pell-holf_factorization.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 18 February 2019
# https://github.com/trizen

# A simple integer factorization method, using square root convergents, combined with the HOLF method.

# See also:
#   https://en.wikipedia.org/wiki/Pell%27s_equation
#   https://wrap.warwick.ac.uk/54707/1/WRAP_Hart_S1446788712000146a.pdf

func pell_holf_factorization(n) {

    var x = n.isqrt
    var y = x
    var z = 1
    var r = 2*x

    return n if n.is_prime
    return x if n.is_square

    var (f1, f2) = (1, x)

    for k in (1..Inf) {

        y = (r*z - y)
        z = ((n - y*y) / z)
        r = round((x + y) / z)

        var s = 1+isqrt(n * 4*k)
        var m = powmod(s, 2, n)

        if (m.is_square) {
            var g = gcd(n, s - m.isqrt)

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

        (f1, f2) = (f2, (r*f2 + f1)%n)

        if (z.abs.is_square) {
            var g = gcd(f1 - z.abs.isqrt, n)
            if (g.is_between(2, n-1)) {
                return g
            }
        }
    }
}

for n in (1..10) {
    var n = 2.of { 15.random_nbit_prime }.prod
    say ("PellFactor(#{n}) = ", pell_holf_factorization(n))
}

say [763546828801, 6031047559681, 184597450297471, 732785991945841, 18641350656000001, 55212580317094201, 9969815738350374661, 73410179782535364796052059].map(pell_holf_factorization) #=> [2621431, 4122691, 60761401, 121060801, 409599991, 452852401, 35723051521, 34271896307617]

say pell_holf_factorization(501683471371554052574395593328002385856815909090992391730837714087618687789563199570570904748766683429134165931903849390638583070098709002977060135024453968329425015984667453956539152122050098453561508979991634026587581234275280113197333815352093406638207274145779974221097800552970426675930071039999999999999999999999999)
say pell_holf_factorization(126727335216251935155511386856292269344039435369339223952841818402341178946842787391524421251641584095380245040090295288781100410936392638772851127857739674291628104658325634644906472821071273024005952940961745631724673242565456694421601734604687078622903224894582381431189998689129881876425605119999999999999999999999999)

__END__
PellFactor(700619593) = 29201
PellFactor(454862633) = 21817
PellFactor(691249777) = 30631
PellFactor(430147321) = 23819
PellFactor(586983343) = 32371
PellFactor(935383093) = 30161
PellFactor(411982057) = 21673
PellFactor(822287573) = 31139
PellFactor(631566191) = 19763
PellFactor(793815643) = 27749

44796583413093193287215634651008016235543664766903178304924622669704447996750359492291902708555128777401775001997067403960568901240094719999999999999999999999999
90526428980625828101248261690578699475994489216450172824535174978361071993433018140673220056871822737666086983202407045503649654589358079999999999999999999999999