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620_research_with_D.pl
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#!/usr/bin/perl
# Author: Daniel "Trizen" Șuteu
# Date: 22 September 2018
# https://github.com/trizen
# A new algorithm for generating Fibonacci pseudoprimes.
# See also:
# https://oeis.org/A081264 -- Odd Fibonacci pseudoprimes.
# https://oeis.org/A212424 -- Frobenius pseudoprimes with respect to Fibonacci polynomial x^2 - x - 1.
# For more info, see:
# https://trizenx.blogspot.com/2018/08/investigating-fibonacci-numbers-modulo-m.html
use 5.020;
use warnings;
use experimental qw(signatures);
use Math::AnyNum qw(fibmod prod powmod lucasmod isqrt);
use ntheory qw(kronecker forcomb forprimes kronecker divisors factor_exp divisor_sum euler_phi lucas_sequence random_prime is_square_free);
sub find_D {
my ($n) = @_;
for (my $k = 2 ; ; ++$k) {
my $D = (-1)**$k * (2 * $k + 1);
if (kronecker($D, $n) == -1) {
return ((1 - $D) / 4);
}
}
}
sub fibonacci_pseudoprimes ($limit, $callback) {
my %common_divisors;
#~ #forprimes {
#~ while (<>) {
#~ my (undef, $p) = split(' ', $_);
#~ $p = Math::AnyNum->new($p);
#~ $p || next;
#~ say $p;
#~ #my $p = $_;
#~ foreach my $d (divisors($p - kronecker($p, 5))) {
#~ if (lucasmod($d, $p) == 0) {
#~ push @{$common_divisors{$d}}, $p;
#~ }
#~ }
#~ } #$limit;
#forprimes {
my %generated;
my $r = random_prime(10**7);
#my $r = 1;
forprimes {
my $p = $_;
$generated{$p} = 1;
foreach my $d (divisors($p - kronecker($p, 5))) {
#if (lucasmod($d, $p) == 0) {
#if (lucas_sequence($p,
my ($U, $V) = lucas_sequence($p, 1, find_D($p), $d);
#if ($U == 0 and $V==2) {
#if ($V == 0) {
#if (lucasmod($d, $p) == 0) {
#if (fibmod($d, $p) == 0) {
if ($U == 0 or $V == 0) {
push @{$common_divisors{$d}}, $p;
}
}
} $r, $r+1e6;
#~ while (<>) {
#~ my (undef, $p) = split(' ', $_);
#~ $p || next;
#~ next if $generated{$p}++;
#~ $p = Math::GMPz->new($p);
#~ say $p;
#~ #my $p = $_;
#~ foreach my $d (divisors($p - kronecker($p, 5))) {
#~ my ($U, $V) = lucas_sequence($p, 1, find_D($p), $d);
#~ #if ($U == 0 and $V==2) {
#~ #if ($V == 0) {
#~ #if (lucasmod($d, $p) == 0) {
#~ #if (fibmod($d, $p) == 0) {
#~ if ($U == 0 or $V == 0) {
#~ push @{$common_divisors{$d}}, $p;
#~ }
#~ }
#~ }
foreach my $n(1..1e6) {
my $p = random_prime(10**7);
next if $generated{$p}++;
foreach my $d (divisors($p - kronecker($p, 5))) {
#if (lucasmod($d, $p) == 0) {
#if (lucas_sequence($p,
my ($U, $V) = lucas_sequence($p, 1, find_D($p), $d);
#if ($U == 0 and $V==2) {
#if ($V == 0) {
#if (lucasmod($d, $p) == 0) {
#if (fibmod($d, $p) == 0) {
if ($U == 0 or $V == 0) {
push @{$common_divisors{$d}}, $p;
}
}
}
say "Found ", scalar(keys%common_divisors), " relations";
my %seen;
foreach my $arr (values %common_divisors) {
my $l = $#{$arr} + 1;
foreach my $k (2 .. $l) {
forcomb {
my $n = prod(@{$arr}[@_]);
$callback->($n, @{$arr}[@_]) if !$seen{$n}++;
} $l, $k;
}
}
}
sub is_fibonacci_pseudoprime($n) {
my ($U) = lucas_sequence($n, 1, -1, $n - kronecker($n, 5));
$U == 0;
#fibmod($n - kronecker($n, 5), $n) == 0;
}
sub BPSW_primality_test ($n) {
$n = Math::GMPz->new("$n");
