sidef-scripts/Math/frobenius_pseudoprimes_generation.sf

#!/usr/bin/ruby

# Author: Daniel "Trizen" Șuteu
# Date: 21 September 2018
# Edit: 17 August 2020
# https://github.com/trizen

# A new algorithm for generating Frobenius pseudoprimes to Fibonacci polynomial x^2 - x - 1.

# OEIS:
#   https://oeis.org/A081264 -- Odd Fibonacci pseudoprimes.
#   https://oeis.org/A212424 -- Frobenius pseudoprimes with respect to Fibonacci polynomial x^2 - x - 1.

# See also:
#   https://trizenx.blogspot.com/2020/08/pseudoprimes-construction-methods-and.html

func frobenius_pseudoprimes(limit, callback, P=1, Q=-1) {

    var D = (P*P - 4*Q)
    var table = Hash()

    for p in (primes(3, limit)) {
        table{ p - kronecker(D,p) -> divisors.first_by {|d| lucasUmod(P, Q, d, p) == 0 } } := [] << p
    }

    table.each_v { |a|

        var len = a.len
        len > 1 || next

        for k in (2..len) {
            a.combinations(k, {|*t|
                callback(t.prod)
            })
        }
    }
}

func is_frobenius_pseudoprime(n, P=1, Q=-1) {

    var D = (P*P - 4*Q)
    var k = kronecker(D, n)

    lucasUmod(P, Q, n - k, n) == 0 || return false
    lucasUmod(P, Q, n + k, n) == 1 || return false

    return true
}

var pseudoprimes = []

frobenius_pseudoprimes(10000, {|n|

    assert(is_frobenius_pseudoprime(n))

    if (kronecker(5, n) == -1){
        if (powmod(2, n-1, n) == 1) {
            die "Found a special pseudoprime: #{n}"
        }
    }

    pseudoprimes << n
})

say pseudoprimes.sort

__END__
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