|
Packit |
5c3484 |
/* Use mpz_kronecker_ui() to calculate an estimate for the quadratic
|
|
Packit |
5c3484 |
class number h(d), for a given negative fundamental discriminant, using
|
|
Packit |
5c3484 |
Dirichlet's analytic formula.
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
Copyright 1999-2002 Free Software Foundation, Inc.
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
This file is part of the GNU MP Library.
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
This program is free software; you can redistribute it and/or modify it
|
|
Packit |
5c3484 |
under the terms of the GNU General Public License as published by the Free
|
|
Packit |
5c3484 |
Software Foundation; either version 3 of the License, or (at your option)
|
|
Packit |
5c3484 |
any later version.
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
This program is distributed in the hope that it will be useful, but WITHOUT
|
|
Packit |
5c3484 |
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
|
|
Packit |
5c3484 |
FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
|
|
Packit |
5c3484 |
more details.
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
You should have received a copy of the GNU General Public License along with
|
|
Packit |
5c3484 |
this program. If not, see https://www.gnu.org/licenses/. */
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
/* Usage: qcn [-p limit] <discriminant>...
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
A fundamental discriminant means one of the form D or 4*D with D
|
|
Packit |
5c3484 |
square-free. Each argument is checked to see it's congruent to 0 or 1
|
|
Packit |
5c3484 |
mod 4 (as all discriminants must be), and that it's negative, but there's
|
|
Packit |
5c3484 |
no check on D being square-free.
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
This program is a bit of a toy, there are better methods for calculating
|
|
Packit |
5c3484 |
the class number and class group structure.
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
Reference:
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
Daniel Shanks, "Class Number, A Theory of Factorization, and Genera",
|
|
Packit |
5c3484 |
Proc. Symp. Pure Math., vol 20, 1970, pages 415-440.
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
*/
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
#include <math.h>
|
|
Packit |
5c3484 |
#include <stdio.h>
|
|
Packit |
5c3484 |
#include <stdlib.h>
|
|
Packit |
5c3484 |
#include <string.h>
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
#include "gmp.h"
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
#ifndef M_PI
|
|
Packit |
5c3484 |
#define M_PI 3.14159265358979323846
|
|
Packit |
5c3484 |
#endif
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
/* A simple but slow primality test. */
|
|
Packit |
5c3484 |
int
|
|
Packit |
5c3484 |
prime_p (unsigned long n)
|
|
Packit |
5c3484 |
{
|
|
Packit |
5c3484 |
unsigned long i, limit;
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
if (n == 2)
|
|
Packit |
5c3484 |
return 1;
|
|
Packit |
5c3484 |
if (n < 2 || !(n&1))
|
|
Packit |
5c3484 |
return 0;
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
limit = (unsigned long) floor (sqrt ((double) n));
|
|
Packit |
5c3484 |
for (i = 3; i <= limit; i+=2)
|
|
Packit |
5c3484 |
if ((n % i) == 0)
|
|
Packit |
5c3484 |
return 0;
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
return 1;
|
|
Packit |
5c3484 |
}
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
/* The formula is as follows, with d < 0.
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
w * sqrt(-d) inf p
|
|
Packit |
5c3484 |
h(d) = ------------ * product --------
|
|
Packit |
5c3484 |
2 * pi p=2 p - (d/p)
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
(d/p) is the Kronecker symbol and the product is over primes p. w is 6
|
|
Packit |
5c3484 |
when d=-3, 4 when d=-4, or 2 otherwise.
