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On the number of solutions of the generalized Ramanujan-Nagell equation D<sub>1</sub><i>x</i><sup>2</sup> + D<sup><i>m</i></sup><sub>2</sub> = 2<sup><i>n</i>+2</sup>
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.
Keywords: Exponential diophantine equation, generalized Ramanujan-Nagell equation, number of solutions, upper bound
