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 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. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 29 19:17 EDT 2023. Contains 365776 sequences. (Running on oeis4.)