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 $k\equiv 0(\mathrm{mod}4)$, $30<k<724$ and $2k-1$ is an odd prime power, then under the GRH, the equation $x^2+(2k-1{)}^y=k^z$ 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 this equation.

10 pages, accepted for publication in Indagationes Mathematicae