On some conjectural determinants of Sun involving residues

TL;DR

This paper explores determinants formed from residues modulo primes, generalizes Sun's results, and proves conjectures about their properties and divisibility, advancing understanding of residue determinants in number theory.

Contribution

It generalizes Sun's residue determinant results, proves related conjectures, and investigates prime divisibility properties of these determinants.

Findings

Residue properties of determinants $S_{m,k}(d,p)$ established.

Conjectures of Sun regarding square roots modulo $p$ confirmed.

Number of primes dividing specific determinants analyzed.

Abstract

For an odd prime $p$ and integers $d, k, m$ with gcd$(p,d)=1$ and $2\leq k\leq \frac{p-1}{2}$, we consider the determinant \begin{equation*} S_{m,k}(d,p) = \left|(\alpha_i - \alpha_j)^m\right|_{1 \leq i,j \leq \frac{p-1}{k}}, \end{equation*} where $\alpha_i$ are distinct $k$-th power residues modulo $p$. In this paper, we deduce some residue properties for the determinant $S_{m,k}(d,p)$ as a generalization of certain results of Sun. Using these, we further prove some conjectures of Sun related to $$\left(\frac{\sqrt{S_{1+\frac{p-1}{2},2}(-1,p)}}{p}\right) \text{ and } \left(\frac{\sqrt{S_{3+\frac{p-1}{2},2}(-1,p)}}{p}\right).$$ In addition, we investigate the number of primes $p$ such that $p\ |\ S_{m+\frac{p-1}{k},k}(-1,p)$, and confirm another conjecture of Sun related to $S_{m+\frac{p-1}{2},2}(-1,p)$.