On the number of solutions of the generalized Ramanujan-Nagell equation $D_{1}x^2+D_{2}^{m}=2^{n+2}$}

TL;DR

This paper investigates the solutions of a generalized Ramanujan-Nagell equation involving coprime odd integers, establishing upper bounds on the number of solutions and identifying specific exceptions.

Contribution

It proves that the number of positive solutions is at most two for most cases, with explicit exceptions where the solution count is higher.

Findings

Most cases have at most two solutions.

Explicit exceptions with up to four solutions identified.

Provides bounds on solutions for a class of exponential Diophantine equations.

Abstract

Let $D_{1}$, $D_{2}$ be coprime odd integers with min$(D_{1},D_{2})>1$, and let $N(D_{1},D_{2})$ denote the number of positive integer solutions $(x, m, n)$ of the equation $D_{1}x^2+D_{2}^{m}=2^{n+2}$. In this paper, we prove that $N(D_{1},D_{2})\leq 2$ except for $N(3,5)=N(5,3)=4$ and $N(13,3)=N(31,97)=3$.