Legendre symbols related to certain determinants

Abstract

Let $p$ be an odd prime. For $b,c\in \mathbb{Z}$, Sun introduced the determinant $$D_p(b,c)={\left|(i^2+bij+c{j}^2{)}^{p-2}\right|}_{1⩽i,j⩽p-1},$$ and investigated the Legendre symbol $(\frac{D_p(b,c)}{p})$. Recently Wu, She and Ni proved that $(\frac{D_p(1,1)}{p})=(\frac{-2}{p})$ if $p\equiv 2(\mathrm{mod}3)$, which confirms a previous conjecture of Sun. In this paper we determine $(\frac{D_p(1,1)}{p})$ in the case $p\equiv 1(\mathrm{mod}3)$. Sun proved that $D_p(2,2)\equiv 0(\mathrm{mod}p)$ if $p\equiv 3(\mathrm{mod}4)$, in contrast we prove that $(\frac{D_p(2,2)}{p})=1$ if $p\equiv 1(\mathrm{mod}8)$, and $(\frac{D_p(2,2)}{p})=0$ if $p\equiv 5(\mathrm{mod}8)$. Our tools include generalized trinomial coefficients and Lucas sequences.

arXiv comment20 pages,