On the Diophantine equation x^2+7^{alpha}.11^{beta}=y^n

TL;DR

This paper completely solves a specific class of Diophantine equations involving squares and prime powers, providing all solutions under certain coprimality and parity conditions.

Contribution

It offers a complete characterization of solutions to the equation x^2 + 7^alpha * 11^beta = y^n for nonnegative integers with coprimality and parity restrictions, extending previous partial results.

Findings

All solutions for x^2 + 7^alpha * 11^beta = y^n are determined under specified conditions.

The solutions exclude cases where alpha.x is odd and beta is even.

The paper clarifies the structure of solutions for this class of exponential Diophantine equations.

Abstract

In this paper, we give all the solutions of the Diophantine equation x^2+7^{alpha}.11^{beta}=y^n, in nonnegative integers x, y, n>=3 with x and y coprime, except for the case when alpha.x is odd and beta is even.