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620_strong_generator.pl
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#!/usr/bin/perl
# Author: Daniel "Trizen" Șuteu
# Date: 07 October 2018
# https://github.com/trizen
# A simple algorithm for generating a subset of strong-Lucas pseudoprimes.
# 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.
use 5.020;
use warnings;
use experimental qw(signatures);
use Math::AnyNum qw(prod powmod);
use ntheory qw(forcomb forprimes kronecker divisors is_strong_lucas_pseudoprime lucas_sequence random_prime);
use List::Util qw(uniq);
sub fibonacci_pseudoprimes ($limit, $callback) {
my %common_divisors;
my $r = random_prime(1e8);
my $r2 = random_prime(1e9);
die 'error' if $r <= 1e7;
die 'error' if $r2+1e7 <= $r;
while (<>) {
my $p = (split(' ', $_))[-1];
$p || next;
$p = Math::AnyNum->new($p);
foreach my $d (divisors($p - kronecker($p, 5))) {
if ((lucas_sequence($p, 1, -1, $d))[0] == 0) {
push @{$common_divisors{$d}}, $p;
}
}
}
forprimes {
my $p = $_;
foreach my $d (divisors($p - kronecker($p, 5))) {
if ((lucas_sequence($p, 1, -1, $d))[0] == 0) {
push @{$common_divisors{$d}}, $p;
}
}
} 1e7;
forprimes {
my $p = $_;
foreach my $d (divisors($p - kronecker($p, 5))) {
if ((lucas_sequence($p, 1, -1, $d))[0] == 0) {
push @{$common_divisors{$d}}, $p;
}
}
} $r, $r+1e7;
forprimes {
my $p = $_;
foreach my $d (divisors($p - kronecker($p, 5))) {
if ((lucas_sequence($p, 1, -1, $d))[0] == 0) {
push @{$common_divisors{$d}}, $p;
}
}
} $r2, $r2+1e7;
my %seen;
foreach my $arr (values %common_divisors) {
@$arr = uniq(@$arr);
my $l = $#{$arr} + 1;
foreach my $k (2 .. $l) {
forcomb {
my $n = prod(@{$arr}[@_]);
$callback->($n, @{$arr}[@_]) if !$seen{$n}++;
} $l, $k;
}
}
}
my @pseudoprimes;
fibonacci_pseudoprimes(
10_000,
sub ($n, @f) {
if (is_strong_lucas_pseudoprime($n)) {
say $n;
#push @pseudoprimes, $n;
if (powmod(2, $n-1, $n) == 1) {
die "Found a BPSW counter-example: $n = prod(@f)";
}
}
if (kronecker($n, 5) == -1) {
if (powmod(2, $n-1, $n) == 1) {
die "Found a Fibonacci special number: $n = prod(@f)";
}
}
}
);
@pseudoprimes = sort { $a <=> $b } @pseudoprimes;
say join(', ', @pseudoprimes);
__END__
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