The Diophantine equation $2 x^2 + 1 = 3^n$

Abstract:

Let $p$ be a rational prime and $D$ a positive rational integer coprime with $p$. Denote by $N(D, 1,p)$ the number of solutions $(x, n)$ of the equation $D x^2 + 1 = p^n$ in rational integers $x \geq 1$ and $n \geq 1$. In a paper of Le, he claimed that $N(D, 1, p) \leq 2$ without giving a proof. Furthermore, the statement $N(D, 1, p) \leq 2$ has been used by Le, Bugeaud and Shorey in their papers to derive results on certain Diophantine equations. In this paper we point out that the statement $N(D, 1, p) \leq 2$ is incorrect by proving that $N(2, 1, 3)=3$.