The Diophantine equation (axk−1)(byk−1)=abzk−1(axk−1)(byk−1)=abzk−1

Let a,ba,b be positive integers. In this paper, we prove that the equation (axk−1)(byk−1)=abzk−1(axk−1)(byk−1)=abzk−1 has no solutions in integers x,y,zx,y,z and k   with |x|>1|x|>1, |y|>1|y|>1 and k⩾4k⩾4.