On some determinant conjectures

Abstract

Let $p$ be a prime and $c,d\in \mathbb{Z}$. Sun introduced the determinant $D_p^{-}(c,d):=\det [(i^2+cij+d{j}^2{)}^{p-2}{]}_{1<i,j<p-1}$ for $p>3$. In this paper, we confirm three conjectures on $D_p^{-}(c,d)$ proposed by Zhi-Wei Sun.