有关Euler函数φ(n)的几个非线性方程

Several Nonlinear Equations on Euler Function φ(n)

令φ(n)是Euler函数,它是数论中重要的数论函数之一.包含Euler函数φ(n)的线性方程整数解的研究成果极为丰富.本文考虑了当b取某些整数时的包含Euler函数φ(n)非线性方程φ(xy)=k1φ(x)+k2φ(y)±b.对于奇数b,利用初等的方法证明了该方程有整数解时b,k1与k2的一些条件.并结合所给出的条件讨论了几个具体方程的整数解,给出了它们的各自的整数解.对于偶数b,讨论了一个具体形式的方程的整数解,利用初等的方法给出了其全部的整数解.

φ(n)is defined as the Euler function,which is one of the most important number-theoretic functions in number theory.The research results of integer solutions of linear equations involving Euler functionφ(n)are very rich.When b takes certain integers,the nonlinear equationsφ(xy)=k1φ(x)+k2φ(y)±b involving Euler function φ(n)are considered in this note.For odd b,it is proved that the equation has integer solutions under some conditions for b,k1 and k2 by the elementary method.Combined with the given conditions in the note,the integer solutions of several specific equations are discussed,and their respective integer solutions are given.For even b,the integer solutions of a special case is discussed,and all the integer solutions are given by using elementary method.