The diophantine equation $\left(2^{k}-1\right)\left(3^{k}-1\right)=x^{n}$

TL;DR

This paper proves that the Diophantine equation (2^k - 1)(3^k - 1) = x^n has no solutions in positive integers for k, x, n > 2, clarifying the equation's solution set.

Contribution

The paper establishes the non-existence of solutions to the specific exponential Diophantine equation for all positive integers greater than 2.

Findings

No solutions for k, x, n > 2

The equation has no positive integer solutions

Clarifies the solution set for the equation

Abstract

In this paper, we investigate the Diophantine equation \[ (2^k - 1)(3^k - 1) = x^n \] and prove that it has no solutions in positive integers $k, x, n > 2$.