与Euler函数有关的一个五元不定方程的正整数解  

The positive integer solutions of a five-variable Diophantine equation on Euler function

作　　者：姜莲霞 张四保 JIANG Lianxia;ZHANG Sibao(School of Mathematics and Statistics,Kashi University,Kashi 844000,Xinjiang,China;Research Center of Modern Mathematics and Its Application,Kashi University,Kashi 844000,Xinjiang,China)

机构地区：[1]喀什大学数学与统计学院,新疆喀什844000 [2]喀什大学现代数学及其应用研究中心,新疆喀什844000

出　　处：《华中师范大学学报（自然科学版）》2024年第4期424-432,共9页Journal of Central China Normal University：Natural Sciences

基　　金：新疆维吾尔自治区自然科学基金项目(2022D01A14);新疆维吾尔自治区高校基本科研业务费项目(XJEDU2022P088).

摘　　要：对于任意正整数n,数论函数φ(n)为Euler函数.该文讨论了与Euler函数有关的一个五元不定方程φ(x_(1)x_(2)x_(3)x_(4)x 5)=3φ(x_(1))φ(x_(2))φ(x_(3))+4φ(x_(4))φ(x 5)的正整数解,利用Euler函数φ(n)的相关性质以及初等方法,得到了这一方程共有501组正整数解,并给出了满足x_(1)≤x_(2)≤x_(3)与x_(4)≤x 5的61组正整数解.For any positive integer n,arithmetic functionφ(n)is Euler function.The positive integer solutions of a five-variable Diophantine equationφ(x_(1)x_(2)x_(3)x_(4)x 5)=3φ(x_(1))φ(x_(2))φ(x_(3))+4φ(x_(4))φ(x 5)on Euler functionφ(n)were discussed in this paper.By using the properties of Euler functionφ(n)and the elementary method,the 501 positive integer solutions of this equation were obtained,and the 61 positive integer solutions satisfied x_(1)≤x_(2)≤x_(3) and x_(4)≤x 5 were given.

关 键 词：EULER函数 不定方程 正整数解 

分 类 号：O156[理学—数学]