On the Diophantine equation \(px^2+3^n=y^p\) (Q2719892)

Let \(p\) be an odd prime with \(p>3\). In this paper the author proves that if \(p\not\equiv 7\pmod 8\), then the equation \(px^2+3^{2m}= y^p\), \(x,y,m\in \mathbb{N}\) has no solutions \((x,y,m)\) with \(\gcd (3,y)=1\); if \(p\not\equiv 5\pmod 8\), then the equation \(px^2+3^{2m+1}= y^p\), \(x,y\in \mathbb{N}\) has no solutions \((x,y,m)\) with \(\gcd (3,y)=1\).