There are no Cube-free Descartes Numbers with Exactly Seven Distinct Prime Factors

TL;DR

This paper proves that no cube-free Descartes number exists with exactly seven distinct prime factors, extending previous results that limited such numbers to fewer prime factors.

Contribution

The authors extend existing methods to demonstrate the non-existence of cube-free Descartes numbers with seven distinct prime factors.

Findings

No cube-free Descartes number with exactly seven distinct prime factors exists.

The proof builds on and extends methods from prior work by Banks et al.

The unique known Descartes number remains the only one with fewer than seven prime factors.

Abstract

We call an odd positive integer $n$ a $\textit{Descartes number}$ if there exist positive integers $k,m$ such that $n = km$ and \begin{equation} \sigma(k)(m+1) = 2km \end{equation} Currently, $\mathcal{D} = 3^{2}7^{2}11^{2}13^{2}22021$ is the only known Descartes number. In $2008$, Banks et al. proved that $\mathcal{D}$ is the only cube-free Descartes number with fewer than seven distinct prime factors. In the present paper, we extend the methods of Banks et al. to show that there is no cube-free Descartes number with seven distinct prime factors.