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upper-bounds_with_residue.pl
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upper-bounds_with_residue.pl
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#!/usr/bin/perl
# Author: Daniel "Trizen" Șuteu
# Date: 28 January 2019
# https://github.com/trizen
# A new algorithm for generating Fermat pseudoprimes to base 2.
# See also:
# https://oeis.org/A050217 -- Super-Poulet numbers: Poulet numbers whose divisors d all satisfy d|2^d-2.
# https://oeis.org/A214305 -- Fermat pseudoprimes to base 2 with two prime factors.
# See also:
# https://en.wikipedia.org/wiki/Fermat_pseudoprime
# https://trizenx.blogspot.com/2018/08/investigating-fibonacci-numbers-modulo-m.html
use 5.020;
use warnings;
use strict;
use ntheory qw(:all);
use experimental qw(signatures);
use List::Util qw(uniqnum);
use Math::AnyNum qw(is_smooth prod);
my $N = 14;
my $SMOOTH = 19;
my $P = nth_prime($N);
my $MAX = ~0;
my @primes_bellow = @{primes($P)};
#~ sub prod {
#~ my $prod = 1;
#~ foreach my $n(@_) {
#~ $prod *= $n;
#~ }
#~ $prod;
#~ }
# a(5) = 11530801
# a(7) = 15656266201
my @primes = @{primes(100)};
sub non_residue {
my ($n) = @_;
foreach my $p (@primes) {
#if ($p > $Q) {
# return -1;
# }
sqrtmod($p, $n) // return $p;
}
return -1;
}
#~ #139309114031, a(9) <= 7947339136801, a(10) <= 72054898434289.
#~ say non_residue(139309114031, nth_prime(8));
#~ say non_residue(7947339136801, nth_prime(9));
#~ say non_residue(72054898434289, nth_prime(10));
#~ __END__
#~ say non_residue(72054898434289, nth_prime(10));
#~ __END__
sub isok {
my ($k) = @_;
vecall { powmod($_, ($k-1)>>1, $k) == (kronecker($_, $k) % $k) } 2..$P;
#~ my $res;
#~ my $p = nth_prime($n);
#~ for(my $k = $A000229[$n-1]*$A000229[$n-1]; ; $k+=2) {
#~ if (!is_prime($k) and powmod($p, ($k-1)>>1, $k) == $k-1) {
#~ my $q = non_residue($k, $p);
#~ if ($p == $q) {
#~ return $k;
#~ }
#~ }
#~ }
#~ return $res;
}
sub fermat_pseudoprimes ($limit, $callback) {
say "Collecting primes...";
my %common_divisors;
forprimes {
my $p = $_;
my $common = 0;
# my $res = non_residue($p);
# if ($res >= $P) {
if (is_smooth(($p-1)>>1, $SMOOTH)) {
foreach my $d (divisors(($p - 1)>>1)) {
#my $t = powmod($P, $d, $p);
#if ( == $p-1 or powmod($P, $d, $p) == 1) {
if (vecall { my $t = powmod($_, $d, $p); $t == (kronecker($_, $p)%$p) } 2..$P) {
push @{$common_divisors{$d}}, $p;
$common = 1;
#say "$d $p";
#~ foreach my $g(divisors($d)) {
#~ if ($g > 4*$SMOOTH) {
#~ push @{$common_divisors{$g}}, $p;
#~ }
#~ }
#if ($t == 1 or $t == $p-1) {
#if ($t == (kronecker($P, $d) % $p)) {
# foreach my $g(divisors($d)) {
# $g > sqrtint($d) or next;
# push @{$common_divisors{$g}}, $p;
#}
#$common = 1;
}
}
if ($common) {
foreach my $n(2..50) {
#foreach my $g(divisors($p-1)) {
for my $g($p-1) {
if (is_prime($g*$n + 1)) {
my $q = $g*$n + 1;
foreach my $d (divisors(($q - 1)>>1)) {
#if (powmod($P, $d, $q) == $q-1) {
if (vecall { my $t = powmod($_, $d, $q); $t == (kronecker($_, $q)%$q) } 2..$P) {
push @{$common_divisors{$d}}, $q;
}
}
}
}
}
}
# }
}
} 3, $limit;
say "Generating combinations...";
#~ forprimes {
#~ my $p = $_;
#~ foreach my $d (divisors($p - 1)) {
#~ if (powmod($P, $d, $p) == $p-1) {
#~ push @{$common_divisors{$d}}, $p;
#~ }
#~ }
#~ } $limit>>1, $limit;
my %seen;
foreach my $value (values %common_divisors) {
my $arr = [uniqnum(@$value)];
my $l = $#{$arr} + 1;
foreach my $k (2..$l) {
forcomb {
my $n = prod(@{$arr}[@_]);
if ($n <= $MAX) {
$callback->($n) #if !$seen{$n}++;
}
} $l, $k;
}
}
}
#~ sub is_fermat_pseudoprime ($n, $base) {
#~ powmod($base, $n - 1, $n) == 1;
#~ }
#~ sub is_fibonacci_pseudoprime($n) {
#~ (lucas_sequence($n, 1, -1, $n - kronecker($n, 5)))[0] == 0;
#~ }
my @values;
my %seen;
fermat_pseudoprimes(
11971,
sub ($k) {
# is_fermat_pseudoprime($n, 2) || die "error for n=$n";
#~ if (kronecker($n, 5) == -1) {
#~ if (is_fibonacci_pseudoprime($n)) {
#~ die "Found a special pseudoprime: $n = prod(@f)";
#~ }
#~ }
if (isok($k) and !$seen{$k}++) {
say $k;
push @values, $k;
}
# push @pseudoprimes, $n;
}
);
say "Min: a($N) <= ", vecmin(@values);
#@pseudoprimes = sort { $a <=> $b } @pseudoprimes;
#say join(', ', @pseudoprimes);
__END__
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6027193, 6122551, 6140161, 6226193, 6233977, 6236257, 6278533, 6334351, 6368689, 6386993, 6539527, 6631549, 6658669, 6749021, 6779137, 6787327, 6912079, 6952037, 6955541, 6973063, 7017193, 7232321, 7295851, 7306261, 7306561, 7462001, 7674967, 7725901, 7759937, 7820201, 7883731, 8036033, 8095447, 8180461, 8231653, 8239477, 8362201, 8534233, 8650951, 8725753, 8745277, 8902741, 9006401, 9037729, 9040013, 9056501, 9073513, 9131401, 9273547, 9371251, 9480461, 9533701, 9564169, 9567673, 9591661, 9724177, 9729301, 9863461, 10031653, 10084177, 10331141, 10386241, 10402237, 10403641, 10425511, 10505701, 10610063, 10712857, 10974881, 11115037, 11335501, 11367137, 11541307, 11585293, 12032021, 12263131, 12273769, 12322133, 12498061, 12599233, 12659989, 12711007, 12854437, 13073941, 13295281, 13338371, 13421773, 13448593, 13635289, 13694761, 13747361, 13773061, 13856417, 13991647, 13996951, 14026897, 14179537, 14282143, 14324473, 14589901, 14794081, 14899751, 15101893, 15139199, 15162941, 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