experimental-projects/pseudoprimes/generators/erdos_carmichael_generate_from_prefix.pl

#!/usr/bin/perl

# Erdos construction method for Carmichael numbers:
#   1. Choose an even integer L with many prime factors.
#   2. Let P be the set of primes d+1, where d|L and d+1 does not divide L.
#   3. Find a subset S of P such that prod(S) == 1 (mod L). Then prod(S) is a Carmichael number.

# Alternatively:
#   3. Find a subset S of P such that prod(S) == prod(P) (mod L). Then prod(P) / prod(S) is a Carmichael number.

use 5.020;
use warnings;
use ntheory qw(:all);
use List::Util qw(shuffle);
use experimental qw(signatures);
use Math::GMPz;

# Modular product of a list of integers
sub vecprodmod ($arr, $mod) {

    #~ if ($mod > ~0) {
        #~ my $prod = Math::GMPz->new(1);
        #~ foreach my $k(@$arr) {
            #~ $prod = ($prod * $k) % $mod;
        #~ }
        #~ return $prod;
    #~ }

    if ($mod < ~0) {
        my $prod = 1;
        foreach my $k(@$arr) {
            $prod = mulmod($prod, $k, $mod);
        }
        return $prod;
    }

    my $prod = 1;

    foreach my $k (@$arr) {
        $prod = Math::Prime::Util::GMP::mulmod($prod, $k, $mod);
    }

    Math::GMPz->new($prod);
}

# Primes p such that p-1 divides L and p does not divide L
sub lambda_primes ($L) {
    #grep { ($L % $_) != 0 } grep { $_ > 2 and is_prime($_) } map { $_ + 1 }
        #map  { Math::GMPz->new($_)}
     #   divisors($L);

    grep { ($_ > 2) and (($L % $_) != 0) and is_prime($_) } map { ($_ >= ~0) ? (Math::GMPz->new($_)+1) : ($_ + 1) } divisors($L);
}

#my @prefix = (3, 5, 17, 23, 29, 53, 83, 89);
#my @prefix = (3,5,17,23, 29);

#my @prefix = (3, 5, 17, 23, 29, 53, 83, 89);
#my @prefix = (3, 5, 17, 23, 29, 53, 83, 89, 113, 353, 617, 2003, 2549);
#my @prefix = (3, 5, 17, 23, 29, 53, 83, 89, 113, 353, 617);
#my @prefix = (3, 5, 17, 23, 29, 53, 83, 89, 113, 419, 449, 617);

#my @prefix = (3, 5, 17, 23, 29, 53, 83, 89, 113, 257, 353, 449, 617, 1409, 2003);
#my @prefix = (3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 1409, 2003);

#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 257, 353, 617, 1409, 2003, 2549, 3137, 9857, 10193, 16073, 68993, 202049, 275969, 1500929, 18386369]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 2003, 2297, 2549, 3137, 9857, 10193, 68993, 88397, 93809, 5850209, 8044037, 18386369]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 2003, 2549, 3137, 4019, 4289, 10193, 16073, 21977, 38459, 50513, 52529, 76649, 93809, 97553]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 1409, 2003, 2549, 8009, 9857, 23297, 50513, 68993, 275969, 375233, 1500929, 3232769, 18386369]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 1409, 2003, 2297, 2549, 3329, 8009, 10193, 16073, 23297, 50177, 93809, 202049, 275969, 656657, 18386369]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019, 7547, 9857, 10193, 16073, 17837, 23297, 68993, 88397, 93809, 202049, 896897, 1500929, 2475089, 8044037, 18386369]

#my @prefix = (3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019);
#my @prefix = (3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 257, 353);
#my @prefix = (3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019);
#my @prefix = (3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019);

# primes(2, 15500).grep{ .dec.is_smooth(43) }.grep{ gcd(.phi, [3, 5, 17, 23, 29].prod) == 1 }
# => [2, 3, 5, 17, 23, 29, 53, 83, 89, 113, 149, 173, 197, 257, 353, 419, 449, 593, 617, 677, 683, 947, 1217, 1373, 1409, 1559, 1613, 2003, 2129, 2237, 2297, 2357, 2543, 2549, 2663, 2729, 2753, 2927, 3137, 3257, 3329, 3389, 3527, 3719, 4019, 4733, 4817, 5333, 5477, 6449, 6689, 6917, 7253, 7547, 7937, 8009, 8429, 8513, 8867, 9473, 9857, 9923, 10169, 10193, 12959, 13313, 13469, 13553, 13859, 14897, 15137, 15377, 15467]

