|
%I #14 Mar 11 2023 08:05:56
%S 2,3,5,7,13,17,19,23,31,41,43,47,67,71,73,97,101,127,151,157,167,191,
%T 199,239,257,311,313,367,409,439,479,521,587,599,739,839,887,1031,
%U 1063,1151,1319,2351,2999,3119
%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 nonresidues modulo p.
%C No more terms below 10^10.
%C 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}.
%C See also A344620 for a similar conjecture.
%e 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}.
%t 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]
%Y Cf. A000040, A239957, A260911, A344620.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, May 24 2021
|
