A344620 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 residues modulo p.

2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 79, 89, 97, 101, 113, 151, 173, 281, 283, 313, 461, 739, 827

Offset: 1

Author

Zhi-Wei Sun, May 24 2021

Keywords

• nonn

Comments

No more terms below 10^10. For any prime p > 11, one of 1^1+1 = 2, 2^2+1 = 5 and 3^2+1 = 10 is a quadratic residue modulo p.

Conjecture: No term is greater than 827. In other words, for any prime p > 828, 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 residue modulo p}.

See also A344621 for a similar conjecture.

Examples

			a(5) = 13. The prime 11 is not a term since 2*2 = 4, 2^2-1 = 3 and 2^2+1 = 5 belong to the set {0 < r < 11: r is a quadratic residue modulo 11} = {1, 3, 4, 5, 9}.
		

Crossrefs

Cf. A000040, A239957, A260911, A344621.

Programs

Mathematica

tab={}; Do[p:=p=Prime[k]; Do[If[p>2&&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,1,150}]; Print[tab]