与两个数论函数有关的一个三元线性方程的解

The solutions of a ternary linear equation relation to two arithmetical functions

令n为一正整数,数论函数φ(n)为Euler函数,数论函数φe(n)为广义Euler函数,其中e是一正整数.讨论了方程φ(abc)=3φ2(a)+4φ2(b)+5φ2(c)的可解性,基于Euler函数φ(n)与广义Euler函数φ2(n)的性质,利用初等方法给出了该方程全部的54组正整数解.

Let n be a positive integer,the arithmetic functionφ(n)is Euler function,and arithmetic functionφ_(e)(n)is generalized Euler function,where e is a positive integer.The solvability of linear equationφ(abc)=3φ_(2)(a)+4φ_(2)(b)+5φ_(2)(c)was discussed,based on the properties of Euler functionφ(n)and generalized Euler functionφ_(2)(n),the all54 sets of positive integer solution of it were given by using the elementary method.