3, l:11, h:23)                   #=> 166585508879747
say carmichael_factorization(61881629277526932459093227009982733523969186747, k:3, l:3, h:11)                   #=> 1233150073853267
say carmichael_factorization(173315617708997561998574166143524347111328490824959334367069087, k:3, l:3, h:11)   #=> 173823271649325368927

# Works even with larger numbers
say carmichael_factorization(131754870930495356465893439278330079857810087607720627102926770417203664110488210785830750894645370240615968198960237761, k:4)      #=> 245960883729518060519840003581