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

TL;DR

This paper completely solves the exponential Diophantine equation involving squares and prime powers, providing all positive integer solutions under specific coprimality and exponent conditions.

Contribution

It offers a complete characterization of solutions for the equation with coprimality and exponent constraints, extending existing results in exponential Diophantine equations.

Findings

All solutions with gcd(x, y) = 1 are determined.

Solutions are characterized for various values of exponents.

The paper extends known results to a specific form involving prime powers.

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 $\text{gcd}(x, y) = 1$.