{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2004:G6KLMU2NGTYU3HKHCITN2YMEDJ","short_pith_number":"pith:G6KLMU2N","schema_version":"1.0","canonical_sha256":"3794b6534d34f14d9d471226dd61841a7e797fa691511550f4a90d7a5277b4e2","source":{"kind":"arxiv","id":"math/0411305","version":4},"attestation_state":"computed","paper":{"title":"A connection between covers of the integers and unit fractions","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2004-11-15T15:56:26Z","abstract_excerpt":"For integers a and n>0, let a(n) denote the residue class {x\\in Z: x=a (mod n)}. Let A be a collection {a_s(n_s)}_{s=1}^k of finitely many residue classes such that A covers all the integers at least m times but {a_s(n_s)}_{s=1}^{k-1} does not. We show that if n_k is a period of the covering function w_A(x)=|{1\\le s\\le k: x\\in a_s(n_s)}| then for any r=0,...,n_k-1 there are at least m integers in the form $\\sum_{s\\in I}1/n_s-r/n_k$ with I contained in {1,...,k-1}."},"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":"math/0411305","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2004-11-15T15:56:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5f654ba5fda91863dec34104ec6adda06c08e4d01f5b2d2dbff782e760f8d856","abstract_canon_sha256":"236a00ad1591e18113c863828e42e3f79c8bdaa9567492c5f7993cda43ee12d9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:50:08.355532Z","signature_b64":"DjMlxsEmPxu0L/WLu0cty49H4J0A8Ha4lZ7Hv9ildtAvAAEuPIiIKr24EmrmXd0wYVhk4TBnSdQfrgShASVcAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3794b6534d34f14d9d471226dd61841a7e797fa691511550f4a90d7a5277b4e2","last_reissued_at":"2026-07-04T14:50:08.355171Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:50:08.355171Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A connection between covers of the integers and unit fractions","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2004-11-15T15:56:26Z","abstract_excerpt":"For integers a and n>0, let a(n) denote the residue class {x\\in Z: x=a (mod n)}. Let A be a collection {a_s(n_s)}_{s=1}^k of finitely many residue classes such that A covers all the integers at least m times but {a_s(n_s)}_{s=1}^{k-1} does not. We show that if n_k is a period of the covering function w_A(x)=|{1\\le s\\le k: x\\in a_s(n_s)}| then for any r=0,...,n_k-1 there are at least m integers in the form $\\sum_{s\\in I}1/n_s-r/n_k$ with I contained in {1,...,k-1}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0411305","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/math/0411305/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":"math/0411305","created_at":"2026-07-04T14:50:08.355230+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0411305v4","created_at":"2026-07-04T14:50:08.355230+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0411305","created_at":"2026-07-04T14:50:08.355230+00:00"},{"alias_kind":"pith_short_12","alias_value":"G6KLMU2NGTYU","created_at":"2026-07-04T14:50:08.355230+00:00"},{"alias_kind":"pith_short_16","alias_value":"G6KLMU2NGTYU3HKH","created_at":"2026-07-04T14:50:08.355230+00:00"},{"alias_kind":"pith_short_8","alias_value":"G6KLMU2N","created_at":"2026-07-04T14:50:08.355230+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/G6KLMU2NGTYU3HKHCITN2YMEDJ","json":"https://pith.science/pith/G6KLMU2NGTYU3HKHCITN2YMEDJ.json","graph_json":"https://pith.science/api/pith-number/G6KLMU2NGTYU3HKHCITN2YMEDJ/graph.json","events_json":"https://pith.science/api/pith-number/G6KLMU2NGTYU3HKHCITN2YMEDJ/events.json","paper":"https://pith.science/paper/G6KLMU2N"},"agent_actions":{"view_html":"https://pith.science/pith/G6KLMU2NGTYU3HKHCITN2YMEDJ","download_json":"https://pith.science/pith/G6KLMU2NGTYU3HKHCITN2YMEDJ.json","view_paper":"https://pith.science/paper/G6KLMU2N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0411305&json=true","fetch_graph":"https://pith.science/api/pith-number/G6KLMU2NGTYU3HKHCITN2YMEDJ/graph.json","fetch_events":"https://pith.science/api/pith-number/G6KLMU2NGTYU3HKHCITN2YMEDJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/G6KLMU2NGTYU3HKHCITN2YMEDJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/G6KLMU2NGTYU3HKHCITN2YMEDJ/action/storage_attestation","attest_author":"https://pith.science/pith/G6KLMU2NGTYU3HKHCITN2YMEDJ/action/author_attestation","sign_citation":"https://pith.science/pith/G6KLMU2NGTYU3HKHCITN2YMEDJ/action/citation_signature","submit_replication":"https://pith.science/pith/G6KLMU2NGTYU3HKHCITN2YMEDJ/action/replication_record"}},"created_at":"2026-07-04T14:50:08.355230+00:00","updated_at":"2026-07-04T14:50:08.355230+00:00"}