Mathematics

CLC number: O156

Counting the Unit Solutions of Certain Quartic Diagonal Congruence Modulo n

一类模n四次对角同余方程的单位解的个数

¹ School of Mathematics and Physics, Nanyang Institute of Technology, Nanyang 473004, Henan, China
² School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

^† Corresponding author. E-mail: hushuangnian@163.com

Received: 6 May 2024

Abstract

Given a positive integer $n$ and the residue class ring ${\mathbf{\mathbb{Z}}}_n:=\mathbf{\mathbb{Z}}/n\mathbf{\mathbb{Z}}$, we set ${\mathbf{\mathbb{Z}}}_n^{\times }$ to be the group of units in${\mathbf{\mathbb{Z}}}_n$, i.e., ${\mathbf{\mathbb{Z}}}_n^{\times }:=\{x\in {\mathbf{\mathbb{Z}}}_n:\mathrm{g}\mathrm{c}\mathrm{d}\left(x,n\right)=\mathrm{1}\}$. Let $N_m(n)$ be the number of solutions of$x_{\mathrm{1}}^{\mathrm{4}}+\cdots +x_m^{\mathrm{4}}\equiv \mathrm{0}(\mathrm{m}\mathrm{o}\mathrm{d}n)$ with $x_{\mathrm{1}},\cdots ,x_m\in {\mathbf{\mathbb{Z}}}_n^{\times }$. In this note, we determine an explicit expression of $N_m(n)$. This extends the results of Sun and Yang in 2014.

摘要

给定正整数$n$和剩余类环${\mathbf{\mathbb{Z}}}_n:=\mathbf{\mathbb{Z}}/n\mathbf{\mathbb{Z}}$, 记${\mathbf{\mathbb{Z}}}_n^{\times }:=\{x\in {\mathbf{\mathbb{Z}}}_n:\mathrm{g}\mathrm{c}\mathrm{d}\left(x,n\right)=\mathrm{1}\}$为${\mathbf{\mathbb{Z}}}_n$中所有单位构成的群。设$N_m(n)$表示$x_{\mathrm{1}}^{\mathrm{4}}+\cdots +x_m^{\mathrm{4}}\equiv \mathrm{0}(\mathrm{m}\mathrm{o}\mathrm{d}$$n)$, $x_{\mathrm{1}},\cdots ,x_m\in {\mathbf{\mathbb{Z}}}_n^{\times }$的解的个数, 本文给出了$N_m(n)$的一个明确表达式, 这推广了孙和杨于2014年给出的结论。