Products of prime powers in binary recurrence sequences. I. The hyperbolic case, with an application to the generalized Ramanujan-Nagell equation

Source abstract

We show how the Gelfond-Baker theory and diophantine approximation techniques can be applied to solve explicitly the diophantine equation G n = w p 1 m 1 ⋯ p t m t {G_n} = wp_1^{{m_1}} \cdots p_t^{{m_t}} (where { G n } n = 0 ∞ \{ {G_n}\} _{n = 0}^\infty is a binary recurrence sequence with positive discriminant), for arbitrary values of the parameters. We apply this to the equation x 2 + k = p 1 z 1 ⋯ p t z t {x^2} + k = p_1^{{z_1}} \cdots p_t^{{z_t}} , which is a generalization of the Ramanujan-Nagell equation x 2 + 7 = 2 z {x^2} + 7 = {2^z} . 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.