cp's OEIS Frontend

A263722 Integers k > 0 such that k^2 + p^2 is composite for all primes p.

%I A263722 #35 Apr 21 2021 11:52:46
%S A263722 9,11,19,21,23,25,29,31,39,41,43,49,51,53,55,59,61,63,69,71,75,77,79,
%T A263722 81,83,89,91,93,99,101,105,107,109,111,113,119,121,123,127,129,131,
%U A263722 133,139,141,143,145,149,151,153,157,159,161,165,169,171,173,175,179,181,185,187,189,191,195,197,199
%N A263722 Integers k > 0 such that k^2 + p^2 is composite for all primes p.
%C A263722 Conjecture: All terms are odd. An equivalent conjecture in A263977 is that if k > 0 is even, then k^2 + p^2 is prime for some prime p. We have checked this for all k <= 12*10^7.
%C A263722 An odd number k is a term if and only if k^2 + 2^2 is composite, since k^2 + p^2 is even and > 2 (hence composite) for all odd primes p. Thus if the Conjecture is true, then the sequence is simply odd k such that k^2 + 4 is composite. The sequence of odd k, such that k^2 + 4 is prime, is A007591.
%C A263722 The complementary sequence is A263977. Given k in it, the smallest prime p, such that k^2 + p^2 is prime, is in A263726. These numbers k^2 + p^2 form A185086, the Fouvry-Iwaniec primes.
%H A263722 Stephan Baier and Liangyi Zhao, <a href="http://arxiv.org/abs/math/0703284">On Primes Represented by Quadratic Polynomials</a>, arXiv:math/0703284 [math.NT], 2007-2008; Anatomy of Integers, CRM Proc. & Lecture Notes, Vol. 46, Amer. Math. Soc. 2008, pp. 169 - 166.
%H A263722 Étienne Fouvry and Henryk Iwaniec, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa79/aa7935.pdf">Gaussian primes</a>, Acta Arithmetica 79:3 (1997), pp. 249-287.
%e A263722 9^2 + 2^2 = 85 = 5*17, 11^2 + 2^2 = 125 = 5^3, and 23^2 + 2^2 = 533 = 13*41 are composite, so 9, 11, and 23 are members.
%e A263722 1^2 + 2^2 = 5 and 2^2 + 3^3 = 13 are prime, so 1, 2, and 3 are not members.
%Y A263722 Cf. A002313, A045637, A062324, A185086, A007591, A240130, A240131, A263466, A263726, A263977.
%K A263722 nonn
%O A263722 1,1
%A A263722 _Jonathan Sondow_ and _Robert G. Wilson v_, Oct 24 2015