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

1,1

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.

Links

Table of n, a(n) for n=1..44.

Example

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}.

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]

Crossrefs

Cf. A000040, A239957, A260911, A344620.

Sequence in context: A339342 A052015 A042992 * A255203 A356445 A233577

Adjacent sequences: A344618 A344619 A344620 * A344622 A344623 A344624

Keyword

nonn

Author

Zhi-Wei Sun, May 24 2021

Status

approved