my @prefix = (

    #3, 5, 17, 23, 29, 53, 83, 89, 113, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019, 7547, 9857, 10193, 16073, 17837, 23297, 68993, 88397, 93809, 202049, 896897
    #3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019, 9923, 10193, 13553, 308309, 486179, 656657
    #3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019, 9923, 10193, 13553, 308309, 486179, 656657, 896897, 902903, 1500929, 1610753, 1944713, 2063777, 2540033, 2818817, 3232769, 6071297, 18386369

    #3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019, 9923, 10193, 13553, 308309, 486179, 656657, 896897, 902903, 1500929, 1610753, 1944713, 2063777, 2540033, 2818817, 3232769, 6071297, 18386369, 22629377, 25281257, 31115393, 33020417

    #3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 2003, 2297, 2549, 3137, 9857, 10193, 68993, 88397, 93809, 5850209, 8044037, 18386369
    #3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 2003, 2297, 2549, 3137, 9857, 10193, 68993, 88397, 93809, 5850209, 8044037, 18386369, 2059273217
    # 3, 5, 17, 23, 29, 53, 83, 89, 113, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019, 7547, 9857, 10193, 16073, 17837, 23297, 68993, 88397, 93809, 202049, 896897, 1500929, 2475089, 8044037, 18386369, 2059273217

    #3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 10193, 3232769, 6071297, 18386369
    # 3, 5, 17, 23, 29, 53, 83, 89, 113, 353, 449, 617, 2003, 2297, 2549, 3137, 10193, 18386369

    #3, 5, 17, 23, 29, 53, 83, 89, 113, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019, 7547, 9857, 10193, 16073, 17837, 23297,

    #3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 617, 1409, 2003, 2549, 3137, 9857, 10193, 16073, 68993, 18386369
    #3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 4019, 4289, 7547, 9857, 10193, 16073, 68993, 88397, 93809

    #3, 5, 17, 23, 29, 53, 89, 197
     #3, 5, 17, 23, 29, 53, 89, 197, 617
    # 3, 5, 17, 23, 29, 197, 617, 1217, 6689, 14897, 16073, 20483, 46817, 50513, 68993, 76343, 81929,

    #3, 5, 17, 23, 29, 197, 419, 449, 617, 1217, 2129, 3137, 9857, 10193, 16073, 68993, 18386369

    #3, 5, 17, 23, 29, 43, 53, 89, 113, 127, 157, 257, 353, 397, 449, 617, 1093, 1499, 2731
    #3, 5, 17, 23, 29, 53, 83, 89, 113, 353,
    #3, 5, 17, 23, 29, 53, 83, 89, 113, 149, 173, 197, 257, 263, 269, 293, 353, 359, 383
    # 3, 5, 17, 23, 29, 53, 83, 89, 113, 149, 173, 197, 257, 353

    # 3, 5, 17, 23, 29, 53, 83, 89, 113, 149, 173, 197, 257, 353, 419, 449, 593, 617, 677, 683, 947, 1217, 1373,
    # 3, 5, 17, 23, 29, 53, 83, 89, 113, 149, 197, 257, 353, 419, 449, 593, 617, 677, 683, 1217, 1373, 1409, 1559, 1613

   # 3, 5, 17, 23, 29, 53, 83, 89, 113

    #~ 3, 5, 17, 23, 29, 53, 83, 89, 113, 149, 173, 197, 257, 263, 269, 293, 353, 359, 383
#~ A328691
#~ A329460
#~ #Constructing Carmichael numbers throughimproved subset-product algorithms
#~ 1.893
#~ The smallest Abundant number which is also pseudoprime (base-2).
#~ Shyam Sunder Gupta
#~ 171800042106877185
#~ 10-11-2019
#~ 3470207934739664512679701940114447720865
    #3, 5, 17, 23, 29, 53, 83, 89, 113, 149, 197, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019
    #3, 5, 17, 23, 29, 53, 83, 89, 2237, 2297, 3527, 8009, 15137, 24683, 26489,
    #3, 5, 17, 23, 29, 53, 83, 89, 2237, 2297, 3527, 8009, 15137, 24683, 26489, 49193, 50513, 56417, 77573, 93809, 98729, 105953, 202049, 250433
    #3, 5, 17, 23, 29, 53, 83, 89, 113, 149, 173, 197, 257, 353, 419, 449, 593, 617, 677, 683, 947, 1217, 1373, 1409, 1559

