The Diophantine equation (xk − 1)(yk − 1) = (zk − 1)t

We sharpen work of Bugeaud to show that the equation of the title has, for t = 1 or 2, no solutions in positive integers x, y, z and k with z > 1 and k > 3. The proof utilizes a variety of techniques, including the hypergeometric method of Thue and Siegel, as well as an assortment of gap principles.