For an odd prime p and integers d, k, m with gcd \((p,d)=1\) , \(p\equiv 1\pmod {k}\) , and \(2\le k\le \frac{p-1}{2}\) , we consider the determinant \(\begin{aligned} S_{m,k}(d,p) = \left| (\alpha _i +d\alpha _j)^m\right| _{1 \le i,j \le \frac{p-1}{k}}, \end{aligned}\) 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 \(\begin{aligned} \left( \frac{\sqrt{S_{1+\frac{p-1}{2},2}(-1,p)}}{p}\right) \text { and } \left( \frac{\sqrt{S_{3+\frac{p-1}{2},2}(-1,p)}}{p}\right) . \end{aligned}\) 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 \(S_{m+\frac{p-1}{2},2}(-1,p)\) .
