On the Addition of Two Cubes of Units and Nonunits mod $p^{\alpha}$

Abstract:

      Let $p\equiv 2\pmod{3}$ be an odd prime and $\alpha$ be a positive integer. In this paper, for any integer $c$, we obtain a formula for the number of solutions of the cubic congruence $x^{3}+y^{3}\equiv c~({\rm mod}\,p^{\alpha})$ with $x, y$ units, nonunits and mixed pairs, respectively. We resolve a problem posed by Yang and Tang.