A note on the number of solutions of the generalized Ramanujan-Nagell equation x 2 − D = p n

Abstract

Let D be a positive integer, and let p be an odd prime with p ∤ D. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M.A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for N(D, p), and also prove that if the equation U 2 − DV 2 = −1 has integer solutions (U, V), the least solution (u 1, v 1) of the equation u 2 − pv 2 = 1 satisfies p ∤ v 1, and D > C(p), where C(p) is an effectively computable constant only depending on p, then the equation x 2 − D = p n has at most two positive integer solutions (x, n). In particular, we have C(3) = 107.