A275939 Consider the prime race mod q (where q >= 2) between q*k+1 and q*k-1. Terms are numbers k where q*k+1 first takes lead over q*k-1.

3, 608981813029, 26861, 11, 608981813017, 71, 192252423729713, 37, 11, 23

Offset: 2

Author

Andy Martin, Aug 12 2016

Keywords

• nonn
• more

Comments

Values are available for all 2 <= q <= 999 except for 12 and 24. If q is odd and > 3 then 2*q will have the same value in the sequence as q.

Additional terms starting with q = 12 are:

unknown, 53, 71, 331, 17, 239, 37, 213827, 1381, 673, 23, 47, unknown, 101, 53, 379, 29, 59, 331

The longest q*k+1 versus q*k-1 races up to q = 999 are for q = 3,6,8,12,24 and 168. When q = 168 the race ends at prime 273084304417.

The mod 12 and 24 races were checked by computer to 1.1 * 10^14 without q*k+1 ever leading.

Kevin Ford (private communication) provides the following information on these races: "My paper with Richard Hudson contains a lot of information about the location of sign changes for pi(x,q,a)-pi(x,q,b). Corollary 4 has rigorous upper bounds, but these will likely not be useful to you. The information in Tables 2 and 3 will be more helpful, as these provide the most likely places to look for the first sign change. In the case of the mod 12 race, it is probably around exp(187.536), or about 2.79 x 10^{81}. For the mod 24 race, it's about exp(43.453)=7.437... x 10^{18}".

Examples

			For the fourth term q is 5. For primes 2,3,5 and 7 the mod 5 values are 2,3,0 and 2 respectively, so there is no change in the race. For the next prime 11, mod 5 gives 1, q*k+1 now leads 1 to 0, and the race is over.
		

References

• Ford, Kevin; Konyagin, Sergei; Chebyshev's conjecture and the prime number race. IV International Conference "Modern Problems of Number Theory and its Applications": Current Problems, Part II (Russian) (Tula, 2001), 67-91.
• Paulo Ribenboim, The Little Book of Big Primes, Springer 1991

Links

• Kevin Ford and Richard H. Hudson, Sign changes in pi q,a(x) - pi q,b(x), Acta Arithmetica 100 (2001), 297-314.
• Kevin Ford and Sergei Konyagin, Chebyshev's conjecture and the prime number race
• Andrew Granville and Greg Martin, Prime Number Races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
• Andy Martin, Values for 2 to 999 as far as they are known

Crossrefs

Cf. A007350, A007352.

Programs

C

/*
C language program used to investigate prime number races.
Computes the first lead of qn+1 over qn-1 for q from 2 to 999.
By Andy Martin [e-mail] 8/12/2016.
Requires Kim Walisch's primesieve library from http://primesieve.org
Iteration based on the primesieve_iterator.c example.
*/
#include 
#include 
#include 
#define UPDATE_COUNT 10000000000ull
void race(uint64_t q)
{
  uint64_t prime  = 0;
  uint64_t m1     = 0;
  uint64_t m_1    = 0;
  uint64_t rem    = 0;
  uint64_t update = UPDATE_COUNT;
  primesieve_iterator pi;
  primesieve_init(&pi);
  while (prime = primesieve_next_prime(&pi)) {
    if ((rem = prime % q) == 1){
      m1 += 1;
    } else if (rem == q-1) {
      m_1 += 1;
    }
    if (m1 > m_1){
      printf("Race mod %3llu ends at %12llu with %11llu pi(x;%llu,1) and %11llu pi(x;%llu,%llu)\n",
             q, prime, m1, q, m_1, q, q-1);
      break;
    }
    /* Enable for update on long races where q = 3,6,8,12,24,168 */
    if (prime > update) {
      printf("  Race mod %llu ongoing at prime %llu with m1 %llu and m_1 %llu diff: %llu\n",
             q, prime, m1, m_1, m_1 - m1);
      update += UPDATE_COUNT;
    }
  }
  primesieve_free_iterator(&pi);
}
int main()
{
  uint64_t i;
  for(i=2; i<1000; i++){ race(i); }
  return(0);
}

Extensions

a(8)-a(11) from Andy Martin, Aug 15 2016