关于模$p^{\alpha}$的单位与非单位的立方和

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

中文关键词:  剩余类  单位的立方和  指数和

英文关键词:residue classes  cubes of units  exponential sum

中文摘要:

      设奇素数$p\equiv 2~({\rm mod}\,3)$, $\alpha$是正整数. 对于任意整数$c$, 本文研究了$x, y$分别是单位,非单位,以及两者混合时,三次同余方程$x^{3}+y^{3}\equiv c~({\rm mod}\, p^{\alpha})$解的个数公式. 我们解决了杨全会和汤敏提出的一个问题.

英文摘要:

      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.