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pseudoprimes_comb.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 Math::GMPz;
use Math::Prime::Util::GMP qw(vecprod divisors lucas_sequence is_lucas_pseudoprime powmod);
use ntheory
qw(forcomb forprimes kronecker is_power factor is_strong_lucas_pseudoprime random_prime valuation);
use List::Util qw(uniq shuffle);
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 <= $r;
my sub generate($p) {
#~ my $P;
#~ for (my $k = 1 ; ; ++$k) {
#~ if (kronecker($p, $k * $k + 4) == -1) {
#~ $P = $k;
#~ last;
#~ }
#~ }
#~ foreach my $d (divisors($p - 1)) {
#~ if (powmod(2, $d, $p) == 1) {
#~ push @{$common_divisors{$d}}, $p;
#~ }
#~ }
#~ foreach my $d (divisors($p - kronecker($p, 5))) {
#~ if ((lucas_sequence($p, 1, -1, $d))[0] == 0) {
#~ push @{$common_divisors{$d}}, $p;
#~ }
#~ }
#~ foreach my $d (divisors($p - 1)) {
#~ if (powmod(2, $d, $p) == 1) {
#~ push @{$common_divisors{$d}}, $p;
#~ }
#~ }
#~ foreach my $d (divisors($p - kronecker($p, 5))) {
#~ if ((lucas_sequence($p, 1, -1, $d))[1] == 2) {
#~ push @{$common_divisors{$d}}, $p;
#~ }
#~ }
foreach my $d (divisors($p-1)) {
if ((lucas_sequence($p, 1, -1, $d))[0] == 0) {
push @{$common_divisors{$d}}, $p;
}
}
foreach my $d (divisors($p+1)) {
if ((lucas_sequence($p, 1, -1, $d))[0] == 0) {
push @{$common_divisors{$d}}, $p;
}
}
#~ foreach my $d (divisors($p - 1)) {
#~ if (powmod(2, $d, $p) == 1) {
#~ push @{$common_divisors{$d}}, $p;
#~ }
#~ }
#~ foreach my $d (divisors($p + 1)) {
#~ if (powmod(2, $d, $p) == 1) {
#~ push @{$common_divisors{$d}}, $p;
#~ }
#~ }
}
my @files = shuffle(glob('../oeis-pseudoprimes/*.txt'));
#die "@files";
#~ say for @files;
#~ exit;
#~ open my $fh, '<', '$620 prime list - b018188.txt';
#~ while (<$fh>) {
#~ my $p = (split(' ', $_))[-1];
#~ $p || next;
#~ $p = Math::GMPz->new($p);
#~ generate($p);
#~ }
#~ close $fh;
my %seen_factor;
#my @pseudoprimes;
my $primes = 0;
foreach my $file(@files) {
open my $fh, '<', $file;
while (<$fh>) {
next if /^#/;
next if !/\S/;
last if ($. > 1000);
rand() < 0.5 or next;
my $p = (split(' ', $_))[-1];
$p || next;
next if $p eq '1';
#next if length($p) < 10;
length($p) <= 30 or next;
next if (is_power($p));
say "Pseudoprime: $p";
$p = Math::GMPz->new($p);
generate($p);
++$primes;
# last if $primes > 1000;
#~ my @factors = grep{!$seen_factor{$_}++} uniq(factor($p));
#~ foreach my $p(@factors) {
#~ $p = Math::GMPz->new($p);
#~ generate($p);
#~ ++$primes;
#~ }
# last if ($primes > 1e6);
}
close $fh;
# last if ($primes > 1e6);
}
say "There are ", scalar(values(%common_divisors)), " divisors.";
my %seen;
foreach my $arr (values %common_divisors) {
@$arr = uniq(@$arr);
my $l = $#{$arr} + 1;
$l = 12 if ($l > 12);
foreach my $k (2 .. $l) {
say "Combinations: $k out of $l";
forcomb {
my $nstr = vecprod(@{$arr}[@_]);
my $n = Math::GMPz->new($nstr);
$callback->($n, @{$arr}[@_]); #if !$seen{$nstr}++;
}
$l, $k;
}
}
}
my @pseudoprimes;
sub is_fibonacci_pseudoprime($n) {
(lucas_sequence($n, 1, -1, $n - kronecker($n, 5)))[0] == 0;
}
fibonacci_pseudoprimes(
1e3,
sub ($n, @f) {
if (is_lucas_pseudoprime($n)) {
say "Lucas: $n";
#push @pseudoprimes, $n;
if (powmod(2, $n - 1, $n) == 1) {
die "Found a BPSW counter-example: $n = prod(@f)";
}
}
elsif (powmod(2, $n - 1, $n) == 1) {
say "Fermat: $n";
if (kronecker($n, 5) == -1) {
if (is_fibonacci_pseudoprime($n)) {
die "Found a special pseudoprime: $n = prod(@f)";
}
}
}
#~ if (kronecker($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__
