A192579 - OEIS

%I #11 May 30 2026 16:40:35

%S 2,3,5,7,17

%N Primes p for which there is no prime q == 3 (mod 4) that is smaller than p and is a quadratic residue modulo p.

%C Gica proved that if p is a prime different from 2, 3, 5, 7, 17, then there exists a prime q < p which is a quadratic residue modulo p and q == 3 (mod 4).

%C This is the unique set of primes answering the question in the Mathematics Stack Exchange link. - _Rick L. Shepherd_, May 29 2016

%H A. Gica, <a href="https://doi.org/10.1216/rmjm/1181069349">Quadratic residues of certain types</a>, Rocky Mt. J. Math. 36 (2006), 1867-1871.

%H A. Gica, <a href="http://atlas-conferences.com/cgi-bin/abstract/cbcw-66">Quadratic residues of certain types</a>, Journées Arithmétiques 2011.

%H Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/250584/x-y-xy-and-x-y-are-prime-numbers-what-is-their-sum">x, y, x - y and x + y are prime numbers. What is their sum?</a>

%e p = 17 is a member, because the primes q < p with q == 3 (mod 4) are q = 3, 7, 11, and they are not quadratic residues modulo 17.

%e 11 is not a member, because 3 < 11 and 3 == 5^2 (mod 11).

%Y Cf. A192578.

%K nonn,fini,full

%O 1,1

%A _Jonathan Sondow_, Jul 04 2011