On The Diophantine Equation Of Euler Function

We consider the equations involving Euler's totient function 0 and Lucas type sequences.In 2015,Faye and Luca proved that,if(m,n,x)is a solution of?(xm-1)=xn-1 or ?(xm-1/x-1)=xn-1/x-1)in positive integers x,m,n with m>n,thenx<ee8000.In this paper,we mainly consider the equation?(zxm-ym/x-y)=zxn-yn/x-yin positive integers x,y,z,m,n with x>y.In particular,we prove the following main theorem that the equation has no nontrivial solutions in positive integers x,y,m,n with 1 ?z 4?x-y.So we can get the following theorems by taking z=x-y and z=1.(1)The equation ?(xm-ym)=xn-yn 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.(2)The equation ?((xm-ym)/(x-y))=(xn-yn)/(x-y)has no solu-tions 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? 1.