On the solutions of certain Lebesgue-Ramanujan-Nagell equations

TL;DR

This paper completely solves a class of Lebesgue-Ramanujan-Nagell equations involving powers of 2, 11, and 19, providing a comprehensive set of solutions and extending previous results in the field.

Contribution

The paper offers a complete solution to a specific family of exponential Diophantine equations, filling gaps in existing literature and handling previously unresolved cases.

Findings

Explicit solutions for the equation in most cases

Identification of exceptional cases where solutions are not found

Extension of earlier results in the same area

Abstract

We completely solve the Diophantine equation $x^2+2^k11^\ell19^m=y^n$ in integers $x,y\geq 1;~ k,\ell, m\geq 0~$ and $n\geq 3$ 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.