On the sums of squares of exceptional units in residue class rings

Abstract

Let $n\ge 1,e\ge 1,k\ge 2$ and $c$ be integers. An integer $u$ is called a unit in the ring ${\mathbb{Z}}_n$ of residue classes modulo $n$ if $\gcd (u,n)=1$. A unit $u$ is called an exceptional unit in the ring ${\mathbb{Z}}_n$ if $\gcd (1-u,n)=1$. We denote by ${\mathcal{N}}_{k,c,e}(n)$ the number of solutions $(x_1,...,x_k)$ of the congruence $x_1^e+...+x_k^e\equiv c(\mathrm{mod}n)$ with all $x_i$ being exceptional units in the ring ${\mathbb{Z}}_n$. In 2017, Mollahajiaghaei presented a formula for the number of solutions $(x_1,...,x_k)$ of the congruence $x_1^2+...+x_k^2\equiv c(\mathrm{mod}n)$ with all $x_i$ being the units in the ring ${\mathbb{Z}}_n$. Meanwhile, Yang and Zhao gave an exact formula for ${\mathcal{N}}_{k,c,1}(n)$. In this paper, by using Hensel's lemma, exponential sums and quadratic Gauss sums, we derive an explicit formula for the number ${\mathcal{N}}_{k,c,2}(n)$. Our result extends Mollahajiaghaei's theorem and that of Yang and Zhao.

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