     #3, 5, 17, 23, 29, 53, 89, 113, 197, 257, 353, 419, 449, 617, 677, 1217, 1373, 1409, 2003, 2129, 2549, 2663, 2927, 3137, 3329, 3389, 3719, 4733, 6689, 6917, 7547, 8009, 8513, 9857, 10193, 13313, 13553, 14897

    #3, 5, 17, 23, 29, 53, 89, 113, 197, 257, 353, 419, 449, 617, 677, 1217, 1373, 1409, 2003, 2129, 2549, 2663, 2927, 3137, 3329, 3389, 3719, 4733, 6689, 6917, 7547, 8009, 8513, 9857, 10193, 13313, 13553, 14897, 15809, 17837, 18773, 19457, 20483, 21737, 23297, 25169, 27437, 30977, 31769, 34607, 35153, 38039, 40433, 46817,
    #3, 5, 17, 23, 29, 89, 113, 197, 257, 353, 449, 617, 1373, 1409, 2663, 3137, 3389, 7547, 9857, 13553, 30977, 50177, 65537, 68993, 114689, 120737, 130439, 166013, 170369, 268913, 275969, 470597, 495617, 521753, 614657, 913067, 1075649, 1103873

     #3, 5, 17, 23, 29, 53, 83, 89, 113, 149, 197, 257, 353, 419, 449, 593, 617, 677, 683, 1217, 1373, 1409, 1559, 1613, 2003, 2129, 2297, 2357, 2543, 2549, 2663, 2729, 2927, 3137, 3257, 3329, 3389, 3719, 4019, 4733, 5477, 6449, 6689

     #256392945019638502140697184374123648403371443140710905

     3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019, 9923
     #3, 5, 17, 23, 29, 53, 83, 89, 113, 149, 173, 197, 257, 263, 269, 293, 353, 359, 383
);

#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 257, 353, 617, 1409, 2003, 2549, 3137, 9857, 10193, 16073, 68993, 202049, 275969, 1500929, 18386369]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 2003, 2297, 2549, 3137, 9857, 10193, 68993, 88397, 93809, 5850209, 8044037, 18386369]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 2003, 2549, 3137, 4019, 4289, 10193, 16073, 21977, 38459, 50513, 52529, 76649, 93809, 97553]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 1409, 2003, 2549, 8009, 9857, 23297, 50513, 68993, 275969, 375233, 1500929, 3232769, 18386369]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 2003, 2297, 2549, 3137, 9857, 10193, 68993, 88397, 93809, 5850209, 8044037, 18386369, 2059273217]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 1409, 2003, 2297, 2549, 3329, 8009, 10193, 16073, 23297, 50177, 93809, 202049, 275969, 656657, 18386369]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 1409, 2003, 2549, 8009, 9857, 23297, 50513, 68993, 275969, 375233, 1500929, 3232769, 18386369, 1176727553]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019, 7547, 9857, 10193, 16073, 17837, 23297, 68993, 88397, 93809, 202049, 896897, 1500929, 2475089, 8044037, 18386369]


#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019, 9923, 10193, 13553, 308309, 486179, 656657, 896897, 902903, 1500929, 1610753, 1944713, 2063777, 2540033, 2818817, 3232769, 6071297, 18386369, 22629377, 25281257, 31115393, 33020417]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 2003, 2297, 2549, 3137, 9857, 10193, 68993, 88397, 93809, 5850209, 8044037, 18386369]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 2003, 2549, 3137, 4019, 4289, 10193, 16073, 21977, 38459, 50513, 52529, 76649, 93809, 97553]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 1409, 2003, 2549, 8009, 9857, 23297, 50513, 68993, 275969, 375233, 1500929, 3232769, 18386369]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 197, 353, 449, 617, 1409, 2003, 2297, 2549, 3329, 8009, 10193, 16073, 23297, 50177, 93809, 202049, 275969, 656657, 18386369]
#~ [3, 5, 17, 23, 29, 53, 83, 89, 113, 257, 353, 449, 617, 1409, 2003, 2297, 2549, 3137, 3329, 4019, 7547, 9857, 10193, 16073, 17837, 23297, 68993, 88397, 93809, 202049, 896897, 1500929, 2475089, 8044037, 18386369]

