carmichael_from_multiple_recursive_mpz.pl

#!/usr/bin/perl

# Daniel "Trizen" Șuteu
# Date: 17 March 2023
# https://github.com/trizen

# Generate Carmichael numbers from a given multiple.

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

use 5.036;
use Math::GMPz;
use ntheory qw(:all);

sub carmichael_from_multiple ($m, $callback, $reps = 1e3) {

    my $t = Math::GMPz::Rmpz_init();
    my $u = Math::GMPz::Rmpz_init();
    my $v = Math::GMPz::Rmpz_init();

    is_square_free($m) || return;

    my $L = lcm(map { subint($_, 1) } factor($m));

    $m = Math::GMPz->new("$m");
    $L = Math::GMPz->new("$L");

    Math::GMPz::Rmpz_invert($v, $m, $L) || return;

    for (my $p = Math::GMPz::Rmpz_init_set($v) ; --$reps >= 0 ; Math::GMPz::Rmpz_add($p, $p, $L)) {

        Math::GMPz::Rmpz_gcd($t, $m, $p);
        Math::GMPz::Rmpz_cmp_ui($t, 1) == 0 or next;

        my @factors = factor_exp($p);
        (vecall { $_->[1] == 1 } @factors) || next;

        Math::GMPz::Rmpz_mul($v, $m, $p);
        Math::GMPz::Rmpz_sub_ui($u, $v, 1);

        Math::GMPz::Rmpz_set_str($t, lcm(map { subint($_->[0], 1) } @factors), 10);

        if (Math::GMPz::Rmpz_divisible_p($u, $t)) {
            $callback->(Math::GMPz::Rmpz_init_set($v));
        }
    }
}

foreach my $p (@{primes(3, 100)}) {
    say "# Sieving with p = $p";

    my @list = ($p);

    while (@list) {
        my $m = shift(@list);
        carmichael_from_multiple(
            $m,
            sub ($n) {
                if ($n > $m) {
                    if ($n > ~0) {
                        say $n;
                    }
                    push @list, $n;
                }
            }
        );
    }
}