On the Addition of Two Cubes of Units and Nonunits mod $p^{\alpha}$ 
Received:February 19, 2016 Revised:March 23, 2016 
Key Words:
residue classes cubes of units exponential sum

Fund Project:Supported by the Research Culture Fundation of Anhui Normal University (Grant No.2014xmpy11) and the National Natural Science Foundation of China (Grant No.11471017). 

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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. 
Citation: 
DOI:10.3770/j.issn:20952651.2017.03.004 
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