On certain determinants and the square root of some determinants involving Legendre Symbols

TL;DR

This paper investigates properties of specific determinants involving Legendre symbols and square roots, confirming some conjectures and extending understanding of these mathematical objects for certain parameters.

Contribution

It introduces new properties of determinants involving Legendre symbols and confirms related conjectures by Zhi-Wei Sun.

Findings

Derived properties of determinants T_m(d,p) and D_p^{(m)}

Confirmed conjectures posed by Zhi-Wei Sun

Extended understanding of determinants involving Legendre symbols

Abstract

Let $p>3$ be a prime and $(\frac{.}{p})$ be the Legendre symbol. For any integer $d$ with $p\nmid d$ and any positive integer $m$, Sun introduced the determinants $$T_m(d,p)=\det\left[(i^2+dj^2)^m\left(\frac{i^2+dj^2}{p}\right)\right]_{1\leqslant i,j \leqslant (p-1)/2},$$ and $$D_p^{(m)}= \det\left[(i^2-j^2)^m\left(\frac{i^2-j^2}{p}\right)\right]_{1\leq i,j\leq (p-1)/2} .$$ In this paper, we obtain some properties of $T_m (d,p)$ and $ \sqrt{D_p^{(m)}}$ for some $m$. We also confirm some related conjectures posed by Zhi-Wei Sun.