Determinants concerning Legendre symbols

TL;DR

This paper investigates determinants with Legendre symbol entries, proving conjectures related to their properties and revealing new insights into their structure over finite fields.

Contribution

It proves conjectures by Sun and others on determinants with Legendre symbols and explores variants, advancing understanding of their algebraic properties.

Findings

Proved that certain determinants with Legendre symbol entries are perfect squares.

Confirmed conjecture that a specific determinant divided by a parameter is an integer square.

Extended results to variants involving primes expressed as sums of squares.

Abstract

The evaluations of determinants with Legendre symbol entries have close relation with character sums over finite fields. Recently, Sun posed some conjectures on this topic. In this paper, we prove some conjectures of Sun and also study some variants. For example, we show the following result: Let $p=a^2+4b^2$ be a prime with $a,b$ integers and $a\equiv1\pmod4$. Then for the determinant $$S(1,p):={\rm det}\bigg[\left(\frac{i^2+j^2}{p}\right)\bigg]_{1\le i,j\le \frac{p-1}{2}},$$ the number $S(1,p)/a$ is an integral square, which confirms a conjecture posed by Cohen, Sun and Vsemirnov.