Proof of three conjectures on determinants related to quadratic residues

TL;DR

This paper proves three conjectures by Z.-W. Sun regarding divisibility properties of determinants involving quadratic residues and provides explicit evaluations of related Legendre symbols.

Contribution

It confirms three conjectures on determinants related to quadratic residues, offering new divisibility results and explicit evaluations of Legendre symbols for these determinants.

Findings

Odd integers greater than 3 divide specific determinants involving quadratic residues.

Divisibility results for determinants of matrices with entries (i+dj)^n and (i^2+dj^2)^n.

Complete determination of Legendre symbols for determinants involving quadratic forms modulo primes.

Abstract

In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer $n>3$ divides the determinant $$\left|(i^2+dj^2)\left(\frac{i^2+dj^2}n\right)\right|_{0\le i,j\le (n-1)/2},$$ where $d$ is any integer and $(\frac{\cdot}n)$ is the Jacobi symbol. Then we prove some divisibility results concerning $|(i+dj)^n|_{0\le i,j\le n-1}$ and $|(i^2+dj^2)^n|_{0\le i,j\le n-1}$, where $d\not=0$ and $n>2$ are integers. Finally, for any odd prime $p$ and integers $c$ and $d$ with $p\nmid cd$, we determine completely the Legendre symbol $(\frac{S_c(d,p)}p)$, where $S_c(d,p):=|(\frac{i^2+dj^2+c}p)|_{1\le i,j\le(p-1)/2}$.