三类包含Euler函数的方程

Three Kinds of Equations Involving Euler Function

讨论了三类包含Euler函数的方程x-ψ(x)=2^(ω(x)),x-ψ(ψ(x))=2^(ω(x))与ψ(x^k)=2^(ω(x^k))的可解性,利用初等方法给出这三类方程的所有正整数解,其中ψ(x)为Euler函数,ω(x)为x的相异素因子个数.

The solvability of three kinds of equations involving Euler functionsx-φ（x） =2^ω（x）,x-φ（φ（x）） = 2^ω（x）andφ（x^k） = 2^ω（x^k）were discussed,and all the positive integer solutions of theirs were given by using elementary method,where φ（x） is Euler function andω（x） is the number of distinct prime factors of x.