return 0 if Math::GMPz::Rmpz_cmp_ui($n, 1) <= 0;
return 1 if Math::GMPz::Rmpz_cmp_ui($n, 2) == 0;
return 0 if Math::GMPz::Rmpz_perfect_power_p($n);
my $d = Math::GMPz::Rmpz_init();
my $t = Math::GMPz::Rmpz_init_set_ui(2);
# Fermat base-2 test
Math::GMPz::Rmpz_sub_ui($d, $n, 1);
Math::GMPz::Rmpz_powm($t, $t, $d, $n);
Math::GMPz::Rmpz_cmp_ui($t, 1) and return 0;
my ($Q, $D) = (1, 0);
for (my $k = 2 ; ; ++$k) {
$D = (-1)**$k * (2 * $k + 1);
if (Math::GMPz::Rmpz_si_kronecker($D, $n) == -1) {
$Q = (1 - $D) / 4;
last;
}
}
Math::GMPz::Rmpz_add_ui($d, $d, 2);
my $s = Math::GMPz::Rmpz_scan1($d, 0);
my $U1 = Math::GMPz::Rmpz_init_set_ui(1);
my ($V1, $V2) = (Math::GMPz::Rmpz_init_set_ui(2), Math::GMPz::Rmpz_init_set_ui(1));
my ($Q1, $Q2) = (Math::GMPz::Rmpz_init_set_ui(1), Math::GMPz::Rmpz_init_set_ui(1));
foreach my $bit (split(//, substr(Math::GMPz::Rmpz_get_str($d, 2), 0, -$s - 1))) {
Math::GMPz::Rmpz_mul($Q1, $Q1, $Q2);
Math::GMPz::Rmpz_mod($Q1, $Q1, $n);
if ($bit) {
Math::GMPz::Rmpz_mul_si($Q2, $Q1, $Q);
Math::GMPz::Rmpz_mul($U1, $U1, $V2);
Math::GMPz::Rmpz_mul($V1, $V1, $V2);
Math::GMPz::Rmpz_powm_ui($V2, $V2, 2, $n);
Math::GMPz::Rmpz_sub($V1, $V1, $Q1);
Math::GMPz::Rmpz_submul_ui($V2, $Q2, 2);
Math::GMPz::Rmpz_mod($V1, $V1, $n);
Math::GMPz::Rmpz_mod($U1, $U1, $n);
}
else {
Math::GMPz::Rmpz_set($Q2, $Q1);
Math::GMPz::Rmpz_mul($U1, $U1, $V1);
Math::GMPz::Rmpz_mul($V2, $V2, $V1);
Math::GMPz::Rmpz_sub($U1, $U1, $Q1);
Math::GMPz::Rmpz_powm_ui($V1, $V1, 2, $n);
Math::GMPz::Rmpz_sub($V2, $V2, $Q1);
Math::GMPz::Rmpz_submul_ui($V1, $Q2, 2);
Math::GMPz::Rmpz_mod($V2, $V2, $n);
Math::GMPz::Rmpz_mod($U1, $U1, $n);
}
}
Math::GMPz::Rmpz_mul($Q1, $Q1, $Q2);
Math::GMPz::Rmpz_mul_si($Q2, $Q1, $Q);
Math::GMPz::Rmpz_mul($U1, $U1, $V1);
Math::GMPz::Rmpz_mul($V1, $V1, $V2);
Math::GMPz::Rmpz_sub($U1, $U1, $Q1);
Math::GMPz::Rmpz_sub($V1, $V1, $Q1);
Math::GMPz::Rmpz_mul($Q1, $Q1, $Q2);
if (Math::GMPz::Rmpz_congruent_ui_p($U1, 0, $n)) {
return 1;
}
if (Math::GMPz::Rmpz_congruent_ui_p($V1, 0, $n)) {
return 1;
}
for (1 .. $s) {
Math::GMPz::Rmpz_mul($U1, $U1, $V1);
Math::GMPz::Rmpz_mod($U1, $U1, $n);
Math::GMPz::Rmpz_powm_ui($V1, $V1, 2, $n);
Math::GMPz::Rmpz_submul_ui($V1, $Q1, 2);
Math::GMPz::Rmpz_powm_ui($Q1, $Q1, 2, $n);
if (Math::GMPz::Rmpz_congruent_ui_p($V1, 0, $n)) {
return 1;
}
}
return 0;
}
#my @pseudoprimes;
fibonacci_pseudoprimes(
10_000,
sub ($n, @f) {
if (is_fibonacci_pseudoprime($n)) {
say "Fibonacci pseudoprime: $n";
}
else {
#if (is_square_free($n)) {
warn "Not a fib: $n\n";
#}
}
#~ say "Found a pseudoprime: $n";
#~ #or die "Not a Fibonacci pseudoprime: $n";
#~ #say join(' * ', @f), " = $n";
#~ # push @pseudoprimes, $n;
#~ }else {
#~ #warn "Not a fibonacci: $n\n";
#~ }
if (powmod(2, $n-1, $n) == 1) {
warn "Found a Fermat pseudoprime: $n\n";
if (kronecker($n, 5) == -1) {
#if (powmod(2, $n - 1, $n) == 1) {
die "\n\n**** Found a special pseudoprime: $n with factors (@f) *** \n\n";
#}
}
if (BPSW_primality_test($n)) {
die "Found a counter-example to BPSW: $n\n";
}
}
}
);
#@pseudoprimes = sort {$a <=> $b} @pseudoprimes;
#say join(', ', @pseudoprimes);