A059772 Smallest prime p such that n is a solution mod p of x^2 = 2, or 0 if no such prime exists.

0, 7, 7, 23, 17, 47, 31, 79, 0, 17, 71, 167, 97, 223, 127, 41, 23, 359, 199, 439, 241, 31, 41, 89, 337, 727, 0, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103, 0, 47, 73, 881, 1847, 967, 0, 151, 2207, 1151, 2399, 1249, 113, 193, 401, 0, 3023, 1567, 191, 0, 71

Offset: 2

Author

Klaus Brockhaus, Feb 21 2001

Keywords

• nonn

Comments

Solutions mod p are represented by integers from 0 to p-1. The following equivalences hold for n > 1: There is a prime p such that n is a solution mod p of x^2 = 2 iff n^2-2 has a prime factor > n; n is a solution mod p of x^2 = 2 iff p is a prime factor of n^2-2 and p > n. n^2-2 has at most one prime factor > n, consequently such a factor is the only prime p such that n is a solution mod p of x^2 = 2. For n such that n^2-2 has no prime factor > n (the zeros in the sequence), cf. A060515.

Examples

			a(11) = 17, since 11 is a solution mod 17 of x^2 = 2 and 11 is not a solution mod p of x^2 = 2 for primes p < 17. Although 11^2 = 2 mod 7, prime 7 is excluded because 7 < 11 and 11 = 4 mod 7.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A038873, A059770, A059771, A060515.

Programs

Maple

f:= proc(n) local P;
  P:= select(`>`,numtheory:-factorset(n^2-2),n);
  if P = {} then 0 else min(P) fi
end proc:
map(f, [$2..100]); # Robert Israel, Feb 23 2016

Mathematica

a[n_] := Module[{P}, P = Select[FactorInteger[n^2 - 2][[All, 1]], # > n&]; If[P == {}, 0, Min[P]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Apr 10 2019, from Maple *)

Formula

If n^2-2 has a (unique) prime factor p > n, then a(n) = p, else a(n) = 0.

Extensions

Offset corrected by R. J. Mathar, Aug 21 2009