三变元非线性欧拉函数方程Φ(xyz)=Φ(x)(Φ(y)+Φ(z))的可解性

The Solvability of Nonlinear Euler Function Equation Φ(xyz)=Φ(x)(Φ(y)+Φ(z))

利用初等数学方法和欧拉函数的若干性质,研究了三变元非线性欧拉函数方程Φ(xyz)=Φ(x)(Φ(y)+Φ(z))的可解性问题,证明了该方程的正整数解存在性定理,给出了该方程的正整数解的一般形式,并求出了当x≤20时该方程的所有正整数解。

In this paper,the solvability of Nonlinear Euler function equation with three variables Φ(xyz)=Φ(x)(Φ(y)+Φ(z))is studied by using primary mathematical methods and some properties of Euler function.The existence theorem of positive integer solution of the equation is proved.A general form of positive integer solution of the equation is given,and all positive integer solutions of the equation are obtained when x≤20.