near-power_factorization_method.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 03 June 2019
# https://github.com/trizen

# A simple factorization method for numbers close to a perfect power.

# Very effective for numbers of the form:
#
#   n^k - 1
#
# where k has many divisors.

func near_power_factorization(n, bound=100) {

    var orig = n

    func f(r, e, k) {
        var factors = gather {
            e.divisors.each {|d|

                for j in (1, -1) {

                    var t = (r**d - k*j)
                    var g = gcd(t, n)

                    if (g.is_between(2, n-1)) {
                        while (g.divides(n)) {
                            n /= g
                            take(g)
                        }
                    }
                }
            }
        }

        factors << orig/factors.prod
        factors.sort
    }

    for j in (1..bound) {
        for k in (1, -1) {

            var u = (k * j**2)

            if (is_power(n + u)) {

                var r = perfect_root(n + u)
                var e = perfect_power(n + u)

                return f(r, e, j)
            }
        }
    }

    return [n]
}

if (ARGV) {
    say near_power_factorization(Num(ARGV[0]))
    return 1
}

say near_power_factorization(2**256 - 1)        #=> [3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457]
say near_power_factorization(10**120 + 1)       #=> [100000001, 9999999900000001, 99999999000000009999999900000001, 10000000099999999999999989999999899999999000000000000000100000001]
say near_power_factorization(10**120 - 1)       #=> [3, 9, 11, 37, 91, 101, 9091, 9901, 10001, 11111, 90090991, 99009901, 99990001, 109889011, 9999000099990001, 10099989899000101, 100009999999899989999000000010001]
say near_power_factorization(10**120 - 25)      #=> [3, 5, 5, 29, 2298850574712643678160919540229885057471264367816091954023, 199999999999999999999999999999999999999999999999999999999999]