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 + D^m₂ = 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.

Keywords: Exponential diophantine equation, generalized Ramanujan-Nagell equation, number of solutions, upper bound