Some Determinants Involving Quadratic Residues Modulo Primes

In this paper we evaluate several determinants involving quadratic residues modulo primes. For example, for any prime p > 3 with p ≡ 3 (mod 4) and a, b ∈ ℤ with p ∤ ab, we prove that $$ {[ {1 + tan\,π {{a{j^2} + b{k^2}}}{p}} ]_{1 j,k {{p - 1}}{2}}} = \{ {{array}{*{20}{c}} { - {2^{{{p - 1}}{2}}}{p^{{{p - 3}}{4}}},}&{if ( {{{ab}}{p}} ) = 1,} \\ {{p^{{{p - 3}}{4}}},}&{if ( {{{ab}}{p}} ) = - 1,} {array}} .$$ denotes the Legendre symbol. We also pose some conjectures for further research.