On the number of unit solutions of cubic congruence modulo n

2021

AIMS Mathematics, Vol 6, Iss 12, Pp 13515-13524 (2021)

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 $ x_1^3+\cdots +x_m^3\equiv 0\pmod n\ \ \ {\rm with} \ x_1, \ldots, x_m \in \mathbb{Z}_n^{\times}. $

article

English

AIMS Press, 2021.

cubic congruence, exponential sums, unit solutions, Mathematics, QA1-939