关于一个三元变系数欧拉函数方程的正整数解

On the positive integer solution of a ternary variable coefficient Euler function equation

设φ(n)为Euler函数,探究一个系数为特殊勾股数的三元不定方程φ(abc)=3φ(a)+4φ(b)+5φ(c)的可解性.利用初等方法与技巧,对主要结论进行了完整的阐明,通过分析筛选后得到该不定方程共有40组正整数解.文中所用分类方法以及将系数选定为特殊勾股数的思想,为同类型方程的研究提供了新的思路.

let φ(n) be a Euler function, the positive integer solution of a ternary variable coefficient Diophantine equations φ(abc)=3φ(a)+4φ(b)+5φ(c) is explored. By using elementary methods and techniques, the main conclusions are expounded completely, and a total of 40 groups of positive integer solutions are obtained after analysis and screening. The classification method used in this paper and the idea of selecting coefficients as special Pythagorean numbers provide a new idea for the study of the same type equation.