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frobenius_pseudoprimes_generation.pl
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frobenius_pseudoprimes_generation.pl
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#!/usr/bin/perl
# Author: Daniel "Trizen" Șuteu
# Date: 07 October 2018
# Edit: 19 August 2020
# https://github.com/trizen
# A new algorithm for generating Frobenius pseudoprimes to Fibonacci polynomial x^2 - x - 1.
# See also:
# https://oeis.org/A217120 -- Lucas pseudoprimes
# https://oeis.org/A217255 -- Strong Lucas pseudoprimes
# https://oeis.org/A177745 -- Semiprimes n such that n divides Fibonacci(n+1).
# https://oeis.org/A212423 -- Frobenius pseudoprimes == 2,3 (mod 5) with respect to Fibonacci polynomial x^2 - x - 1.
# See also:
# https://trizenx.blogspot.com/2020/08/pseudoprimes-construction-methods-and.html
use 5.020;
use warnings;
use experimental qw(signatures);
use ntheory qw(:all);
use Math::AnyNum qw(prod);
sub frobenius_pseudoprimes ($limit, $callback, $P = 1, $Q = -1) {
my %table;
my $D = $P*$P - 4*$Q;
forprimes {
my $p = $_;
foreach my $d (divisors($p - kronecker($D, $p))) {
if ((lucas_sequence($p, $P, $Q, $d))[0] == 0) {
push @{$table{$d}}, $p;
last;
}
}
} 3, $limit;
foreach my $arr (values %table) {
my $l = $#{$arr} + 1;
foreach my $k (2 .. $l) {
forcomb {
my $n = prod(@{$arr}[@_]);
$callback->($n, @{$arr}[@_]);
} $l, $k;
}
}
}
my @pseudoprimes;
frobenius_pseudoprimes(
100_000,
sub ($n, @f) {
is_frobenius_pseudoprime($n, 1, -1) or die "error: $n";
push @pseudoprimes, $n;
if (kronecker(5, $n) == -1 and powmod(2, $n-1, $n) == 1) {
die "Found a BPSW counter-example: $n = prod(@f)";
}
}
);
@pseudoprimes = sort { $a <=> $b } @pseudoprimes;
say join(', ', @pseudoprimes);
__END__
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