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

TL;DR

This paper investigates the arithmetic properties of a determinant introduced by Z.-W. Sun, establishing a Legendre symbol relation under certain conditions and confirming related conjectures.

Contribution

The paper proves a new Legendre symbol relation for the determinant D_p(b,1) and confirms several of Sun's conjectures.

Findings

Proves that (D_p(b,1)/p) = (2b/p) under specified conditions.

Establishes a connection between the determinant and Legendre symbols.

Confirms multiple conjectures posed by Sun.

Abstract

Let $p$ be an odd prime. For any $b,c\in\mathbb{Z}$, Z.-W. Sun introduced the new-type determinant $$D_p(b,c)=|(i^2+bij+cj^2)^{p-2}|_{1\leqslant i,j\leqslant p-1},$$ 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 $p\equiv1\pmod 4$. As an application of our result, we confirm several conjectures of Sun.