-
Notifications
You must be signed in to change notification settings - Fork 34
/
strong_fermat_pseudoprimes_in_range_mpz.pl
175 lines (123 loc) · 6.48 KB
/
strong_fermat_pseudoprimes_in_range_mpz.pl
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
#!/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 5.036;
use Math::GMPz;
use ntheory qw(:all);
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));
$A = Math::GMPz->new("$A");
$B = Math::GMPz->new("$B");
my $u = Math::GMPz::Rmpz_init();
my $v = Math::GMPz::Rmpz_init();
my %seen;
my @list;
my $generator = sub ($m, $L, $lo, $j, $k_exp, $congr) {
Math::GMPz::Rmpz_tdiv_q($u, $B, $m);
Math::GMPz::Rmpz_root($u, $u, $j);
my $hi = Math::GMPz::Rmpz_get_ui($u);
if ($lo > $hi) {
return;
}
if ($j == 1) {
Math::GMPz::Rmpz_invert($v, $m, $L);
if (Math::GMPz::Rmpz_cmp_ui($v, $hi) > 0) {
return;
}
if (Math::GMPz::Rmpz_fits_ulong_p($L)) {
$L = Math::GMPz::Rmpz_get_ui($L);
}
my $t = Math::GMPz::Rmpz_get_ui($v);
$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 Math::GMPz::Rmpz_gcd_ui($Math::GMPz::NULL, $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 % $p) or next;
Math::GMPz::Rmpz_mul_ui($v, $m, $p);
if ($k == 1 and is_prime($p) and Math::GMPz::Rmpz_cmp_ui($m, 1) == 0) {
## ok
}
elsif (Math::GMPz::Rmpz_cmp($v, $A) >= 0) {
Math::GMPz::Rmpz_sub_ui($u, $v, 1);
if (Math::GMPz::Rmpz_divisible_ui_p($u, znorder($base, $p))) {
push(@list, Math::GMPz::Rmpz_init_set($v)) if !$seen{Math::GMPz::Rmpz_get_str($v, 10)}++;
}
}
}
}
return;
}
my $u = Math::GMPz::Rmpz_init();
my $v = Math::GMPz::Rmpz_init();
my $lcm = Math::GMPz::Rmpz_init();
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, $p);
Math::GMPz::Rmpz_gcd_ui($Math::GMPz::NULL, $m, $z) == 1 or next;
Math::GMPz::Rmpz_lcm_ui($lcm, $L, $z);
Math::GMPz::Rmpz_set_ui($u, $p);
for (Math::GMPz::Rmpz_mul_ui($v, $m, $p) ; Math::GMPz::Rmpz_cmp($v, $B) <= 0 ; Math::GMPz::Rmpz_mul_ui($v, $v, $p)) {
__SUB__->($v, $lcm, $p + 1, $j - 1, $k_exp, $congr);
Math::GMPz::Rmpz_mul_ui($u, $u, $p);
powmod($base, $z, $u) == 1 or last;
}
}
};
# Case where 2^d == 1 (mod p), where d is the odd part of p-1.
$generator->(Math::GMPz->new(1), Math::GMPz->new(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->(Math::GMPz->new(1), Math::GMPz->new(1), 2, $k, $v, -1);
}
return sort { $a <=> $b } @list;
}
# Generate all the strong 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) * 4;
my $hi = mulint($lo, 1000);
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__
121, 703, 1891, 3281, 8401, 8911, 10585, 12403, 16531, 18721, 19345, 23521, 31621, 44287, 47197, 55969, 63139, 74593, 79003, 82513, 87913, 88573, 97567