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%I #18 Aug 28 2013 02:59:23
%S 1,-1,-1,0,1,-1,-1,0,1,-1,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,1,-1,-1,0,4,
%T -16,-9,25,4,-81,-81,81,841,-5929,-3969,19600,69169,-667489,-285156,
%U 80656,276676,-790321,-60025,3136,10816,-40000,-45369,221841,86436,-168100,-12100,13225,11881,-87616,-71289,729
%N Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j and 4*(i + j)^2 + 1 are both prime or not.
%C Conjecture: a(n) is nonzero for each n > 28.
%C This implies that there are infinitely many primes p with 4*p^2 + 1 also prime. Note also that (-1)^{n*(n-1)/2}*a(n) is always a square in view of the comments in A228591.
%H Zhi-Wei Sun, <a href="/A228561/b228561.txt">Table of n, a(n) for n = 1..300</a>
%e a(1) = 1 since 1 + 1 = 2 and 4*2^2 + 1 = 17 are both prime.
%t a[n_]:=a[n]=Det[Table[If[PrimeQ[i+j]==True&&PrimeQ[4(i+j)^2+1]==True,1,0],{i,1,n},{j,1,n}]]
%t Table[a[n],{n,1,30}]
%Y Cf. A052291, A069191, A071524, A228591, A228552, A228557, A228559, A228574, A228578, A228615, A228616.
%K sign
%O 1,29
%A _Zhi-Wei Sun_, Aug 25 2013
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