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-12, we consider the determinant equation* S₌, ₊ (d, p) = | (ᵢ - ⱼ) ᵐ|₁ ₈, ₉ ₏-₁₊, equation* where ᵢ are distinct k-th power residues modulo p. In this paper, we deduce some residue properties for the determinant S₌, ₊ (d, p) as a generalization of certain results of Sun. Using these, we further prove some conjectures of Sun related to (S₁+{₏-₁{₂, 2 (-1, p) }}p) and (S₃+{₏-₁{₂, 2 (-1, p) }}p). In addition, we investigate the number of primes p such that p\ |\ S₌+₏-₁₊, k (-1, p), and confirm another conjecture of Sun related to S₌+₏-₁₂, 2 (-1, p).