A modular approach to the generalized Ramanujan-Nagell equation

TL;DR

This paper uses the modular approach to prove the uniqueness of solutions for a generalized Ramanujan-Nagell equation under specific conditions, advancing the understanding of Terai's conjecture in number theory.

Contribution

It establishes new results on the solutions of the generalized Ramanujan-Nagell equation for certain parameters, under the assumption of GRH, solving some difficult cases of Terai's conjecture.

Findings

Proves the equation has only one positive solution under specified conditions

Solves some difficult cases of Terai's conjecture

Utilizes the modular approach and GRH assumptions

Abstract

Let $k$ be a positive integer. In this paper, using the modular approach, we prove that if $k\equiv 0 \pmod{4}$, $30< k<724$ 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.