A344621 Primes p such that there is no positive integer a with 2*a, a^2-1 and a^2+1 not only smaller than p but also quadratic nonresidues modulo p.

2, 3, 5, 7, 13, 17, 19, 23, 31, 41, 43, 47, 67, 71, 73, 97, 101, 127, 151, 157, 167, 191, 199, 239, 257, 311, 313, 367, 409, 439, 479, 521, 587, 599, 739, 839, 887, 1031, 1063, 1151, 1319, 2351, 2999, 3119

Offset: 1

Author

Zhi-Wei Sun, May 24 2021

Keywords

• nonn

Comments

No more terms below 10^10.

Conjecture: No term is greater than 3119. In other words, for any prime p > 3120, there is a Pythagorean triple (2*a,a^2-1,a^2+1) with 2*a, a^2-1 and a^2+1 in the set {0 < r < p: r is a quadratic nonresidue modulo p}.

See also A344620 for a similar conjecture.

Examples

			a(5) = 13. The prime 11 is not a term since 2*3 = 6, 3^2-1 = 8 and 3^2+1 = 10 belong to the set {0 < r < 11: r is a quadratic nonresidue modulo 11} = {2, 6, 7, 8, 10}.
		

Crossrefs

Cf. A000040, A239957, A260911, A344620.

Programs

Mathematica

tab={};Do[p:=p=Prime[k];Do[If[JacobiSymbol[2a,p]==-1&&JacobiSymbol[a^2-1,p]==-1&&JacobiSymbol[a^2+1,p]==-1,Goto[aa]],{a,1,Sqrt[p-2]}];tab=Append[tab,p];Label[aa],{k,1,450}];Print[tab]