sophie_germain_factorization_method.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 26 July 2019
# https://github.com/trizen

# A simple factorization method, based on Sophie Germain's identity:
#   x^4 + 4y^4 = (x^2 + 2xy + 2y^2) * (x^2 - 2xy + 2y^2)

# This method is also effective for numbers of the form: n^4 + 4^(2k+1).

# See also:
#   https://oeis.org/A227855 -- Numbers of the form x^4 + 4*y^4.
#   https://www.quora.com/What-is-Sophie-Germains-Identity

func sophie_germain_factorization (n) {

    func sophie(A, B) {
        var factors = []

        for f in (
            A**2 + 2*B*B - 2*A*B,
            A**2 + 2*B*B + 2*A*B,
        ) {
            var g = gcd(f, n)

            if (g.is_between(2, n-1)) {
                while (n%g == 0) {
                    n /= g
                    factors << g
                }
            }
        }

        factors
    }

    var orig = n
    var sophie_params = []

    func sophie_check_root(r1) {

        var x  = (4 * r1**4)
        var dx = (n - x)

        if (dx.is_power(4)) {
            var r2 = dx.iroot(4)
            sophie_params << [r2, r1]
        }

        var y  = (r1**4)
        var dy = (n - y)

        if ((dy%4 == 0) && (dy/4).is_power(4)) {
            var r2 = (dy/4).iroot(4)
            sophie_params << [r1, r2]
        }
    }

    # Try to find n = x^4 + 4*y^4, for x or y small.
    for r in (2..n.ilog2) {
        sophie_check_root(r)
    }

    # Try to find n = x^4 + 4*y^4 for x,y close to floor(n/5)^(1/4).
    var k = floor(n/5).iroot(4)

    for j in (0..100) {
        sophie_check_root(k + j)
    }

    var factors = []

    for args in (sophie_params) {
        factors << sophie(args...)...
    }

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

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

say sophie_germain_factorization(ipow(43,        4) + 4*ipow(372485613, 4))     #=> [277491031750210669, 277491095817736105]
say sophie_germain_factorization(ipow(372485613, 4) + 4*ipow(97,        4))     #=> [138745459629795665, 138745604154213509]
say sophie_germain_factorization(ipow(372485613, 4) + 4*ipow(372485629, 4))     #=> [138745543811525897, 693727695218548205]
say sophie_germain_factorization(ipow(372485629, 4) + 4*ipow(372485511, 4))     #=> [138745455904945045, 693727455337830721]

say '';

say sophie_germain_factorization(ipow(4, 117) + ipow(53,  4))       #=> [166153499473114453560556010453601017, 166153499473114514665395754616490745]
say sophie_germain_factorization(ipow(4, 213) + ipow(240, 4))       #=> [13164036458569648337239753460419861813422875717854660184319779072, 13164036458569648337239753460497746266300898132282617629258080512]
say sophie_germain_factorization(ipow(4, 251) + ipow(251, 4))       #=> [3618502788666131106986593281521497099061968496512379043906292883903830095385, 3618502788666131106986593281521497141767405545090156208559806116590740633113]