A191088 Least number k such that the first n primes have Kronecker symbol (p|k) = -1.

3, 5, 43, 43, 67, 67, 163, 163, 163, 163, 163, 163, 74093, 77683, 214037, 214037, 214037, 2004917, 36254563, 51599563, 96295483, 96295483, 114148483, 163520117, 163520117, 261153653, 261153653, 585811843

Offset: 1

Author

T. D. Noe, May 25 2011

Keywords

• nonn

Comments

This implies, but is not equivalent to, that the first n primes are quadratic non-residues mod k. [Corrected by M. F. Hasler, Jan 18 2016]

The first 13 terms are prime. Sequence A191089 is similar, but forces k to be prime.

In case k is restricted to be a prime, the condition (p|k) = -1 becomes equivalent to saying that p is a quadratic non-residue (mod k). - M. F. Hasler, Jan 18 2016

Examples

			a(1) = 3 is the least number k such that prime(1) = 2 is not a square mod k (since for k=1 and k=2, p=2 would be zero, thus a square, mod k).
a(2) = 4 is the least number k such that prime(1) = 2 and prime(2) = 3 are not squares mod k (the only squares mod 4 are 0 = 0^1 = 2^2 and 1 = 1^2 = 3^3 (mod 4); while for k=2, p=2 would be zero, thus a square, and for k=3 this would be the case for p=3).
		

Crossrefs

Cf. A191089.

Programs

Mathematica

Table[k = 2; While[Length[Select[Prime[Range[n]], JacobiSymbol[#, k] == -1 &]] < n, k++]; k, {n, 15}]

PARI

q=2;for(k=3,1e9,forprime(p=2,q,if(kronecker(p,k)>=0,next(2)));print1(k", ");q=nextprime(q+1);k--) \\ Charles R Greathouse IV, Oct 10 2011

Extensions

a(16)-a(28) from Charles R Greathouse IV, Oct 10 2011