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

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Abstract

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 \(k=2\).

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Acknowledgments

This work was supported by the National Natural Science Foundation for Youth of China, Grant No. 11501299, the Natural Science Foundation of Jiangsu Province, Grant Nos. BK20150889, 15KJB110014 and the Startup Foundation for Introducing Talent of NUIST, Grant No. 2014r029. We would like to thank Professor K. Gyory for pointing out the reference [3] and the anonymous referee for his/her useful comments.

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Correspondence to Quan-Hui Yang.

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Communicated by J. S. Wilson.

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Yang, QH., Zhao, QQ. On the sumsets of exceptional units in \(\mathbb {Z}_n\) . Monatsh Math 182, 489–493 (2017). https://doi.org/10.1007/s00605-015-0872-y

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