A modular approach to the generalized Ramanujan–Nagell equation

Abstract

Let k be a positive integer. In this paper, using the modular approach, we prove that if k≡0(mod4), 30<k<724 and 2k−1 is an odd prime power, then under the GRH, the equation x2+(2k−1)y=kz has only one positive integer solution (x,y,z)=(k−1,1,2). The above results solve some difficult cases of Terai’s conjecture concerning this equation.