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Let D1 and D2 be coprime positive integers with min(D1, D2) > 1, and let p be an odd prime with p 6 |D1D2. Further, let N(D1, D2, p) denote the number of positive integer solutions (x,m, n) of the equation D1x 2 + D 2 = p . In this paper, we prove that N(D1, D2, p) ≤ 2 except for N(2, 7, 3) = N(10, 3, 13) = N(10, 3, 37) = N((32l−1 − 1)/a, 3, 4 · 32l−1 − 1) = 3, where a, l are positive integers.

On the number of solutions of the generalized Ramanujan-Nagell equation $x²-D = 2^{n+2}$