An elementary approach to the generalized Ramanujan-Nagell equation

TL;DR

This paper uses elementary number theory techniques to establish criteria that determine when the generalized Ramanujan-Nagell equation has no positive integer solutions for specific exponents.

Contribution

It provides new elementary criteria for the non-existence of solutions to a generalized form of the Ramanujan-Nagell equation with fixed parameters.

Findings

Criteria for no solutions when y=3 or 5

Application of elementary methods in number theory

Conditions depending on the fixed integer k

Abstract

Let $k$ be a fixed positive integer with $k>1$. In this paper, using various elementary methods in number theory, we give criteria under which the equation $x^2+(2k-1)^y=k^z$ has no positive integer solutions $(x,y,z)$ with $y\in\{3,5\}$.