5777, 10877, 75077, 100127, 113573, 161027, 162133, 231703, 430127, 635627, 851927, 1033997, 1106327, 1256293, 1388903, 1697183, 2263127, 2435423, 2662277, 3175883, 3399527, 3452147, 3774377, 3900797, 4109363, 4226777, 4403027, 4828277, 4870847, 5208377, 5942627, 6003923, 7353917, 8518127, 9401893, 9713027, 9793313, 9922337, 10054043, 11637583, 13277423, 13455077, 13695947, 14015843, 14985833, 15754007, 16485493, 16685003, 17497127, 19168477, 20018627, 22361327, 23307377, 24157817, 25948187, 27854147, 29395277, 29604893, 30299333, 31673333, 32702723, 34134407, 34175777, 36061997, 39247393, 39850127, 40928627, 41177993, 42389027, 42525773, 47297543, 49219673, 49476377, 50075027, 51931333, 53697953, 57464207, 59268827, 62133377, 64610027, 67237883, 70894277, 73295777, 73780877, 74580767, 75239513, 75245777, 75983627, 83241013, 83963177, 85015493, 85903277, 86023943, 87471017, 90686777, 91418543, 93400277, 98385377, 104943827, 106728053, 110734667, 116853827, 117772877, 122879063, 124477513, 131017577, 131990627, 136579127, 139904627, 142593827, 144967877, 146278373, 148472347, 153256277, 154308527, 157132127, 158197577, 163578827, 166850777, 168018353, 171579883, 177991277, 179295443, 184135673, 185504633, 186003827, 192227027, 202368143, 207023087, 210089303, 211099877, 213361937, 226525883, 229206347, 231437957, 247030877, 247882963, 253755053, 254194877, 257815277, 259179527, 264250367, 264689963, 276795217, 277932113, 280075277, 284828777, 290256947, 293485877, 306219377, 311387693, 312189697, 316701527, 320234777, 334046627, 344107133, 360783793, 375578683, 376682627, 386628527, 387009737, 400091327, 400657277, 401790377, 403675973, 409245563, 420717527, 432988877, 437118527, 438894377, 439744397, 443146057, 443969063, 448504697, 450825377, 455039027, 456780193, 461700077, 461807147, 464407883, 465964127, 468245207, 469721647, 475167377, 480053573, 480891143, 485326403, 495101777, 500337713, 504097397, 523827527, 540136277, 544339637, 558030527, 562046627, 570122027, 574181327, 577647017, 583031693, 584238563, 598147577, 623709217, 634888253, 638227127, 657665777, 659936423, 664939277, 670042903, 670786877, 686258627, 691455077, 692726473, 704907377, 727615877, 729645563, 731349233, 734498627, 747587777, 768614027, 772719947, 780421277, 788342777, 797102627, 799500077, 811541327, 812957903, 825393997, 839350363, 847053323, 847887823, 856901267, 863097377, 869420473, 873933527, 878330573, 922483693, 923962577, 930039743, 961095923, 969210377, 978920627, 979805777, 985125077, 1011449753, 1015183343, 1032469817, 1034663713, 1055586377, 1085197577, 1113330077, 1171643027, 1173580127, 1194143443, 1203809777, 1218575027, 1226486627, 1230253133, 1280000357, 1295786777, 1296805127, 1308489103, 1309056527, 1326270203, 1345118777, 1364001113, 1371177527, 1387768397, 1435476803, 1437954377, 1477822433, 1524039373, 1538321777, 1541651627, 1546097027, 1561706327, 1598226167, 1599941027, 1620370127, 1663923827, 1689403127, 1749213377, 1757470643, 1770571277, 1783687127, 1826950127, 1839059627, 1885440527, 1897742027, 1909027273, 1966151713, 1986232877, 2021685077, 2031527803, 2044641377, 2056699133, 2087064527, 2093530277, 2132534777, 2152172027, 2179815377, 2189069027, 2207635127, 2231621027, 2271885527, 2273233877, 2336003647, 2382397877, 2407312577, 2411416883, 2444927627, 2509684127, 2525294777, 2535254027, 2564590757, 2630493643, 2641736327, 2660668877, 2767644017, 2774193827, 2775683777, 2807065127, 2817978767, 2832598277, 2834103827, 2837116127, 2865812777, 2887050077, 2915997527, 2985547447, 3045706127, 3059770877, 3092714627, 3174423947, 3181427027, 3226253627, 3234291377, 3279487577, 3316826083, 3331524377, 3333157127, 3342962027, 3400444277, 3424906253, 3435168827, 3501798827, 3525270527, 3558410813, 3560625077, 3582599627, 3585571907, 3601246277, 3643805027, 3690049277, 3744599777, 3752161877, 3782019617, 3797343377, 3860348777, 3881456123, 3923872577, 3966509777, 4041679277, 4068854957, 4092184277, 4097614127, 4131655097, 4141182527, 4188641627, 4223494277, 4249267577, 4256645777, 4265877527, 4484755277, 4601042627, 4622170877, 4639493627, 4681974527, 4840050077, 4887391277, 4893325127, 4938314813, 4968798827, 4998750077, 5294024413, 6039541727, 11851534697, 22200933343, 35646833933, 68055160643, 92402327687, 98831168617, 101590045727, 192900153617, 353348357933, 353833078717, 671092578683, 1118047771487, 2270927963303, 3357827162143, 3601866154427, 3703263099587, 5324864903273, 7973122223753, 8932423389707, 18846129954107, 25022761143923, 29469429987317, 29536817792327, 61561639243505213