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A227968 Determinant of the (p_n+1)/2 X (p_n+1)/2 matrix with (i,j)-entry (i,j=0,...,(p_n-1)/2) equal to the Legendre symbol((i^2+j^2)/p_n), where p_n is the n-th prime.

%I A227968 #38 Aug 25 2021 07:02:18
%S A227968 -1,2,-12,-80,-162,3528,-9216,-11264,482230,-206684160,1488942450,
%T A227968 976835722500,-1420648513536,-12993312063488,-4001622478404278,
%U A227968 -738964000238206976,73685520670239843750,-13192753286712605540352,-505689449431040
%N A227968 Determinant of the (p_n+1)/2 X (p_n+1)/2 matrix with (i,j)-entry (i,j=0,...,(p_n-1)/2) equal to the Legendre symbol((i^2+j^2)/p_n), where p_n is the n-th prime.
%C A227968 Conjecture: 2*a(n) is always a quadratic residue mod p_n.
%C A227968 Zhi-Wei Sun also made the following general conjecture:
%C A227968 Let p be any odd prime. For each integer d let T(d,p) be the determinant of the (p+1)/2 X (p+1)/2 matrix whose (i,j)-entry (i,j=0,...,(p-1)/2) is the Legendre symbol ((i^2+d*j^2)/p). Then T(-d,p) == (2/p)*T(d,p) (mod p). If d is a quadratic residue mod p, then so is 2*T(d,p). If d is a quadratic non-residue mod p, then T(d,p) is a quadratic residue mod p.
%C A227968 The author has proved the following results for any odd prime p:
%C A227968 (1) If c and d are integers with c not divisible by p, and d' == c^2*d (mod p), then T(d',p) = (c/p)^((p+1)/2)*T(d,p).
%C A227968 (2) If p is congruent to 1 mod 4 and d is not divisible by p, then T(-d,p) = (2/p)*T(d,p).
%C A227968 The general conjecture mentioned above was proved in a recent preprint of Zhi-Wei Sun.
%H A227968 Zhi-Wei Sun, <a href="/A227968/b227968.txt">Table of n, a(n) for n = 2..70</a>
%H A227968 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1308.2900">On some determinants with Legendre symbol entries</a>, preprint, arXiv:1308.2900 [math.NT], 2013-2019.
%e A227968 a(2) = -1 since the determinant |((i^2 + j^2)/3)|_{i=0,1; j=0,1} equals -1.
%t A227968 a[n_] := Det[Table[JacobiSymbol[i^2+j^2, Prime[n]], {i, 0, (Prime[n]-1)/2}, {j, 0, (Prime[n]-1)/2}]]; Table[a[n], {n, 2, 20}]
%o A227968 (PARI) a(n) = my(p=prime(n)); matdet(matrix((p+1)/2, (p+1)/2, i, j, i--; j--; kronecker(i^2+j^2, p))); \\ _Michel Marcus_, Aug 25 2021
%Y A227968 Cf. A227609, A227971.
%K A227968 sign
%O A227968 2,2
%A A227968 _Zhi-Wei Sun_, Aug 01 2013