# A note on the Diophantine equation $2^{n-1}(2^{n}-1)=x^3+y^3+z^3$

## TL;DR

This paper proves that for certain values of n, the Diophantine equation involving powers of two and sums of three cubes has multiple solutions, linking perfect numbers to sums of three cubes and providing computational insights.

## Contribution

It establishes the existence of multiple solutions for the equation when n ≡ ±1 mod 6 and connects perfect numbers to sums of three cubes, extending previous results.

## Key findings

- At least two solutions exist for n ≡ ±1 mod 6
- Every even perfect number can be expressed as a sum of three cubes
- Computational results and conjectures related to the equation

## Abstract

Motivated by the recent result of Farhi we show that for each $n\equiv \pm 1\pmod{6}$ the title Diophantine equation has at least two solutions in integers. As a consequence, we get that each (even) perfect number is a sum of three cubes of integers. Moreover, we present some computational results concerning the considered equation and state some questions and conjectures.