标题： 涉及欧拉函数的丢番图方程 显示英文标题

标题： Diophantine equations involving Euler's totient function

摘要： 在本文中，我们考虑涉及欧拉函数$\phi$和卢卡斯型序列的方程。 特别地，我们证明方程$\phi (x^m-y^m)=x^n-y^n$在正整数$x, y, m, n$中除了平凡解$(x, y, m , n)=(a+1, a, 1, 1)$外没有解，其中$a$是一个正整数，而方程$\phi ((x^m-y^m)/(x-y))=(x^n-y^n)/(x-y)$在正整数$x, y, m, n$中除了平凡解$(x, y, m , n)=(a, b, 1, 1)$外没有解，其中$a, b$是整数且$a>b\ge 1$。 显示英文摘要

摘要： In this paper, we consider the equations involving Euler's totient function $\phi$ and Lucas type sequences. In particular, we prove that the equation $\phi (x^m-y^m)=x^n-y^n$ has no solutions in positive integers $x, y, m, n$ except for the trivial solutions $(x, y, m , n)=(a+1, a, 1, 1)$, where $a$ is a positive integer, and the equation $\phi ((x^m-y^m)/(x-y))=(x^n-y^n)/(x-y)$ has no solutions in positive integers $x, y, m, n$ except for the trivial solutions $(x, y, m , n)=(a, b, 1, 1)$, where $a, b$ are integers with $a>b\ge 1$.