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 $S_{ m+\frac{ p-1}{ 2},2}(-1,p)$.
Categories: math.NT