An Exponential Diophantine equation $x^2+3^{\alpha} 113^{\beta}=y^{\mathfrak{n}}$

Abstract

The objective of the paper is to determine the complete solutions for the Diophantine equation $x^2 + 3^{\alpha}113^{\beta} = y^{\mathfrak{n}}$ in positive integers $x$ and $y$ (where $x, y \geq 1$), non-negative exponents $\alpha$ and $\beta$, and an integer $\mathfrak{n}\geq 3$, subject to the condition $\text{gcd}(x, y) = 1$.