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 $\gcd (x,y)=1$, except the case $2\mid k,2\nmid \ell m$ and $5\mid n$. We use this result to recover some earlier results in the same direction.

16 pages. Minor corrections have been made. To appear in Rocky Mountain Journal of Mathematics