On the Diophantine Equation $x^{2}+5^{a}\cdot 11^{b}=y^{n} $

TL;DR

This paper completely solves the Diophantine equation involving squares and prime powers for coprime integers, except when all variables are odd, providing a comprehensive classification of solutions.

Contribution

It offers the first complete solution to the specific exponential Diophantine equation under the coprimality condition, excluding the odd case.

Findings

Complete solutions for coprime integers when gcd(x,y)=1

Identification of solutions excluding the case when x, a, b are all odd

Advancement in understanding equations involving prime powers and squares

Abstract

We give the complete solution in integers $(n,a,b,x,y)$ of the title equation when $\gcd(x,y)=1$, except for the case when $xab$ is odd.