A192578 Primes p for which there is no prime q == 1 (mod 4) that is smaller than p and is a quadratic residue modulo p.

2, 3, 5, 7, 13, 37

Offset: 1

Author

Jonathan Sondow, Jul 04 2011

Keywords

• nonn
• fini
• full

Comments

Gica proved that if p is a prime different from 2, 3, 5, 7, 13, 37, then there exists a prime q < p which is a quadratic residue modulo p and q == 1 (mod 4).

Examples

			p = 7 is a member, because the only prime q < p with q == 1 (mod 4) is q = 5, but 5 is not a quadratic residue modulo 7.
11 is not a member, because 11 > 5 == 1 (mod 4) and 5 == 4^2 (mod 11).
		

References

• A. Gica, Quadratic residues of certain types, Rocky Mt. J. Math. 36 (2006), 1867-1871.

Links

• A. Gica, Quadratic residues of certain types, Journées Arithmétiques 2011.

Crossrefs

Cf. A192579.

Programs

Mathematica

maxPrimePi = 100; nextPrime[q_] := If[q1 = NextPrime[q]; Mod[q1, 4] != 1, nextPrime[q1], q1]; foundQ[p_] := (q = 2; found = False; While[q < p, q = nextPrime[q]; If[JacobiSymbol[q, p] == 1, found = True; Break[]]]; found); Select[ Prime[ Range[ maxPrimePi]], Not[foundQ[#]] &] (* Jean-François Alcover, Oct 18 2011 *)