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

TL;DR

This paper derives an explicit formula for counting solutions to a sum of squares of exceptional units in residue class rings, extending previous results by using advanced number theory techniques.

Contribution

It introduces a new explicit formula for the number of solutions involving squares of exceptional units, generalizing earlier formulas for units and specific exponents.

Findings

Derived an explicit formula for al N_{k,c,2}(n).

Extended previous theorems by Mollahajiaghaei and Yang-Zhao.

Utilized Hensel's lemma, exponential sums, and quadratic Gauss sums.

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 $x_1^e+...+x_k^e\equiv c \pmod n$ 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 $x_1^2+...+x_k^2\equiv c\pmod n$ 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.