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There have been many papers concerned with upper bounds for N(D, k). Let Ci (i = 1, 2, . . .) denote effectively computable absolute constants. The known results include the following: 1 (Apéry [1, 2]). If D < 0, k is a prime and (D, k) 6= (−7, 2), then N(D, k) ≤ 2. 2 (Beukers [3]). If D < −7, then N(−23, 2) = N(−2r + 1, 2) = 2 for some r ∈ N, otherwise N(D, 2) ≤ 1. 3 (Le [10]). If D < 0, k is an odd prime and |D| > C1, then N(−3s2 − 1, 4s2 + 1) = 2 for some s ∈ N, otherwise N(D, k) ≤ 1. 4 (Xu and Le [15]). If D < 0, 2 k and |D| > C2, then

A note on the number of solutions of the generalized Ramanujan-Nagell equation $x²-D = k^n$