一类Euler函数方程的解的上界

Upper Bounds of the Solutions of a Class of Euler' Functional Equation

设φ(n)为正整数n的Euler函数,讨论了Euler函数方程φ(x1…xn-1xn)=m(φ(x1)+…+φ(xn-1)+φ(xn))的求解问题,给出了该方程的所有正整数解的较为精确的上界.作为应用,对于一些给定的正整数m和n,求出了此时方程的全部正整数解.

Let φ（n） is Euler function of a positive integer n. The nature of the Euler function is interesting research topics in number theory. On the basis of the literature, we discuss Euler functional equationφ（x1…xn-1xn）=m（φ（x1）＋…＋φ（xn-1）＋φ（xn））, and the more accurate upper bounds of all positive integer solutions of the equation are given. As the application, we obtain all solutions of the equation for some given positive integers m and n.