A193064 -id:A193064 - cp's OEIS frontend

A193063 Primes p for which there is no prime q == 1 (mod 3) that is smaller than p and is a quadratic residue modulo p.

2, 3, 5, 7, 11, 13

Offset: 1

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Jonathan Sondow, Jul 15 2011

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Gica proved that if p is a prime different from 2, 3, 5, 7, 11, 13, then a prime q < p exists which is a quadratic residue modulo p and q == 1 (mod 3). His paper has not yet been published, but see A192578 for a reference, link, and examples of a similar result.

Cf. A192578, A192579, A193064.

piMax = 100; quadQ[q_, p_] := Length[ Select[ Table[x^2, {x, 1, Floor[p/2]}], Mod[#, p] == q & , 1]] == 1; noPrimeQ[p_] := Select[ Table[ Prime[n], {n, 1, PrimePi[p]}], Mod[#, 3] == 1 && quadQ[#, p] &] === {}; Select[ Table[ Prime[n], {n, 1, piMax}], noPrimeQ] (* Jean-François Alcover, Oct 03 2011 *)

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A193063 Primes p for which there is no prime q == 1 (mod 3) that is smaller than p and is a quadratic residue modulo p.

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