一类欧拉函数方程的解

Solutions of a Class of Euler Function Equations

摘要: 设n是正整数,  ϕ(n)是著名的欧拉函数， 它的值等于序列0, 1, 2...n − 1中与n互素的整数的个数。 对于涉及欧拉函数的方程解的讨论是一个富有意义的课题，而欧拉函数的性质在讨论欧 拉函数方程的解中至关重要。 本文利用欧拉函数性质的相关结论， 讨论了一类欧拉函数方 程ϕ(mn) = aϕ(m) + bϕ(n) + c 存在整数解的必要条件，并给出了当a = 5, b = 6, c = 16 时，亥欧拉函数方程的全部解。

Abstract: Let n be a positive integer, ϕ(n) is Euler function, the value is equal to the sequence 0, 1, 2, ..., n − 1 which are prime to n.  In fact, discussing the solutions of Euler function equation is a meaningful work, moreover, the properties of the function are very important to discuss the solution. In this paper, using the properties of the Euler function, we discuss the necessity of integer solution of the Euler function equation ϕ(mn) = aϕ(m) + bϕ(n) + c, and then gives all solutions if a = 5, b = 6, C = 16.