On the solutions of a Lebesgue - Nagell type equation

TL;DR

This paper completely characterizes positive integer solutions to a specific Lebesgue-Nagell type equation involving powers of 19 and a parameterized exponent, expanding understanding of such Diophantine equations.

Contribution

It provides a complete solution set for the equation $x^2 + 19^{2k+1} = 4 y^n$ for all non-negative integers $k$, which was previously unresolved.

Findings

All solutions for the equation are explicitly determined.

The solutions depend on the parity of $k$ and the value of $n$.

The paper establishes conditions under which solutions exist.

Abstract

We find all positive integer solutions in $x, y$ and $n$ of $x^2+19^{2k+1}=4y^{n}$ for any non-negative integer $k$.