Legendre symbols related to $D_p(b,1)$

Abstract

Let $p$ be an odd prime. For any $b,c\in \mathbb{Z}$, Z.-W. Sun introduced the new-type determinant $$D_p(b,c)=|(i^2+bij+c{j}^2{)}^{p-2}{|}_{1⩽i,j⩽p-1},$$ and studied its arithmetic properties. In this paper we mainly prove that $$\left(\frac{D_p(b,1)}{p}\right)=\left(\frac{2b}{p}\right)$$ when $(\frac{b^2-4}{p})=-1$ and $p\equiv 1(\mathrm{mod}4)$. As an application of our result, we confirm several conjectures of Sun.

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