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