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strong_fermat_pseudoprimes_in_range.pl
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strong_fermat_pseudoprimes_in_range.pl
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#!/usr/bin/perl
# Daniel "Trizen" Șuteu
# Date: 24 September 2022
# https://github.com/trizen
# Generate all the k-omega strong Fermat pseudoprimes in range [A,B]. (not in sorted order)
# Definition:
# k-omega primes are numbers n such that omega(n) = k.
# See also:
# https://en.wikipedia.org/wiki/Almost_prime
# https://en.wikipedia.org/wiki/Prime_omega_function
# https://trizenx.blogspot.com/2020/08/pseudoprimes-construction-methods-and.html
=for comment
# PARI/GP program (slow):
strong_fermat_psp(A, B, k, base) = A=max(A, vecprod(primes(k))); (f(m, l, p, j, k_exp, congr) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), if(base%q != 0, my(tv=valuation(q-1, 2)); if(tv > k_exp && Mod(base, q)^(((q-1)>>tv)<<k_exp) == congr, my(v=m*q, t=q, r=nextprime(q+1)); while(v <= B, my(L=lcm(l, znorder(Mod(base, t)))); if(gcd(L, v) == 1, if(j==1, if(v>=A && if(k==1, !isprime(v), 1) && (v-1)%L == 0, listput(list, v)), if(v*r <= B, list=concat(list, f(v, L, r, j-1, k_exp, congr)))), break); v *= q; t *= q)))); list); my(r=f(1, 1, 2, k, 0, 1)); for(v=0, logint(B, 2), r=concat(r, f(1, 1, 2, k, v, -1))); vecsort(Vec(r));
# PARI/GP program (fast):
strong_check(p, base, e, r) = my(tv=valuation(p-1, 2)); tv > e && Mod(base, p)^((p-1)>>(tv-e)) == r;
strong_fermat_psp(A, B, k, base) = A=max(A, vecprod(primes(k))); (f(m, l, lo, k, e, r) = my(list=List()); my(hi=sqrtnint(B\m, k)); if(lo > hi, return(list)); if(k==1, forstep(p=lift(1/Mod(m, l)), hi, l, if(isprimepower(p) && gcd(m*base, p) == 1 && strong_check(p, base, e, r), my(n=m*p); if(n >= A && (n-1) % znorder(Mod(base, p)) == 0, listput(list, n)))), forprime(p=lo, hi, base%p == 0 && next; strong_check(p, base, e, r) || next; my(z=znorder(Mod(base, p))); gcd(m,z) == 1 || next; my(q=p, v=m*p); while(v <= B, list=concat(list, f(v, lcm(l, z), p+1, k-1, e, r)); q *= p; Mod(base, q)^z == 1 || break; v *= p))); list); my(res=f(1, 1, 2, k, 0, 1)); for(v=0, logint(B, 2), res=concat(res, f(1, 1, 2, k, v, -1))); vecsort(Set(res));
=cut
use 5.020;
use warnings;
use ntheory qw(:all);
use experimental qw(signatures);
sub divceil ($x, $y) { # ceil(x/y)
(($x % $y == 0) ? 0 : 1) + divint($x, $y);
}
sub strong_fermat_pseudoprimes_in_range ($A, $B, $k, $base) {
$A = vecmax($A, pn_primorial($k));
$A > $B and return;
my %seen;
my @list;
my $generator = sub ($m, $L, $lo, $j, $k_exp, $congr) {
my $hi = rootint(divint($B, $m), $j);
if ($lo > $hi) {
return;
}
if ($j == 1) {
if ($L == 1) { # optimization
foreach my $p (@{primes($lo, $hi)}) {
$base % $p == 0 and next;
my $val = valuation($p - 1, 2);
$val > $k_exp or next;
powmod($base, ($p - 1) >> ($val - $k_exp), $p) == ($congr % $p) or next;
for (my $v = (($m == 1) ? ($p * $p) : ($m * $p)) ; $v <= $B ; $v *= $p) {
$v >= $A or next;
powmod($base, $v - 1, $v) == 1 or last;
push(@list, $v) if !$seen{$v}++;
}
}
return;
}
my $t = invmod($m, $L);
$t > $hi && return;
$t += $L * divceil($lo - $t, $L) if ($t < $lo);
for (my $p = $t ; $p <= $hi ; $p += $L) {
if (is_prime_power($p) and gcd($m, $p) == 1 and gcd($base, $p) == 1) {
my $val = valuation($p - 1, 2);
$val > $k_exp or next;
powmod($base, ($p - 1) >> ($val - $k_exp), $p) == ($congr % $p) or next;
my $v = $m * $p;
$v >= $A or next;
($v - 1) % znorder($base, $p) == 0 or next;
push(@list, $v) if !$seen{$v}++;
}
}
return;
}
foreach my $p (@{primes($lo, $hi)}) {
$base % $p == 0 and next;
my $val = valuation($p - 1, 2);
$val > $k_exp or next;
powmod($base, ($p - 1) >> ($val - $k_exp), $p) == ($congr % $p) or next;
my $z = znorder($base, $p);
gcd($m, $z) == 1 or next;
for (my ($q, $v) = ($p, $m * $p) ; $v <= $B ; ($q, $v) = ($q * $p, $v * $p)) {
if ($q > $p) {
powmod($base, $z, $q) == 1 or last;
}
__SUB__->($v, lcm($L, $z), $p + 1, $j - 1, $k_exp, $congr);
}
}
};
# Case where 2^d == 1 (mod p), where d is the odd part of p-1.
$generator->(1, 1, 2, $k, 0, 1);
# Cases where 2^(d * 2^v) == -1 (mod p), for some v >= 0.
foreach my $v (0 .. logint($B, 2)) {
$generator->(1, 1, 2, $k, $v, -1);
}
return sort { $a <=> $b } @list;
}
# Generate all the Fermat pseudoprimes to base 3 in range [1, 10^5]
my $from = 1;
my $upto = 1e5;
my $base = 3;
my @arr;
foreach my $k (1 .. 100) {
last if pn_primorial($k) > $upto;
push @arr, strong_fermat_pseudoprimes_in_range($from, $upto, $k, $base);
}
say join(', ', sort { $a <=> $b } @arr);
# Run some tests
if (0) { # true to run some tests
foreach my $k (1 .. 5) {
say "Testing k = $k";
my $lo = pn_primorial($k);
my $hi = mulint($lo, 10000);
my $omega_primes = omega_primes($k, $lo, $hi);
foreach my $base (2 .. 100) {
my @this = grep { is_strong_pseudoprime($_, $base) and !is_prime($_) } @$omega_primes;
my @that = strong_fermat_pseudoprimes_in_range($lo, $hi, $k, $base);
join(' ', @this) eq join(' ', @that)
or die "Error for k = $k and base = $base with hi = $hi\n(@this) != (@that)";
}
}
}
__END__
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