AN OLD and NEW APPROACH to GOORMAGHTIGH'S EQUATION

We show that if n ≥ 3 is a fixed integer, then there exists an effectively computable constant c(n) such that if x, y, and m are integers satisfying xm-1 x-1 = yn-1 y-1 , y>x>1, m > n, with gcd(m-1, n-1) > 1, then max{x, y,m} < c(n). In case n ∈ {3, 4, 5}, we solve the equation completely, subject to this non-coprimality condition. In case n = 5, our resulting computations require a variety of innovations for solving Ramanujan-Nagell equations of the shape f(x) = yn, where f(x) is a given polynomial with integer coefficients (and degree at least two), and y is a fixed integer.