A344620 - OEIS

%I #16 May 26 2021 02:13:57

%S 2,3,5,7,13,17,19,23,29,31,37,41,43,47,53,79,89,97,101,113,151,173,

%T 281,283,313,461,739,827

%N 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.

%C 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.

%C 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}.

%C See also A344621 for a similar conjecture.

%e 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}.

%t 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]

%Y Cf. A000040, A239957, A260911, A344621.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, May 24 2021