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

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 $N(13,3)=N(31,97)=3$.