On the number of unit solutions of cubic congruence modulo $ n

• Published in: AIMS mathematics Vol. 6; no. 12; pp. 13515 - 13524
• Main Author: Zhao, Junyong
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2021
• Subjects: cubic congruence; exponential sums; unit solutions
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2021784

For any positive integer $ n $, let $ \mathbb Z_n: = \mathbb Z/n\mathbb Z = \{0, \ldots, n-1\} $ be the ring of residue classes module $ n $, and let $ \mathbb{Z}_n^{\times}: = \{x\in \mathbb Z_n|\gcd(x, n) = 1\} $. In 1926, for any fixed $ c\in\mathbb Z_n $, A. Brauer studied the linear congruence $ x_1+\cdots+x_m\equiv c\pmod n $ with $ x_1, \ldots, x_m\in\mathbb{Z}_n^{\times} $ and gave a formula of its number of incongruent solutions. Recently, Taki Eldin extended A. Brauer's result to the quadratic case. In this paper, for any positive integer $ n $, we give an explicit formula for the number of incongruent solutions of the following cubic congruence <disp-formula> <tex-math id="FE1"> \begin{document}$ x_1^3+\cdots +x_m^3\equiv 0\pmod n\ \ \ {\rm with} \ x_1, \ldots, x_m \in \mathbb{Z}_n^{\times}. $\end{document} </tex-math></disp-formula>