On the diophantine equation $px^2+3^n=y^p$

Abstract

Let $p$ be a prime. In this paper we prove that: (1) the equation $px^2+3^{2m}=y^p$, $p\not\equiv 7$ (mod 8) has no solutions in positive integers $(x,y,p)$ with $\hbox{gcd}(3,y)=1$. (2) if $3|y$ then the equation has at most one solution when $p=3$ and for $p>3$ it may have a solution only when $p\equiv 2$ (mod 3) and $\hbox{gcd}(p,m)=1$. (3) the equation $px^2+3^{2m+1}=y^p$, $p\not\equiv 5$ (mod 8), has a solution only when $p=3$, and this solution exists only when $m=3+3M$, and is given by $x=46.3^{3M+1}$, $y=13.3^{2M+1}$.