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

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 $x,y\ge 1$), non-negative exponents $\alpha$ and $\beta$, and an integer $\mathfrak{n}\ge 3$, subject to the condition $\text{gcd}(x,y)=1$.