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%I #52 Aug 25 2021 06:56:09
%S 0,-16,0,0,0,18939904,0,0,-600706205614080,0,-3126394312091238400,
%T 915844279166632469526048000,0,0,
%U 1513783909437524991467008819200000000,0,-6597762875255062617688526826958066024448000,0,0
%N Determinant of the (p_n-1) X (p_n-1) matrix with (i,j)-entry equal to the Legendre symbol ((i^2+6*i*j+j^2)/p_n), where p_n is the n-th prime.
%C Conjecture: We have a(n) = 0 if p_n == 3 (mod 4).
%C For an odd prime p and integers c and d, _Zhi-Wei Sun_ defined (c,d)_p to be the determinant of the (p-1) X (p-1) matrix whose (i,j)-entry is the Legendre symbol ((i^2+c*i*j+d*j^2)/p). It is easy to see that (-c,d)_p = (-1/p)*(c,d)_p. Sun conjectured that for any integer c and nonzero integer d there are infinitely many odd primes p with (c,d)_p = 0, moreover (c,d)_p = 0 if (d/p) = -1. He also formulated some concrete conjectures in the case (d/p) = 1. For example, (3,3)_p = 0 if p == 11 (mod 12), and (10,9)_p = 0 if p == 5 (mod 12); (3,2)_p = (4,2)_p = 0 if p == 7 (mod 8).
%C On August 12 2013, _Zhi-Wei Sun_ conjectured that for any odd prime p and integers c and d with d not divisible by p, if (c,d)_p is nonzero then its p-adic valuation (i.e., p-adic order) must be even.
%H Zhi-Wei Sun, <a href="/A225611/b225611.txt">Table of n, a(n) for n = 2..50</a>
%H Zhi-Wei Sun, <a href="https://arxiv.org/abs/1308.2900">On some determinants with Legendre symbol entries</a>, preprint, arXiv:1308.2900 [math.NT], 2013-2019.
%t a[n_]:=Det[Table[JacobiSymbol[i^2+6*i*j+j^2,Prime[n]],{i,1,(Prime[n]-1)},{j,1,(Prime[n]-1)}]]
%t Table[a[n],{n,2,20}]
%o (PARI) a(n) = my(p=prime(n)); matdet(matrix(p-1, p-1, i, j, kronecker(i^2+6*i*j+j^2, p))); \\ _Michel Marcus_, Aug 25 2021
%Y Cf. A226163, A227609, A227968, A227971, A228005, A228077, A228095.
%K sign
%O 2,2
%A _Zhi-Wei Sun_, Aug 11 2013
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