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

1,1

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.

Links

Table of n, a(n) for n=1..28.

Example

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}.

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]

Crossrefs

Cf. A000040, A239957, A260911, A344621.

Sequence in context: A032758 A106118 A029743 * A042990 A040169 A040165

Adjacent sequences: A344617 A344618 A344619 * A344621 A344622 A344623

Keyword

nonn

Author

Zhi-Wei Sun, May 24 2021

Status

approved