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

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• 作者：Quan-Hui Yang ; Qing-Qing Zhao
• 关键词：Ring of residue classes ; Exceptional units ; Exponential sum
• 刊名：Monatshefte für Mathematik
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：182
• 期：2
• 页码：489-493
• 全文大小：
• 刊物主题：Mathematics, general;
• 出版者：Springer Vienna
• ISSN：1436-5081
• 卷排序：182

文摘

Let R be a commutative ring with \(1\in R\) and \(R^{*}\) be the multiplicative group of its units. In 1969, Nagell introduced the concept of an exceptional unit, namely a unit u such that \(1-u\) is also a unit. 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 \(k=2\).