qcn.c

/* Use mpz_kronecker_ui() to calculate an estimate for the quadratic
   class number h(d), for a given negative fundamental discriminant, using
   Dirichlet's analytic formula.

Copyright 1999, 2000, 2001, 2002 Free Software Foundation, Inc.

This file is part of the GNU MP Library.

This program is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the Free
Software Foundation; either version 2 of the License, or (at your option)
any later version.

This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License for
more details.

You should have received a copy of the GNU General Public License along with
this program; if not, write to the Free Software Foundation, Inc., 51 Franklin
Street, Fifth Floor, Boston, MA 02110-1301, USA. */


/* Usage: qcn [-p limit] <discriminant>...

   A fundamental discriminant means one of the form D or 4*D with D
   square-free.  Each argument is checked to see it's congruent to 0 or 1
   mod 4 (as all discriminants must be), and that it's negative, but there's
   no check on D being square-free.

   This program is a bit of a toy, there are better methods for calculating
   the class number and class group structure.

   Reference:

   Daniel Shanks, "Class Number, A Theory of Factorization, and Genera",
   Proc. Symp. Pure Math., vol 20, 1970, pages 415-440.

*/

#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>

#include "gmp.h"

#ifndef M_PI
#define M_PI  3.14159265358979323846
#endif


/* A simple but slow primality test.  */
int
prime_p (unsigned long n)
{
  unsigned long  i, limit;

  if (n == 2)
    return 1;
  if (n < 2 || !(n&1))
    return 0;

  limit = (unsigned long) floor (sqrt ((double) n));
  for (i = 3; i <= limit; i+=2)
    if ((n % i) == 0)
      return 0;

  return 1;
}


/* The formula is as follows, with d < 0.

               w * sqrt(-d)      inf      p
        h(d) = ------------ *  product --------
                  2 * pi         p=2   p - (d/p)
                             

   (d/p) is the Kronecker symbol and the product is over primes p.  w is 6
   when d=-3, 4 when d=-4, or 2 otherwise.
  
   Calculating the product up to p=infinity would take a long time, so for
   the estimate primes up to 132,000 are used.  Shanks found this giving an
   accuracy of about 1 part in 1000, in normal cases.  */

unsigned long  p_limit = 132000;

double
qcn_estimate (mpz_t d)
{
  double  h;
  unsigned long  p;

  /* p=2 */
  h = sqrt (-mpz_get_d (d)) / M_PI
    * 2.0 / (2.0 - mpz_kronecker_ui (d, 2));

  if (mpz_cmp_si (d, -3) == 0)       h *= 3;
  else if (mpz_cmp_si (d, -4) == 0)  h *= 2;

  for (p = 3; p <= p_limit; p += 2)
    if (prime_p (p))
      h *= (double) p / (double) (p - mpz_kronecker_ui (d, p));

  return h;
}


void
qcn_str (char *num)
{
  mpz_t  z;

  mpz_init_set_str (z, num, 0);

  if (mpz_sgn (z) >= 0)
    {
      mpz_out_str (stdout, 0, z);
      printf (" is not supported (negatives only)\n");
    }
  else if (mpz_fdiv_ui (z, 4) != 0 && mpz_fdiv_ui (z, 4) != 1)
    {
      mpz_out_str (stdout, 0, z);
      printf (" is not a discriminant (must == 0 or 1 mod 4)\n");
    }
  else
    {
      printf ("h(");
      mpz_out_str (stdout, 0, z);
      printf (") approx %.1f\n", qcn_estimate (z));
    }
  mpz_clear (z);
}


int
main (int argc, char *argv[])
{
  int  i;
  int  saw_number = 0;

  for (i = 1; i < argc; i++)
    {
      if (strcmp (argv[i], "-p") == 0)
        {
          i++;
          if (i >= argc)
            {
              fprintf (stderr, "Missing argument to -p\n");
              exit (1);
            }
          p_limit = atoi (argv[i]);
        }
      else
        {
          qcn_str (argv[i]);
          saw_number = 1;
        }
    }

  if (! saw_number)
    {
      /* some default output */
      qcn_str ("-85702502803");           /* is 16259   */
      qcn_str ("-328878692999");          /* is 1499699 */
      qcn_str ("-928185925902146563");    /* is 52739552 */
      qcn_str ("-84148631888752647283");  /* is 496652272 */
      return 0;
    }

  return 0;
}