An elementary approach to the generalized Ramanujan-Nagell equation

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 $y\in \{3,5\}$.

accepted for publication in Indian Journal of Pure and Applied Mathematics (2023)