Solutions of some generalized Ramanujan-Nagell equations

Abstract

in positive integers x, y,D and n > 2 with gcd(x, y)= 1. When D = 1, the equation has no solution by an old result of Lebesgue [14]. We assume from now on that D > 1. Eq. (1) has been extensively studied by many authors, in particular, by Cohn and Le. See [8,10–13] for several results. We also refer to [8] for a survey. The equation is referred as the generalized Ramanujan–Nagell equation who pioneered the study on (1). In his paper, Cohn solved (1) completely for 77 values of D 100. The values D = 74,86 were solved by Mignotte and de Weger [17] and D = 55,95 were solved by Bennett and Skinner [2]. Recently, Bugeaud, Mignotte and Siksek [6] covered the remaining 19 values 100. We write