On the addition of squares of units modulo n

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• 作者：Mohsen Mollahajiaghaei ^{mmollaha@uwo.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11B13 ; 05C50
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：170
• 期：Complete
• 页码：35-45
• 全文大小：305 K

文摘

Let Z_n${\mathbb{Z}}_n$ be the ring of residue classes modulo n  , and let ${\mathbb{Z}}_n^{⁎}$ be the group of its units. 90 years ago, Brauer obtained a formula for the number of representations of c∈Z_n$c\in {\mathbb{Z}}_n$ as the sum of k units. Recently, Yang and Tang (2015) [6] gave a formula for the number of solutions of the equation $x_1^2+x_2^2=c$ with $x_1,x_2\in {\mathbb{Z}}_n^{⁎}$. In this paper, we generalize this result. We find an explicit formula for the number of solutions of the equation $x_1^2+\cdots +x_k^2=c$ with $x_1,\dots ,x_k\in {\mathbb{Z}}_n^{⁎}$.