#my @prefix = (3, 5, 17, 23, 29);
my $prefix_prod = Math::GMPz->new(vecprod(@prefix));
my $multiplier = lcm(map {$_-1} @prefix);

sub method_1 ($L) {     # smallest numbers first

    (vecall { ($L % ($_-1)) == 0 } @prefix) or return;

    my @P = lambda_primes($L);

    @P = grep {
        (($prefix_prod % $_) == 0)
            ? 1
            : Math::Prime::Util::GMP::gcd(Math::Prime::Util::GMP::totient(Math::Prime::Util::GMP::mulint($prefix_prod, $_)), Math::Prime::Util::GMP::mulint($prefix_prod, $_)) eq '1'
    } @P;

    #vecprodmod(@P, 3*5*17*23) == 0 or return;
    #vecprodmod(\@P, 3*5*17*23*29) == 0 or return;
    if (@prefix) {
        vecprodmod(\@P, $prefix_prod) == 0 or return;
    }
    #return if (vecprod(@P) < ~0);

   # @P = grep { $_ > 257 } @P;

    #@P = grep { $_ > $prefix[-1] } @P;
    @P = grep { gcd($prefix_prod, $_) == 1 } @P;

    say "# Testing: $L -- ", scalar(@P);

    my $n = scalar(@P);
    my @orig = @P;

    my $max = 1e3;
    my $max_k = 30;

    my $L_rem = invmod($prefix_prod, $L);

    foreach my $k (1 .. @P) {
        #next if (binomial($n, $k) > 1e6);

        next if ($k > $max_k);

        @P = @orig;

        my $count = 0;
        forcomb {
            if (vecprodmod([@P[@_]], $L) == $L_rem) {
                say vecprod(@P[@_], $prefix_prod);
            }
            lastfor if (++$count > $max);
        } $n, $k;

        next if (binomial($n, $k) < $max);

        @P = reverse(@P);

        $count = 0;
        forcomb {
            if (vecprodmod([@P[@_]], $L) == $L_rem) {
                say vecprod(@P[@_], $prefix_prod);
            }
            lastfor if (++$count > $max);
        } $n, $k;

        for (1..2) {
        @P = shuffle(@P);

        $count = 0;
        forcomb {
            if (vecprodmod([@P[@_]], $L) == $L_rem) {
                say vecprod(@P[@_], $prefix_prod);
            }
            lastfor if (++$count > $max);
        } $n, $k;
    }
    }

    my $B = vecprodmod([@P, $prefix_prod], $L);
    my $T = Math::GMPz->new(Math::Prime::Util::GMP::vecprod(@P));

    foreach my $k (1 .. @P) {
        #next if (binomial($n, $k) > 1e6);

        last if ($k > $max_k);

        @P = @orig;

        my $count = 0;
        forcomb {
            if (vecprodmod([@P[@_]], $L) == $B) {
                my $S = Math::GMPz->new(Math::Prime::Util::GMP::vecprod(@P[@_]));
                say vecprod($prefix_prod, $T / $S) if ($T != $S);
            }
            lastfor if (++$count > $max);
        } $n, $k;

        next if (binomial($n, $k) < $max);

        @P = reverse(@P);

        $count = 0;
        forcomb {
            if (vecprodmod([@P[@_]], $L) == $B) {
                my $S = Math::GMPz->new(Math::Prime::Util::GMP::vecprod(@P[@_]));
                say vecprod($prefix_prod, $T / $S) if ($T != $S);
            }
            lastfor if (++$count > $max);
        } $n, $k;


        for (1..2) {
        @P = shuffle(@P);

        $count = 0;
        forcomb {
            if (vecprodmod([@P[@_]], $L) == $B) {
                my $S = Math::GMPz->new(Math::Prime::Util::GMP::vecprod(@P[@_]));
                say vecprod($prefix_prod, $T / $S) if ($T != $S);
            }
            lastfor if (++$count > $max);
        } $n, $k;
    }
    }
}

use Math::GMPz;

my %seen;
my $count = 0;

foreach my $n(1..1e4) {
    #method_1($n*3236000768);
    #method_1($n* 3236000768);
    #method_1($n * 18386368);

   # ($n == 1) or gcd($n, $multiplier) > 1 or next;

    method_1(mulint($n, $multiplier));
}