Cubic residues and their new distributive properties

Let \(M_{k}(p)\) $M_k(p)$ denote the number of all integers \(1\le a \le p-1\) $1\le a\le p-1$ such that \(a+a^{k}\) $a+a^k$ and \(a-a^{k}\) $a-a^k$ are cubic residues modulo p. We obtain some identities or asymptotic formulae for \(M_{2}(p)\) $M_2(p)$ and \(M_{3}(p)\) $M_3(p)$ by using the properties of Gauss sums and third-order Dirichlet character. Through these results, the distributive properties of cubic residues have been fully characterized.