Diophantine equations involving Euler function

In this paper, we show that the equation $\phi (\mathrm{\mid }{x}^m-y^m\mathrm{\mid })=\mathrm{\mid }{x}^n-y^n\mathrm{\mid }$ has no nontrivial solutions in integers $x,y,m,n$ with $xy\mathrm{\ne }0,m>0,n>0$ except for the solutions $(x,y,m,n)=((2^{t-1}\pm1),-(2^{t-1}\mp1),2,1), (-(2^{t-1}\pm1),(2^{t-1}\mp1),2,1),$where$t$isaintegerwith$t\geq 2.$ The equation $\phi (\mathrm{\mid }\frac{x^m-y^m}{x-y}\mathrm{\mid })=\mathrm{\mid }\frac{x^n-y^n}{x-y}\mathrm{\mid }$ has no nontrivial solutions in integers $x,y,m,n$ with $xy\mathrm{\ne }0,m>0,n>0$ 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 $i=1,2.$