@article{Zhao2024, 
author = {Junyong Zhao},
title = {Counting sums of exceptional units in  Zn},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {9},
pages = {24546-24554},
keywords = {exceptional unit, circulant matrix, residue class rings, linear congruence, exponential sums},
url = {https://www.sciopen.com/article/10.3934/math.20241195},
doi = {10.3934/math.20241195},
abstract = {Let  R be a commutative ring with the identity  1R, and let  R∗ be the multiplicative group of units in  R. An element  a∈R∗ is called an exceptional unit if there exists a  b∈R∗ such that  a+b=1R. We set  R∗∗ to be the set of all exceptional units in  R. In this paper, we consider the residue-class ring  Zn. For any positive integers  n,s, and  c∈Zn, let  Ns(n,c):=♯{(x1,...,xs)∈(Zn∗∗)s:x1+...+xs≡c(modn)}. In 2016, Sander (J.Number Theory 159 (2016)) got a formula for  N2(n,c). Later on, Yang and Zhao (Monatsh. Math. 182 (2017)) extended Sander's theorem to finite terms by using exponential sum theory. In this paper, using matrix theory, we present an explicit formula for  Ns(n,c). This extends and improves earlier results.}
}