A Note on the Generalized Ramanujan-Nagell Equation

Abstract

Let D, D1, D2 be positive integers such that D = D1D2, 1 ≤ D1 ≤ D2, and gcd(D1, D2) = 1, and let p be a prime with p[formula]D. Further, let N(D1, D2, p) denote the number of positive integer solutions (x, n) of the equation D1x2 + D2 = λpn, where λ = 4 or 1 according to whether p = 2 or not. In this paper we prove that if max(D, p) > exp exp exp 1000, then N(D1, D2, p) ≤ 1 except when (D1, D2, p) is exceptional.