On the Diophantine equation xm-1/x-1 = yn-1/y-1

We study the Diophantine equation x(m)-1/x-1 = y(n)-1/y-1 in integers x> 1, y> 1, m> 1, n> 1 with x not equal y. We show that, for given x and y, this equation has at most two solutions. Further, we prove that it has finitely many solutions (x, y, m, n) with m> 2 and n> 2 such that gcd( m - 1, n - 1) > 1 and (m - 1) / (n - 1) is bounded.