On some conjectural determinants of Sun involving residues

For an odd prime p and integers $d, k, m$ with gcd$(p,d)=1$ and 2≤ k≤ p-1/2, we consider the determinant {equation*} Sm,k(d,p) = |(α_i - α_j)^m|1 ≤ i,j ≤ p-1/k, {equation*} where αᵢ are distinct k-th power residues modulo p. In this paper, we deduce some residue properties for the determinant Sm,k(d,p) as a generalization of certain results of Sun. Using these, we further prove some conjectures of Sun related to (√S1+p-1/2,2(-1,p)p) and (√S3+p-1/2,2(-1,p)p). In addition, we investigate the number of primes p such that p\ |\ Sm+p-1/k,k(-1,p), and confirm another conjecture of Sun related to Sm+p-1/2,2(-1,p).