Diophantine equations involving Euler's totient function

Abstract

In this paper, we consider the equations involving Euler's totient function $\varphi$ and Lucas type sequences. In particular, we prove that the equation $\varphi (x^m-y^m)=x^n-y^n$ 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 $\varphi ((x^m-y^m)/(x-y))=(x^n-y^n)/(x-y)$ 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 $a>b\ge 1$.

40 pages