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620_from_big_prime_list_2.pl
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620_from_big_prime_list_2.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::GMPz;
use ntheory qw(forcomb forprimes is_strong_lucas_pseudoprime random_prime);
use Math::Prime::Util::GMP qw(vecprod powmod divisors kronecker lucas_sequence);
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] || next;
$p = Math::GMPz->new($p);
#Math::GMPz::Rmpz_kronecker_ui($p, 5) == -1 or next;
say "Processing $p...";
my $p_str = "$p";
foreach my $d (divisors($p - Math::GMPz::Rmpz_kronecker_ui($p, 5))) {
if ((lucas_sequence($p_str, 1, -1, $d))[0] == 0) {
push @{$common_divisors{$d}}, $p_str;
last;
}
}
}
#~ 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 $n = Math::GMPz->new(vecprod(@{$arr}[@_]));
$callback->($n, @{$arr}[@_]) if !$seen{$n}++;
} $l, $k;
}
}
}
sub PSW_primality_test ($n) {
# Find P such that kronecker(n, P^2 + 4) = -1.
my $P;
for (my $k = 1 ; ; ++$k) {
if (kronecker($n, $k * $k + 4) == -1) {
$P = $k;
last;
}
}
# If LucasU(P, -1, n+1) = 0 (mod n), then n is probably prime.
(lucas_sequence($n, $P, -1, $n + 1))[0] == 0;
}
fibonacci_pseudoprimes(
10_000,
sub ($n, @f) {
if (is_strong_lucas_pseudoprime($n)) {
say "Lucas pseudoprime: $n";
if (powmod(2, $n-1, $n) == 1) {
die "Found a BPSW counter-example: $n = prod(@f)";
}
}
if (powmod(2, $n-1, $n) == 1) {
say "Fermat pseudoprime: $n";
if (PSW_primality_test($n)) {
die "PSW counter-example: $n = prod(@f)";
}
if (kronecker($n, 5) == -1) {
die "Found a Fibonacci special number: $n = prod(@f)";
}
}
#~ if (kronecker($n, 5) == -1) {
#~ if (powmod(2, $n-1, $n) == 1) {
#~ die "Found a Fibonacci special number: $n = prod(@f)";
#~ }
#~ }
}
);