Evaluation of a determinant involving Legendre symbols

TL;DR

This paper fully evaluates the determinant of a matrix defined via Legendre symbols and a parameter, confirming a conjecture for specific primes and extending understanding of such determinants.

Contribution

The paper provides a complete evaluation of the determinant of a Legendre symbol-based matrix, confirming a conjecture for primes congruent to 3 mod 4.

Findings

Confirmed Sun's conjecture for primes p ≡ 3 mod 4

Derived explicit formulas for the determinant for all primes

Extended previous partial results on Legendre symbol matrices

Abstract

Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $A_p(x)$ denote the matrix $[x+a_{ij}]_{1\leqslant i,j\leqslant (p-1)/2}$, where $$ a_{ij}=\begin{cases} (\frac{j}{p}) &\text{if} \ i=1, \$\frac{i+j}{p}) &\text{if} \ i>1. \end{cases}$$ In 2018 Z.-W. Sun conjectured that $\det A_p(0)=-2^{(p-3)/2}$ if $p\equiv 3 \pmod{4}$, which was later confirmed by G. Zaimi. In this paper we evaluate $\det A_p(x)$ completely.