# Complete solution of the Diophantine Equation $x^{2}+5^{a}\cdot 11^{b}=y^{n}$

## TL;DR

This paper completely solves a specific exponential Diophantine equation involving powers of 5 and 11, including the first explicit resolution of a quintic Thue-Mahler equation, providing both results and detailed methodology.

## Contribution

It provides the first explicit solution to a quintic Thue-Mahler equation and fully resolves the Diophantine equation involving powers of 5 and 11.

## Key findings

- Complete classification of solutions for the equation
- First explicit resolution of a quintic Thue-Mahler equation
- Methodological insights into solving complex Thue-Mahler equations

## Abstract

The title equation is completely solved in integers $(n,x,y,a,b)$, where $n\geq 3$, $\gcd(x,y)=1$ and $a,b\geq 0$. The most difficult stage of the resolution is the explicit resolution of a quintic Thue-Mahler equation. Since it is for the first time -to the best of our knowledge- that such an equation is solved in the literature, we make a detailed presentation of the resolution; this gives our paper also an expository character.