Goormaghtigh's equation: small parameters

We explicitly solve the polynomial-exponential Diophantine equation $\frac{x^m-1}{x-1}=\frac{y^n-1}{y-1}$ for various small fixed choices of the parameters $(x,y,m,n)$, generalizing the classical work of Makowski and Schinzel. Rather than elementary arguments, our proofs utilize a wide range of techniques from the theory of Diophantine approximation, together with extensive computer computations.