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In particular, any nonnegative integer is a sum of five pentagonal numbers two of which are equal; this refines a classical result of Cauchy claimed by Fermat."},"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":"2001.05325","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-01-14T14:59:35Z","cross_cats_sorted":[],"title_canon_sha256":"2aba0bae5653b71ad30b01cb88ed0dc4e90b2f185fb9951d7a7efaec98674185","abstract_canon_sha256":"aecadab99e3b9197e86520c7d30987e08df201fceae3cdd77178db76d0690cb2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:51:38.348482Z","signature_b64":"fS/A2lHvu+xnfGZ7XPA/B66VkwFs/PnO1laot89WtUBabGMWMDk9yCEBRe/mQVoU6ney0QRLYa8xL2jLo5edAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dbd9af2d305aaf81bcaf90134c5b2dd80f0369b84659d96e0415683bd0cf1b34","last_reissued_at":"2026-07-05T00:51:38.347944Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:51:38.347944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On sums of four pentagonal numbers with coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dmitry Krachun, Zhi-Wei Sun","submitted_at":"2020-01-14T14:59:35Z","abstract_excerpt":"The pentagonal numbers are the integers given by $p_5(n)=n(3n-1)/2\\ (n=0,1,2,\\ldots)$. Let $(b,c,d)$ be one of the triples $(1,1,2),(1,2,3),(1,2,6)$ and $(2,3,4)$. We show that each $n=0,1,2,\\ldots$ can be written as $w+bx+cy+dz$ with $w,x,y,z$ pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integer is a sum of five pentagonal numbers two of which are equal; this refines a classical result of Cauchy claimed by Fermat."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2001.05325","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2001.05325/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2001.05325","created_at":"2026-07-05T00:51:38.348005+00:00"},{"alias_kind":"arxiv_version","alias_value":"2001.05325v4","created_at":"2026-07-05T00:51:38.348005+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2001.05325","created_at":"2026-07-05T00:51:38.348005+00:00"},{"alias_kind":"pith_short_12","alias_value":"3PM26LJQLKXY","created_at":"2026-07-05T00:51:38.348005+00:00"},{"alias_kind":"pith_short_16","alias_value":"3PM26LJQLKXYDPFP","created_at":"2026-07-05T00:51:38.348005+00:00"},{"alias_kind":"pith_short_8","alias_value":"3PM26LJQ","created_at":"2026-07-05T00:51:38.348005+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/3PM26LJQLKXYDPFPSAJUYWZN3A","json":"https://pith.science/pith/3PM26LJQLKXYDPFPSAJUYWZN3A.json","graph_json":"https://pith.science/api/pith-number/3PM26LJQLKXYDPFPSAJUYWZN3A/graph.json","events_json":"https://pith.science/api/pith-number/3PM26LJQLKXYDPFPSAJUYWZN3A/events.json","paper":"https://pith.science/paper/3PM26LJQ"},"agent_actions":{"view_html":"https://pith.science/pith/3PM26LJQLKXYDPFPSAJUYWZN3A","download_json":"https://pith.science/pith/3PM26LJQLKXYDPFPSAJUYWZN3A.json","view_paper":"https://pith.science/paper/3PM26LJQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2001.05325&json=true","fetch_graph":"https://pith.science/api/pith-number/3PM26LJQLKXYDPFPSAJUYWZN3A/graph.json","fetch_events":"https://pith.science/api/pith-number/3PM26LJQLKXYDPFPSAJUYWZN3A/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3PM26LJQLKXYDPFPSAJUYWZN3A/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3PM26LJQLKXYDPFPSAJUYWZN3A/action/storage_attestation","attest_author":"https://pith.science/pith/3PM26LJQLKXYDPFPSAJUYWZN3A/action/author_attestation","sign_citation":"https://pith.science/pith/3PM26LJQLKXYDPFPSAJUYWZN3A/action/citation_signature","submit_replication":"https://pith.science/pith/3PM26LJQLKXYDPFPSAJUYWZN3A/action/replication_record"}},"created_at":"2026-07-05T00:51:38.348005+00:00","updated_at":"2026-07-05T00:51:38.348005+00:00"}