与两个数论函数有关的一个三元非线性方程的解

The solutions of a ternary nonlinear equation related to two arithmetical functions

令φ(n)为Euler函数,φ_(e)(n)为广义Euler函数。基于Euler函数φ(n)与广义Euler函数φ_(e)(n)的性质,讨论与Euler函数φ(n)和广义Euler函数φ_(e)(n)有关的三元非线性方程φ(abc)=2_(φ_(2))(a)+3_(φ_(2))(b)+4_(φ_(2))(c)+7的可解性,利用初等方法得到该方程的所有52组正整数解。

Letφ(n)be the Euler function andφ_(e)(n)be the generalized Euler function.Based on properties of Euler function and generalized Euler functionφ_(e)(n),the solvability of a ternary nonlinear equationφ(abc)=2_(φ_(2))(a)+3_(φ_(2))(b)+4_(φ_(2))(c)+7 related to Euler functionφ(n)and generalized Euler functionφe(n)was discussed,and the all 52 positive integer solutions were obtained by using elementary theory.