On the generalized Ramanujan-Nagell equation x2

Let k be a fixed positive integer with k > 1. In 2014, N. Terai [6] conjectured that the equation x(2) + (2k - 1)(y) = k(z) has only the positive integer solution (x, y, z) = (k - 1, 1, 2). This is still an unsolved problem as yet. For any positive integer n, let Q(n) denote the squarefree part of n. In this paper, using some elementary methods, we prove that if k equivalent to 3 (mod 4) and Q(k - 1) >= 2.11 log k, then the equation has only the positive integer solution (x, y, z) = (k -1, 1, 2). It can thus be seen that Terai's conjecture is true for almost all positive integers k with k equivalent to 3(mod 4).