On the addition of squares of units modulo n

Abstract

Let $\mathbb{Z}_n$ be the ring of residue classes modulo $n$, and let $\mathbb{Z}_n^{\ast}$ be the group of its units. 90 years ago, Brauer obtained a formula for the number of representations of $c\in \mathbb{Z}_n$ as the sum of $k$ units. Recently, Yang and Tang in [Q. Yang, M. Tang, On the addition of squares of units and nonunits modulo $n$, J. Number Theory., 155 (2015) 1--12] gave a formula for the number of solutions of the equation $x_1^2+x_2^2=c$ with $x_{1},x_{2}\in \mathbb{Z}_n^{\ast}$. In this paper, we generalize this result. We find an explicit formula for the number of solutions of the equation $x^2_{1}+\cdots+x^2_{k}=c$ with $x_{1},\ldots,x_{k}\in \mathbb{Z}_n^{\ast}$.