Diophantine equations involving Euler function

Abstract

In this paper, we show that the equation $\phi (|x^m-y^m|)=|x^n-y^n|$ has no nontrivial solutions in integers $x,y,m,n$ with $xy\ne 0,m>0,n>0$ except for the solutions $(x,y,m,n)=((2^{t-1}\pm 1),-(2^{t-1}\mp 1),2,1),(-(2^{t-1}\pm 1),(2^{t-1}\mp 1),2,1),$ where $t$ is a integer with $t\ge 2.$ The equation $\phi (|\frac{x^m-y^m}{x-y}|)=|\frac{x^n-y^n}{x-y}|$ has no nontrivial solutions in integers $x,y,m,n$ with $xy\ne 0,m>0,n>0$ except for the solutions $(x,y,m,n)=(a\pm 1,-a,1,2),(a\pm i,-a,2,1),$ where $a$ is a integer with $i=1,2.$