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
Calculating the product up to p=infinity would take a long time, so for
|
|
Packit |
5c3484 |
the estimate primes up to 132,000 are used. Shanks found this giving an
|
|
Packit |
5c3484 |
accuracy of about 1 part in 1000, in normal cases. */
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
unsigned long p_limit = 132000;
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
double
|
|
Packit |
5c3484 |
qcn_estimate (mpz_t d)
|
|
Packit |
5c3484 |
{
|
|
Packit |
5c3484 |
double h;
|
|
Packit |
5c3484 |
unsigned long p;
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
/* p=2 */
|
|
Packit |
5c3484 |
h = sqrt (-mpz_get_d (d)) / M_PI
|
|
Packit |
5c3484 |
* 2.0 / (2.0 - mpz_kronecker_ui (d, 2));
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
if (mpz_cmp_si (d, -3) == 0) h *= 3;
|
|
Packit |
5c3484 |
else if (mpz_cmp_si (d, -4) == 0) h *= 2;
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
for (p = 3; p <= p_limit; p += 2)
|
|
Packit |
5c3484 |
if (prime_p (p))
|
|
Packit |
5c3484 |
h *= (double) p / (double) (p - mpz_kronecker_ui (d, p));
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
return h;
|
|
Packit |
5c3484 |
}
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
void
|
|
Packit |
5c3484 |
qcn_str (char *num)
|
|
Packit |
5c3484 |
{
|
|
Packit |
5c3484 |
mpz_t z;
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
mpz_init_set_str (z, num, 0);
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
if (mpz_sgn (z) >= 0)
|
|
Packit |
5c3484 |
{
|
|
Packit |
5c3484 |
mpz_out_str (stdout, 0, z);
|
|
Packit |
5c3484 |
printf (" is not supported (negatives only)\n");
|
|
Packit |
5c3484 |
}
|
|
Packit |
5c3484 |
else if (mpz_fdiv_ui (z, 4) != 0 && mpz_fdiv_ui (z, 4) != 1)
|
|
Packit |
5c3484 |
{
|
|
Packit |
5c3484 |
mpz_out_str (stdout, 0, z);
|
|
Packit |
5c3484 |
printf (" is not a discriminant (must == 0 or 1 mod 4)\n");
|
|
Packit |
5c3484 |
}
|
|
Packit |
5c3484 |
else
|
|
Packit |
5c3484 |
{
|
|
Packit |
5c3484 |
printf ("h(");
|
|
Packit |
5c3484 |
mpz_out_str (stdout, 0, z);
|
|
Packit |
5c3484 |
printf (") approx %.1f\n", qcn_estimate (z));
|
|
Packit |
5c3484 |
}
|
|
Packit |
5c3484 |
mpz_clear (z);
|
|
Packit |
5c3484 |
}
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
int
|
|
Packit |
5c3484 |
main (int argc, char *argv[])
|
|
Packit |
5c3484 |
{
|
|
Packit |
5c3484 |
int i;
|
|
Packit |
5c3484 |
int saw_number = 0;
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
for (i = 1; i < argc; i++)
|
|
Packit |
5c3484 |
{
|
|
Packit |
5c3484 |
if (strcmp (argv[i], "-p") == 0)
|
|
Packit |
5c3484 |
{
|
|
Packit |
5c3484 |
i++;
|
|
Packit |
5c3484 |
if (i >= argc)
|
|
Packit |
5c3484 |
{
|
|
Packit |
5c3484 |
fprintf (stderr, "Missing argument to -p\n");
|
|
Packit |
5c3484 |
exit (1);
|
|
Packit |
5c3484 |
}
|
|
Packit |
5c3484 |
p_limit = atoi (argv[i]);
|
|
Packit |
5c3484 |
}
|
|
Packit |
5c3484 |
else
|
|
Packit |
5c3484 |
{
|
|
Packit |
5c3484 |
qcn_str (argv[i]);
|
|
Packit |
5c3484 |
saw_number = 1;
|
|
Packit |
5c3484 |
}
|
|
Packit |
5c3484 |
}
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
if (! saw_number)
|
|
Packit |
5c3484 |
{
|
|
Packit |
5c3484 |
/* some default output */
|
|
Packit |
5c3484 |
qcn_str ("-85702502803"); /* is 16259 */
|
|
Packit |
5c3484 |
qcn_str ("-328878692999"); /* is 1499699 */
|
|
Packit |
5c3484 |
qcn_str ("-928185925902146563"); /* is 52739552 */
|
|
Packit |
5c3484 |
qcn_str ("-84148631888752647283"); /* is 496652272 */
|
|
Packit |
5c3484 |
return 0;
|
|
Packit |
5c3484 |
}
|
|
Packit |
5c3484 |
|
|
Packit |
5c3484 |
return 0;
|
|
Packit |
5c3484 |
}
|