An old and new approach to Goormaghtigh’s equation

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