On the sumsets of exceptional units in $\mathbb{Z}_n$

TL;DR

This paper derives an exact formula for representing elements of residue class rings as sums of exceptional units, generalizing previous results for the case of two units to arbitrary sums of $k$ units.

Contribution

It provides a new explicit formula for the number of representations of elements as sums of $k$ exceptional units in $Z_n$, extending prior work from the case $k=2$ to general $k \,\geq 2$.

Findings

Derived an exact formula for the number of sum representations.

Generalized previous results from $k=2$ to arbitrary $k\geq 2$.

Enhanced understanding of the structure of exceptional units in residue rings.

Abstract

Let $R$ be a commutative ring with $1\in R$ and $R^{\ast}$ be the multiplicative group of its units. In 1969, Nagell introduced the exceptional unit $u$ if both $u$ and $1-u$ belong to $R^{\ast}$. Let $\mathbb{Z}_n$ be the ring of residue classes modulo $n$. In this paper, given an integer $k\ge 2$, we obtain an exact formula for the number of ways to represent each element of $ \mathbb{Z}_n$ as the sum of $k$ exceptional units. This generalizes a recent result of J. W. Sander for the case $k=2$.