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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1808.10027","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-08-29T20:07:13Z","cross_cats_sorted":[],"title_canon_sha256":"581d295c95254cb10432bdf6a975fafaf8ce9b485f97f91d1cb12c661eb73c3d","abstract_canon_sha256":"f5181ad50f4f35eacc7fca791859c9a8fa5730a0372e49991b8ede77d68f0301"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:50.067631Z","signature_b64":"vkND9z71CQCiKZEvnI7IoUlgCnefijPOm0YwgTAM6OaRCGvRxafi7OHxRZrCh2QtVFSfLv0etELgIrKxDfV5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc8bdc72c786a9a3c0b98e9c4176f822220a855409bceb0cceeed7da191ff86b","last_reissued_at":"2026-05-18T00:06:50.066946Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:50.066946Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1808.10027","created_at":"2026-05-18T00:06:50.067050+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.10027v1","created_at":"2026-05-18T00:06:50.067050+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.10027","created_at":"2026-05-18T00:06:50.067050+00:00"},{"alias_kind":"pith_short_12","alias_value":"3SF5Y4WHQ2U2","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_16","alias_value":"3SF5Y4WHQ2U2HQFZ","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_8","alias_value":"3SF5Y4WH","created_at":"2026-05-18T12:32:05.422762+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3SF5Y4WHQ2U2HQFZR2OEC5XYEI","json":"https://pith.science/pith/3SF5Y4WHQ2U2HQFZR2OEC5XYEI.json","graph_json":"https://pith.science/api/pith-number/3SF5Y4WHQ2U2HQFZR2OEC5XYEI/graph.json","events_json":"https://pith.science/api/pith-number/3SF5Y4WHQ2U2HQFZR2OEC5XYEI/events.json","paper":"https://pith.science/paper/3SF5Y4WH"},"agent_actions":{"view_html":"https://pith.science/pith/3SF5Y4WHQ2U2HQFZR2OEC5XYEI","download_json":"https://pith.science/pith/3SF5Y4WHQ2U2HQFZR2OEC5XYEI.json","view_paper":"https://pith.science/paper/3SF5Y4WH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.10027&json=true","fetch_graph":"https://pith.science/api/pith-number/3SF5Y4WHQ2U2HQFZR2OEC5XYEI/graph.json","fetch_events":"https://pith.science/api/pith-number/3SF5Y4WHQ2U2HQFZR2OEC5XYEI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3SF5Y4WHQ2U2HQFZR2OEC5XYEI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3SF5Y4WHQ2U2HQFZR2OEC5XYEI/action/storage_attestation","attest_author":"https://pith.science/pith/3SF5Y4WHQ2U2HQFZR2OEC5XYEI/action/author_attestation","sign_citation":"https://pith.science/pith/3SF5Y4WHQ2U2HQFZR2OEC5XYEI/action/citation_signature","submit_replication":"https://pith.science/pith/3SF5Y4WHQ2U2HQFZR2OEC5XYEI/action/replication_record"}},"created_at":"2026-05-18T00:06:50.067050+00:00","updated_at":"2026-05-18T00:06:50.067050+00:00"}