On some determinants arising from quadratic residues

Abstract

Let $p>3$ be a prime, and let $d\in \mathbb{Z}$ with $p\nmid d$. For the determinants $$S_m(d,p)=\det {\left[(i^2+d{j}^2{)}^m\right]}_{1⩽i,j⩽(p-1)/2}\left(\frac{p-1}{2}⩽m⩽p-1\right),$$ Sun recently determined $S_m(d,p)$ modulo $p$ when $m\in \{p-2,p-3\}$ and $(\frac{-d}{p})=-1$. In this paper, we obtain $S_{p-2}(d,p)$ modulo $p$ in the remaining case $(\frac{-d}{p})=1$, and determine the Legendre symbols $(\frac{S_{p-3}(d,p)}{p})$ and $(\frac{S_{p-4}(d,p)}{p})$ in some special cases.

14 pages