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⩽j,k⩽\frac{p-1}{2}}=\{\begin{array}{cc}-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}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\det {\left[ {1 + \tan\,\pi \frac{{a{j^2} + b{k^2}}}{p}} \right]_{1 \leqslant j,k \leqslant \tfrac{{p - 1}}{2}}} = \left\{ {\begin{array}{*{20}{c}}{ - {2^{\frac{{p - 1}}{2}}}{p^{\frac{{p - 3}}{4}}},}&{if \left( {\frac{{ab}}{p}} \right) = 1,} \\{{p^{\frac{{p - 3}}{4}}},}&{if \left( {\frac{{ab}}{p}} \right) =- 1,}\end{array}} \right.$$\end{document}

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

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RIGHTS & PERMISSIONS

Peking University