On Certain Determinants and Related Legendre Symbols

Let p be an odd prime. For \(b,c\in {\mathbb {Z}}\) $b,c\in \mathbb{Z}$ , we study the Legendre symbol \(\big (\frac{D_p^*(b,c)}{p}\big )\) $(\frac{D_p^{\ast }(b,c)}{p})$ , where \(D_p^*(b,c)\) $D_p^{\ast }(b,c)$ denotes the determinant of the matrix \([(i^2+bij+cj^2)^{p-3}]_{1\le i,j\le p-1}\) ${[{(i^2+bij+c{j}^2)}^{p-3}]}_{1\le i,j\le p-1}$ . For example, we prove that if \(p\equiv 2\ (\mathrm{{mod}}\ 3)\) $p\equiv 2(\mathrm{mod}3)$ then \(\begin{aligned} D_p^*(1,1)\equiv \det \left[ \frac{1}{(i^2+ij+j^2)^2}\right] _{1\le i,j\le p-1}\equiv -x^2\ (\mathrm{{mod}}\ p) \end{aligned}\) $$\begin{array}{c}\hfill D_p^{\ast }(1,1)\equiv det{\left[\frac{1}{{(i^2+ij+j^2)}^2}\right]}_{1\le i,j\le p-1}\equiv -x^2(\mathrm{mod}p)\end{array}$$ for some integer \(x\not \equiv 0\ (\mathrm{{mod}}\ p)\) $x\not\equiv 0(\mathrm{mod}p)$ . We also show that \(\begin{aligned} \left( \frac{D_p^*(2,2)}{p}\right) =\left( \frac{p}{3}\right) (-1)^{(p+1)/8} \end{aligned}\) $$\begin{array}{c}\hfill \left(\frac{D_p^{\ast }(2,2)}{p}\right)=\left(\frac{p}{3}\right){(-1)}^{(p+1)/8}\end{array}$$ if \(p\equiv 7\ (\mathrm{{mod}}\ 8)\) $p\equiv 7(\mathrm{mod}8)$ .