Products of Prime Powers in Binary Recurrence Sequences Part I: The Hyperbolic Case, with an Application to the Generalized Ramanujan-Nagell Equation

We show how the Gelfond-Baker theory and diophantine approximation techniques can be applied to solve explicitly the diophantine equation C„ = wpTM< ■ ■ ■ p"< (where {G„}^_o is a binary recurrence sequence with positive discriminant), for arbitrary values of the parameters. We apply this to the equation x2 + k = p\' ■ ■ ■ pf', which is a generalization of the Ramanujan-Nagell equation x2 + 7 = 2~. We present algorithms to reduce upper bounds for the solutions of these equations. The algorithms are easy to translate into computer programs. We present an example which shows that in practice the method works well.