Uniform random number is not Uniform!!!

[Torbjorn Granlund]

rohan <rohan076 at gmail.com> writes:

  I used the unsigned long mpz_urandomm(ran, s, N) function to generate a
  random number. the documentation says that the number generated will be
  uniformly distributed between 0 and n-1 inclusive!!! but after running the
  program several times i realised that for smaller N's the random numbers are
  better distributed in the range 0 to n-1
  I wrote this small piece of code, it basically generates random numbers from
  0 to n-1, where n = z^ex
  ---------------------
  #include <stdio.h>
  #include <gmp.h>
  
  int main(void) {
      gmp_randstate_t s;
      unsigned long seed, i, z,ex;
  mpz_t big;
  mpz_init(big);
  mpz_t store;
  ex = 10;
  z=2;
   mpz_ui_pow_ui(big, z, ex);
   gmp_printf("BIG : %Zd", big);
   gmp_randinit_default(s);
      seed = time(NULL); // system time
      gmp_randseed_ui(s, seed);
     for(i = 0; i < 50; ++i) {
          mpz_init(store);
         mpz_urandomm (store, s, big);
      gmp_printf("\nRAN : %Zd", store);
        }
      gmp_randclear(s);
       return 0;
  }
  -------------------------------------------------------
  now the random numbers were evenly distributed for ex=10, but for ex = 256,
  the random numbers generated were mostly in the range on 2^240  to 2^256 ,
  how can this be called evenly distributed?? am I missing something or this
  has something to do with the seed value.

You're confused.

In the range 0..2^256, half of the numbers actually
lie in the range 2^255..2^256, so about half of the uniformly distributed
random numbers should lie there too.

Only about 1 random number in 65536 will be less than 2^240.

End of this discussion, please.

-- 
Torbjörn