On the diophantine equation xn-1/x-1=yq

We prove that if (x, y, n, q) not equal (18, 7, 3, 3) is a solution of the Diophantine equation (x(n) - 1)/(x - 1) = y(q) with q prime, then there exists a prime number p such that p divides x and q divides p - 1. This allows us to solve completely this Diophantine equation for infinitely many values of x. The proofs require several different methods in diophantine approximation together with some heavy computer calculations.