On the generalized Ramanujan—Nagell equation $D_1x^2-D_2=p^n$

Let $D$ be a nonsquare positive integer, and let $D_1$, $D_2$ be positive integers such that $D_1D_2=D$ and $\gcd(D_1,D_2)=1$. Further, let $p$ be an odd prime with $p\nmid D$. Denote by $N(D_1,D_2,p)$ the number of positive integer solutions $(x,n)$ to the generalized Ramanujan—Nagell equation $D_1x^2-D_2=p^n$. In this paper, we prove $N(D_1,D_2,p)\le4$. Moreover, we generalize the case where $D_1=1$ to give the exceptional triple $(D_1,D_2,p)$, which satisfies $$ p^{\nu}=D_1a^2+\eta,\quad D_2=\dfrac{1}{D_1}\left(\dfrac{p^m-\eta}{2a}\right)^2-p^m, \qquad a,\nu,m\in\mathbb{N},\quad \nu＜m,\quad \eta\in\{1,-1\}. $$ In this case, it holds $N(D_1,D_2,p)\ge3$. We prove that if $(D_1,D_2,p)$ is non-exceptional, and the squarefree part of $D_1$ is at most $1001$, then $N(D_1,D_2,p)\le3$.