On certain determinants and the square root of some 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 $$T_m(d,p)=\det {\left[(i^2+d{j}^2{)}^m\left(\frac{i^2+d{j}^2}{p}\right)\right]}_{1⩽i,j⩽(p-1)/2},$$ and $$D_p^{(m)}=\det {\left[(i^2-j^2{)}^m\left(\frac{i^2-j^2}{p}\right)\right]}_{1\le i,j\le (p-1)/2}.$$ 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.

arXiv comment20 pages