A003625 Primes congruent to {3, 5, 6} mod 7.

3, 5, 13, 17, 19, 31, 41, 47, 59, 61, 73, 83, 89, 97, 101, 103, 131, 139, 157, 167, 173, 181, 199, 223, 227, 229, 241, 251, 257, 269, 271, 283, 293, 307, 311, 313, 349, 353, 367, 383, 397, 409, 419, 433, 439, 461, 467, 479, 503, 509, 521, 523, 563, 577, 587, 593

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn

Comments

Inert rational primes in Q(sqrt(-7)).

For terms >= 13, sequence consists of primes p such that Sum_{k=0..p} binomial(2*k,k)^3 == 8 (mod p). - Benoit Cloitre, Aug 10 2003

Primes which cannot be written in the form a^2 + 7*b^2, where a >= 0, b >= 0. - V. Raman, Sep 08 2012

Conjecture: Also such primes p where the polynomial x^2 + x + 2 is irreducible over GF(p). - Federico Provvedi, Jul 21 2018

Primes that have -7 as a quadratic nonresidue, or equivalently, primes that are quadratic nonresidues modulo 7. - Jianing Song, Jul 21 2018

References

• H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 498.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index to sequences related to decomposition of primes in quadratic fields

Programs

Magma

[p: p in PrimesUpTo(1000) | p mod 7 in [3, 5, 6]]; // Vincenzo Librandi, Aug 04 2012

Mathematica

Select[Prime[Range[800]],MemberQ[{3, 5, 6},Mod[#,7]]&] (* Vincenzo Librandi, Aug 04 2012 *)

PARI

{a(n) = local( cnt, m ); if( n<1, return( 0 )); while( cnt < n, if( isprime( m++) && kronecker( -7, m )==-1, cnt++ )); m} /* Michael Somos, Aug 14 2012 */