On the number of solutions of the generalized Ramanujan-Nagell equation D1X2 + DM2 = 2N+2

Abstract

Let D1, D2 be coprime odd integers with min (D1, D2) > 1, and let N (D1, D2) denote the number of positive integer solutions (x, m, n) of the equation D1x2+Dm2 = 2n+2. In this paper, we prove that N (D1, D2) ≤ 2 except for N (3, 5) = N (5, 3) = 4 and N (13, 3) = N (31, 97) = 3.