与Euler函数φ(n)有关的一个四元不定方程的解  

Solutions of a Quaternary Diophantine Equation on Euler Function φ( n)

作　　者：张四保 杨燕妮 姜莲霞 席小忠[2] ZHANG Sibao;YANG Yanni;JIANG Lianxia;XI Xiaozhong(School of Mathematics and Statistics,Kashgar University,Kashgar 844008,China;Institute of Mathematics and Computer Science,Yichun College,Yichun 336000,China)

机构地区：[1]喀什大学数学与统计学院,新疆喀什844008 [2]宜春学院数学与计算机科学学院,江西宜春336000

出　　处：《重庆理工大学学报（自然科学）》2019年第11期214-221,共8页Journal of Chongqing University of Technology：Natural Science

基　　金：国家自然科学基金项目(11201411);新疆维吾尔自治区自然科学基金资助项目(2017D01A13)

摘　　要：讨论了与Euler函数φ(n)有关的四元不定方程φ(xyzw)=4(φ(x)+φ(y)+φ(z)+φ(w))的整数解,基于Euler函数φ(n)的性质及初等的方法,给出了其满足x≤y≤z≤w的整数解,再由对称性确定了其共有1080组整数解,从而确定其全部的整数解。The integer solutions of a quaternary Diophantine equationφ(xyzw)=4(φ(x)+φ(y)+φ(z)+φ(w))on Euler functionφ(n)were discussed,and based on the properties and elementary methods of Euler functionφ(n),the integer solutions satisfying x≤y≤z≤w were given.Then it is determined by symmetry that this equation has 1080 integer solutions.Therefore,all the integer solutions of the equation can be determined.

关 键 词：EULER函数 不定方程 整数解 

分 类 号：O156[理学—数学]