Legendre Symbols Related to Certain Determinants

Abstract

Let p be an odd prime. For $b,c\in {\mathbb {Z}}$, Sun introduced the determinant $D_p(b,c)=\left| (i^2+bij+cj^2)^{p-2}\right| _{1\leqslant i,j \leqslant p-1}$ 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.