A note on the number of solutions of the generalized Ramanujan-Nagell equation x2-D=pn

Let D be a positive integer, and let p be an odd prime with p acurrency sign 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 acurrency sign 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) = 10(7).