On some determinant conjectures

TL;DR

This paper confirms three conjectures related to a specific determinant defined by Sun, involving prime numbers and quadratic forms, advancing understanding in number theory and determinant properties.

Contribution

The paper proves three conjectures proposed by Sun concerning the properties of a particular determinant involving primes and quadratic forms.

Findings

Confirmed three conjectures on the determinant D_p^-(c,d)

Established new properties of determinants related to quadratic forms

Enhanced understanding of determinant behavior in number theory

Abstract

Let $p$ be a prime and $c,d\in\mathbb{Z}$. Sun introduced the determinant $D_p^-(c,d):=\det[(i^2+cij+dj^2)^{p-2}]_{1<i,j<p-1}$ for $p>3$. In this paper, we confirm three conjectures on $D_p^-(c,d)$ proposed by Zhi-Wei Sun.