Publications » Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation

# Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation

## Abstract

In this paper, we refine work of Beukers, applying results from the theory of Padé approximation to $(1 - z)^{1/2}$ to the problem of restricted rational approximation to quadratic irrationals. As a result, we derive effective lower bounds for rational approximation to $\sqrt{m}$ (where m is a positive nonsquare integer) by rationals of certain types. For example, we have $\left|\sqrt{2}-\frac{p}{q}\right|\gg q^{-1.47}\quad {\rm and}\quad\left|\sqrt{3}-\frac{p}{q}\right|\gg q^{-1.62},$ provided q is a power of 2 or 3, respectively. We then use this approach to obtain sharp bounds for the number of solutions to certain families of polynomial-exponential Diophantine equations. In particular, we answer a question of Beukers on the maximal number of solutions of the equation $x^2 + D = p^n$ where D is a nonzero integer and p is an odd rational prime, coprime to D.