Proof of three conjectures on determinants related to quadratic residues

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 $${\left|(i^2+d{j}^2)\left(\frac{i^2+d{j}^2}{n}\right)\right|}_{0\le i,j\le (n-1)/2},$$ 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 $|(i^2+d{j}^2{)}^n{|}_{0\le i,j\le n-1}$, where $d\ne 0$ 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 $S_c(d,p):=|(\frac{i^2+d{j}^2+c}{p}){|}_{1\le i,j\le (p-1)/2}$.

14 pages, accepted by Linear and Multilinear Algebra

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