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Let      D D       be a positive integer which is odd. In this paper we prove that the equation      x 2 −  D = 2 n + 2 {x^2} - D = {2^{n + 2}}       has at most three positive integer solutions      ( x , n ) (x,n)       except when      D = 2 2 m −  3 ⋅  2 m + 1 + 1 D = {2^{2m}} - 3 \cdot {2^{m + 1}} + 1       , where      m m       is a positive integer with      m ≥  3 m \geq 3       .

On the generalized Ramanujan-Nagell equation 𝑥²-𝐷=2ⁿ⁺²