On the generalized Ramanujan-Nagell equation x2 + qm = cn with qr+1=2c2

Let q be an odd prime, and let c, r be positive integers with q(r) + 1 = 2c(2). For any nonnegative integer S, let U2s+1 =(alpha(2s+1) + beta(2s+1)) /2 root 2 and V2s+1 = (alpha(2s+1) - beta(2s+1)) /2 root 2 where alpha = 1 + root 2 and beta = 1- root 2 In this paper we prove the following results: (i) If r > 2, then (q, r, c) = (23,3, 78) and the equation x(2)+23(m) = 78(n) has only the positive integer solution (x, m, n) = (6083, 3, 4). (ii) If r = 2 and (q, c) = (U2s+1,V2s+1) with s not equivalent to 0(mod4), then the equation x(2) + q(m) = c(n) has only the positive integer solution (x, m, n) = (c(2) - 1, 2, 4).