{"paper":{"title":"There are no Cube-free Descartes Numbers with Exactly Seven Distinct Prime Factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Pratik Rathore","submitted_at":"2018-08-29T20:07:13Z","abstract_excerpt":"We call an odd positive integer $n$ a $\\textit{Descartes number}$ if there exist positive integers $k,m$ such that $n = km$ and\n  \\begin{equation} \\sigma(k)(m+1) = 2km \\end{equation}\n  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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.10027","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}