

A344620


Primes p such that there is no positive integer a with 2*a, a^21 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
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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^21, a^2+1) with 2*a, a^21 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



EXAMPLE

a(5) = 13. The prime 11 is not a term since 2*2 = 4, 2^21 = 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^21, p]==1&&JacobiSymbol[a^2+1, p]==1, Goto[aa]], {a, 1, Sqrt[p2]}]; tab=Append[tab, p]; Label[aa], {k, 1, 150}]; Print[tab]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



