Diophantine equations involving Euler function

In this paper, we show that the equation \(\varphi(|x^{m}-y^{m}|)=|x^{n}-y^{n}|\) has no nontrivial solutions in integers \(x,y,m,n\) with \(xy\neq0, m>0, n>0\) 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 \(t\) is a integer with \(t\geq 2.\) The equation \(\varphi(|\frac{x^{m}-y^{m}}{x-y}|)=|\frac{x^{n}-y^{n}}{x-y}|\) has no nontrivial solutions in integers \(x,y,m,n\) with \(xy\neq0, m>0, n>0\) except for the solutions \((x,y,m,n)=(a\pm1, -a, 1, 2), (a\pm i, -a, 2, 1),\) where \(a\) is a integer with \(i=1,2.\)