On the solutions of certain Lebesgue–Ramanujan–Nagell equations

Abstract

We completely solve the Diophantine equation $x^2+2^{k}11^{\ell }19^m=y^n$ in integers $x,y\ge 1$; $k,\ell ,m\ge 0$ and $n\ge 3$ with $gcd\left(x,y\right)=1$, except the case $2|k$, $2\nmid \ell m$ and $5|n$. We use this result to recover some earlier results in the same direction.