174 lines
4.2 KiB
C
174 lines
4.2 KiB
C
/* Use mpz_kronecker_ui() to calculate an estimate for the quadratic
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class number h(d), for a given negative fundamental discriminant, using
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Dirichlet's analytic formula.
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Copyright 1999, 2000, 2001, 2002 Free Software Foundation, Inc.
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This file is part of the GNU MP Library.
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This program is free software; you can redistribute it and/or modify it
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under the terms of the GNU General Public License as published by the Free
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Software Foundation; either version 2 of the License, or (at your option)
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any later version.
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This program is distributed in the hope that it will be useful, but WITHOUT
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ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
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more details.
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You should have received a copy of the GNU General Public License along with
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this program; if not, write to the Free Software Foundation, Inc., 51 Franklin
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Street, Fifth Floor, Boston, MA 02110-1301, USA. */
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/* Usage: qcn [-p limit] <discriminant>...
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A fundamental discriminant means one of the form D or 4*D with D
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square-free. Each argument is checked to see it's congruent to 0 or 1
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mod 4 (as all discriminants must be), and that it's negative, but there's
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no check on D being square-free.
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This program is a bit of a toy, there are better methods for calculating
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the class number and class group structure.
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Reference:
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Daniel Shanks, "Class Number, A Theory of Factorization, and Genera",
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Proc. Symp. Pure Math., vol 20, 1970, pages 415-440.
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*/
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#include <math.h>
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#include <stdio.h>
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#include <stdlib.h>
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#include <string.h>
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#include "gmp.h"
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#ifndef M_PI
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#define M_PI 3.14159265358979323846
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#endif
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/* A simple but slow primality test. */
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int
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prime_p (unsigned long n)
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{
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unsigned long i, limit;
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if (n == 2)
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return 1;
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if (n < 2 || !(n&1))
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return 0;
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limit = (unsigned long) floor (sqrt ((double) n));
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for (i = 3; i <= limit; i+=2)
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if ((n % i) == 0)
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return 0;
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return 1;
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}
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/* The formula is as follows, with d < 0.
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w * sqrt(-d) inf p
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h(d) = ------------ * product --------
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2 * pi p=2 p - (d/p)
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(d/p) is the Kronecker symbol and the product is over primes p. w is 6
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when d=-3, 4 when d=-4, or 2 otherwise.
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Calculating the product up to p=infinity would take a long time, so for
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the estimate primes up to 132,000 are used. Shanks found this giving an
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accuracy of about 1 part in 1000, in normal cases. */
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unsigned long p_limit = 132000;
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double
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qcn_estimate (mpz_t d)
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{
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double h;
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unsigned long p;
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/* p=2 */
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h = sqrt (-mpz_get_d (d)) / M_PI
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* 2.0 / (2.0 - mpz_kronecker_ui (d, 2));
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if (mpz_cmp_si (d, -3) == 0) h *= 3;
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else if (mpz_cmp_si (d, -4) == 0) h *= 2;
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for (p = 3; p <= p_limit; p += 2)
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if (prime_p (p))
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h *= (double) p / (double) (p - mpz_kronecker_ui (d, p));
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return h;
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}
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void
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qcn_str (char *num)
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{
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mpz_t z;
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mpz_init_set_str (z, num, 0);
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if (mpz_sgn (z) >= 0)
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{
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mpz_out_str (stdout, 0, z);
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printf (" is not supported (negatives only)\n");
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}
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else if (mpz_fdiv_ui (z, 4) != 0 && mpz_fdiv_ui (z, 4) != 1)
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{
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mpz_out_str (stdout, 0, z);
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printf (" is not a discriminant (must == 0 or 1 mod 4)\n");
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}
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else
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{
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printf ("h(");
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mpz_out_str (stdout, 0, z);
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printf (") approx %.1f\n", qcn_estimate (z));
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}
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mpz_clear (z);
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}
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int
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main (int argc, char *argv[])
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{
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int i;
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int saw_number = 0;
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for (i = 1; i < argc; i++)
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{
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if (strcmp (argv[i], "-p") == 0)
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{
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i++;
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if (i >= argc)
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{
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fprintf (stderr, "Missing argument to -p\n");
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exit (1);
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}
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p_limit = atoi (argv[i]);
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}
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else
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{
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qcn_str (argv[i]);
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saw_number = 1;
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}
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}
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if (! saw_number)
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{
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/* some default output */
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qcn_str ("-85702502803"); /* is 16259 */
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qcn_str ("-328878692999"); /* is 1499699 */
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qcn_str ("-928185925902146563"); /* is 52739552 */
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qcn_str ("-84148631888752647283"); /* is 496652272 */
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return 0;
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}
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return 0;
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}
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