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%I #84 Aug 25 2021 07:02:10
%S -1,1,-4,-16,-27,441,-1024,-1024,34445,-13778944,82719025,48841786125,
%T -67649929216,-564926611456,-153908556861703,-25481517249593344,
%U 2456184022341328125,-399780402627654713344,-14448269983744,-214168150727821285287075
%N Determinant of the (p_n-1)/2 X (p_n-1)/2 matrix with (i,j)-entry being the Legendre symbol((i^2+j^2)/p_n), where p_n is the n-th prime.
%C Conjecture: p_n never divides a(n), and moreover -a(n) is a quadratic residue mod p_n.
%C Zhi-Wei Sun also made the following conjecture:
%C Let p be any odd prime. For each integer d let S(d,p) be the determinant of the (p-1)/2 X (p-1)/2 matrix whose (i,j)-entry is the Legendre symbol ((i^2+d*j^2)/p). If d is a quadratic residue mod p, then so is -S(d,p). If d is a quadratic non-residue mod p, then we have S(d,p) = 0.
%C These were proved in version 9 of arXiv:1308.2900 (2018). In addition, the author has the following new conjecture.
%C Conjecture: For any prime p == 3 (mod 4), the number -S(1,p) is a positive square divisible by 2^((p-3)/2), i.e., -S(1,p) = (2^((p-3)/4)*m)^2 for some positive integer m. - _Zhi-Wei Sun_, Sep 09 2018
%H Alois P. Heinz, <a href="/A227609/b227609.txt">Table of n, a(n) for n = 2..100</a>
%H 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.
%H Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/310192/">Is -det[Legendre(i^2+j^2,p)]_{i,j=1,...,(p-1)/2} always a square for each prime p == 3 (mod 4)?</a>, Question 310192 in MathOverflow, Sept. 9, 2018.
%e a(2) = -1 since the Legendre symbol ((1^2 + 1^2)/3) is -1.
%p with(numtheory): with(LinearAlgebra):
%p a:= n-> Determinant(Matrix((ithprime(n)-1)/2, (i, j)->
%p jacobi(i^2+j^2, ithprime(n)))):
%p seq(a(n), n=2..20); # _Alois P. Heinz_, Jul 18 2013
%t a[n_]:=Det[Table[JacobiSymbol[i^2+j^2,Prime[n]],{i,1,(Prime[n]-1)/2},{j,1,(Prime[n]-1)/2}]]
%t Table[a[n],{n,2,20}]
%o (PARI) a(n) = my(p=prime(n)); matdet(matrix((p-1)/2, (p-1)/2, i, j, kronecker(i^2+j^2, p))); \\ _Michel Marcus_, Aug 25 2021
%Y Cf. A179071, A179072, A227968, A227971.
%K sign
%O 2,3
%A _Zhi-Wei Sun_, Jul 17 2013
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