On the Addition of Two Cubes of Units and Nonunits mod $p^{\alpha}$
Received:February 19, 2016  Revised:March 23, 2016
Key Word: residue classes   cubes of units   exponential sum
Fund ProjectL:Supported by the Research Culture Fundation of Anhui Normal University (Grant No.2014xmpy11) and the National Natural Science Foundation of China (Grant No.11471017).
 Author Name Affiliation Cuifang SUN School of Mathematics and Computer Science, Anhui Normal University, Anhui 241003, P. R. China Zhi CHENG School of Mathematics and Computer Science, Anhui Normal University, Anhui 241003, P. R. China
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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.