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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.
1
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.
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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 24 2021
STATUS
approved