The Diophantine equation 2𝑥²+1=3ⁿ

Let p p be a rational prime and D D a positive rational integer coprime with p p . Denote by N ( D , 1 , p ) N(D, 1,p) the number of solutions ( x , n ) (x, n) of the equation D x 2 + 1 = p n D x^2 + 1 = p^n in rational integers x ≥ 1 x \geq 1 and n ≥ 1 n \geq 1 . In a paper of Le, he claimed that N ( D , 1 , p ) ≤ 2 N(D, 1, p) \leq 2 without giving a proof. Furthermore, the statement N ( D , 1 , p ) ≤ 2 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 ) ≤ 2 N(D, 1, p) \leq 2 is incorrect by proving that N ( 2 , 1 , 3 ) = 3 N(2, 1, 3)=3 .