Some Determinants Involving Quadratic Residues Modulo Primes

Abstract

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

$\det {\left[ {1 + \tan\,\pi \frac{{a{j^2} + b{k^2}}}{p}} \right]_{1 \leqslant j,k \leqslant \tfrac{{p - 1}}{2}}} = \left\{ {\begin{array}{*{20}{c}} { - {2^{\frac{{p - 1}}{2}}}{p^{\frac{{p - 3}}{4}}},}&{if \left( {\frac{{ab}}{p}} \right) = 1,} \\ {{p^{\frac{{p - 3}}{4}}},}&{if \left( {\frac{{ab}}{p}} \right) = - 1,} \end{array}} \right.$

denotes the Legendre symbol. We also pose some conjectures for further research.

Keywords

Determinants / Legendre symbols / quadratic residues modulo primes / the tangent function / 11A15 / 11C20 / 15A15 / 33B10

RIGHTS & PERMISSIONS

Peking University