数论函数方程kφ(n)=7φ_(2)(n)+S(n^(13))的正整数解  

The Positive Integer Solutions of Arithmetic Function Equation kφ(n)=7φ_(2)(n)+S(n^(13))

作　　者：姜莲霞 杨振志 JIANG Lian-xia;YANG Zhen-zhi(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;Xinjiang Politics and Law Stability Maintenance Information Center,Urumqi 830003,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 [3]新疆政法维稳信息中心,新疆乌鲁木齐830003

出　　处：《喀什大学学报》2023年第3期18-21,共4页Journal of Kashi University

基　　金：新疆维吾尔自治区高校基本科研业务费科研项目“与数论函数相关的方程可解性的研究”(XJEDU2022P088);喀什大学校内科研项目一般课题“广义欧拉函数φe(n)的性质研究”((19)2652).

摘　　要：对于正整数n,数论函数φ(n),φe(n)和S(n)分别为Euler函数、广义Euler函数和Smarandache函数.讨论了包含φ(n),φe(n)和S(n)三个数论函数的方程kφ(n)=7φ_(2)(n)+S(n13)的可解性,基于这三个数论函数方程的性质,用初等方法证明了该数论函数方程只在k=1,4,5,6,7,8,9,11,14,17,23时有正整数解,并给出了具体的正整数解.For positive integer n,the arithmetic functionφ(n)is the Euler function,the arithmetic functionφ_(2)(n)is generalized Euler function and arithmetic function S(n)is the Smarandache function.The solvability of the arithmetic function equation kφ(n)=7φ_(2)(n)+S(n^(13))involving three arithmetic functionsφ(n),φe(n)and S(n)was discussed.Based on the properties of these three arithmetic functions,we obtained the equation has positive integer solutions only when k=1,4,5,6,7,8,9,11,14,17,23,and gave its specific positive integer solu-tions by using the elementary method.

关 键 词：Euler函数φ(n) 广义Euler函数φ2(n) Smarandache函数S(n) 正整数解 

分 类 号：O156[理学—数学]