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A219791 Number of ways to write n=x+y (0 with (xy)^2+1 prime.

%I A219791 #12 Nov 29 2012 14:01:10
%S A219791 0,1,1,1,2,0,2,1,2,2,2,2,2,2,5,0,2,1,2,2,4,2,4,0,6,2,6,2,5,3,6,3,5,4,
%T A219791 7,3,6,2,5,6,6,1,6,5,4,1,6,2,7,5,5,2,9,3,8,4,8,3,6,6,4,3,9,4,13,4,9,4,
%U A219791 5,9,2,1,11,4,14,4,10,3,9,8,4,3,6,5,10,3
%N A219791 Number of ways to write n=x+y (0<x<=y) with (xy)^2+1 prime.
%C A219791 Conjecture: a(n)>0 if n is different from 1, 6, 16, 24.
%C A219791 This conjecture has been verified for n up to 10^7. It implies that there are infinitely many primes of the form x^2+1.
%C A219791 Zhi-Wei Sun also made the following general conjecture: For any positive integer k, each sufficiently large integer n cna be written as x+y (x>0, y>0) with (xy)^{2^k}+1 prime.
%C A219791 For example, for k=2,3,4 it suffices to require that n is greater than 22, 386, 748 respectively.
%H A219791 Zhi-Wei Sun, <a href="/A219791/b219791.txt">Table of n, a(n) for n = 1..10000</a>
%H A219791 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;173d1416.1211">An amazing conjecture on primes</a>, a message to Number Theory List, Nov. 27, 2012.
%H A219791 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%e A219791 a(8)=1 since 8=4+4 with (4*4)^2+1=257 prime.
%e A219791 a(9)=2 since 9=2+7=4+5, and (2*7)^2+1=197 and (4*5)^2+1=401 are prime.
%t A219791 a[n_] := a[n] = Sum[If[PrimeQ[(k(n-k))^2+1] == True, 1, 0], {k, n/2}]; Do[Print[n, " ", a[n]], {n, 100}]
%Y A219791 Cf. A091182, A219782.
%K A219791 nonn
%O A219791 1,5
%A A219791 _Zhi-Wei Sun_, Nov 28 2012