On some conjectural determinants of Sun involving residues

Abstract

For an odd prime $p$ and integers $d,k,m$ with gcd$(p,d)=1$ and $2\le k\le \frac{p-1}{2}$, we consider the determinant \begin{equation*} S_{m,k}(d,p) = \left|(α_i - α_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 $S_{m+\frac{p-1}{2},2}(-1,p)$.

Submitted for publication on 2nd July, 2024. Comments are welcome