On the addition of squares of units modulo n

TL;DR

This paper generalizes previous formulas to count solutions of the equation involving sums of squares of units modulo n, extending known results to any number of terms.

Contribution

It provides an explicit formula for the number of solutions to the sum of k squares of units equaling c modulo n, generalizing earlier specific cases.

Findings

Derived an explicit formula for solutions of sum of k squares of units

Extended previous results from two squares to k squares

Generalized formulas for sums involving 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 $x_{1},\ldots,x_{k}\in \mathbb{Z}_n^{\ast}$.