Let d, k be fixed coprime positive integers with min{d, k} > 1. A class of polynomial-exponential Diophantine equations of the form x^2 + d^y = k^z , x, y, z ∈ Z+ (1) is usually called the generalized Ramanujan-Nagell equation. It has a long history and rich content. In 2014, N. Terai discussed the solution of (1) in the case d = 2k − 1. He conjectured that for any k with k > 1, the equation x^2 + (2k − 1)^y = k^z , x, y, z ∈ Z+ (2) has only one solution (x, y, z) = (k − 1, 1, 2). The above conjecture has been verified in some special cases. In this work, firstly, using the modular approach, we prove that if k ≡ 0 (mod 4), 30 < k < 724 and 2k − 1 is an odd prime power, then under the GRH, the equation (2) has only one positive integer solution (x, y, z) = (k − 1, 1, 2). The above results solve some difficult cases of Terai’s conecture concerning the equation (2). Secondly, using various elementary methods in number theory, we give certain criterions which can make the equation (2) to have no positive integer solutions (x, y, z) with y ∈ {3, 5}. These results make up the defiency of the modular approach when applied to (2). This is a joint work with Maohua Le and Elif Kızıldere Mutlu.