fibonacci_factorization_method.sf

#!/usr/bin/ruby

# Author: Daniel "Trizen" Șuteu
# Date: 09 September 2018
# https://github.com/trizen

# A new integer factorization method, using the Fibonacci numbers.

# It uses the smallest divisor `d` of `p - legendre(p, 5)`, such that `Fibonacci(d) = 0 (mod p)`.

# By selecting a small bound B, we compute `k = lcm(1..B)`, hoping that `k` is a
# multiple of `d`, then `gcd(Fibonacci(k) (mod n), n)` in a non-trivial factor of `n`.

# This method is similar in flavor to Pollard's p-1 and Williams's p+1 methods.

func fibonacci_factorization(n, B=10000) {

    var k = consecutive_lcm(B)      # lcm(1..B)
    var F = fibmod(k, n)            # Fibonacci(k) (mod n)

    return gcd(F, n)
}

say fibonacci_factorization(257221 * 470783,              700);     #=> 470783           (p+1 is  700-smooth)
say fibonacci_factorization(333732865481 * 1632480277613, 3000);    #=> 333732865481     (p-1 is 3000-smooth)

for k in (1..50) {

    var n = (random_prime(2**k) * random_prime(2**k))
    var p = fibonacci_factorization(n, 2 * n.ilog2**2)

    if (p.is_prime) {
        say "#{n} = #{p} * #{n/p}"
    }
}

__END__
3214709 = 1613 * 1993
167569819 = 19001 * 8819
2851899013 = 49807 * 57259
27732054173 = 153607 * 180539
3709275441913 = 1991999 * 1862087
16170839753069 = 1629653 * 9922873
423422405914009 = 10333903 * 40974103
122720571221359159 = 384088241 * 319511399
69782959258128563 = 271895191 * 256653893
35012441184542164741 = 6294003887 * 5562824843
524934717838728509869 = 18711680699 * 28053851831
113042322296238658633 = 3370274783 * 33540980951
28555019079287984329261 = 386638476841 * 73854571621
544812320889004864776853 = 333732865481 * 1632480277613
5070894233593682106381371 = 1927323093737 * 2631055607683