{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:OMEKICKVF47ACU32XQCMOL7PEX","short_pith_number":"pith:OMEKICKV","schema_version":"1.0","canonical_sha256":"7308a409552f3e01537abc04c72fef25d591821b65c8fa2640e4a7555df79f73","source":{"kind":"arxiv","id":"1412.0523","version":2},"attestation_state":"computed","paper":{"title":"Two congruences involving harmonic numbers with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Guo-Shuai Mao, Zhi-Wei Sun","submitted_at":"2014-11-28T14:07:57Z","abstract_excerpt":"The harmonic numbers $H_n=\\sum_{0<k\\le n}1/k\\ (n=0,1,2,\\ldots)$ play important roles in mathematics. Let $p>3$ be a prime. With helps of some combinatorial identities, we establish the following two new congruences: $$\\sum_{k=1}^{p-1}\\frac{\\binom{2k}k}kH_k\\equiv\\frac13\\left(\\frac p3\\right)B_{p-2}\\left(\\frac13\\right)\\pmod{p}$$ and $$\\sum_{k=1}^{p-1}\\frac{\\binom{2k}k}kH_{2k}\\equiv\\frac7{12}\\left(\\frac p3\\right)B_{p-2}\\left(\\frac13\\right)\\pmod{p},$$ where $B_n(x)$ denotes the Bernoulli polynomial of degree $n$. As an application, we determine $\\sum_{n=1}^{p-1}g_n$ and $\\sum_{n=1}^{p-1}h_n$ modulo"},"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":"1412.0523","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-11-28T14:07:57Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d0e8440bbb47382a52968b5b616b692c53ce4e263d872f18c2878856300438c4","abstract_canon_sha256":"ff2bca244718a5c9862ae790070a93a77da16d9bcdb0b899a37b46dd2efde0f0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:03.779557Z","signature_b64":"J9njMq5rPmRZ723+ATzHqdhET/1iy2YAWhMWvSJT5isFH51//oL73tbnMyR02psEo4UU1NETNS/JEKdruYIVAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7308a409552f3e01537abc04c72fef25d591821b65c8fa2640e4a7555df79f73","last_reissued_at":"2026-05-18T01:20:03.779003Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:03.779003Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two congruences involving harmonic numbers with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Guo-Shuai Mao, Zhi-Wei Sun","submitted_at":"2014-11-28T14:07:57Z","abstract_excerpt":"The harmonic numbers $H_n=\\sum_{0<k\\le n}1/k\\ (n=0,1,2,\\ldots)$ play important roles in mathematics. Let $p>3$ be a prime. With helps of some combinatorial identities, we establish the following two new congruences: $$\\sum_{k=1}^{p-1}\\frac{\\binom{2k}k}kH_k\\equiv\\frac13\\left(\\frac p3\\right)B_{p-2}\\left(\\frac13\\right)\\pmod{p}$$ and $$\\sum_{k=1}^{p-1}\\frac{\\binom{2k}k}kH_{2k}\\equiv\\frac7{12}\\left(\\frac p3\\right)B_{p-2}\\left(\\frac13\\right)\\pmod{p},$$ where $B_n(x)$ denotes the Bernoulli polynomial of degree $n$. As an application, we determine $\\sum_{n=1}^{p-1}g_n$ and $\\sum_{n=1}^{p-1}h_n$ modulo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.0523","kind":"arxiv","version":2},"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":"1412.0523","created_at":"2026-05-18T01:20:03.779095+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.0523v2","created_at":"2026-05-18T01:20:03.779095+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.0523","created_at":"2026-05-18T01:20:03.779095+00:00"},{"alias_kind":"pith_short_12","alias_value":"OMEKICKVF47A","created_at":"2026-05-18T12:28:41.024544+00:00"},{"alias_kind":"pith_short_16","alias_value":"OMEKICKVF47ACU32","created_at":"2026-05-18T12:28:41.024544+00:00"},{"alias_kind":"pith_short_8","alias_value":"OMEKICKV","created_at":"2026-05-18T12:28:41.024544+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/OMEKICKVF47ACU32XQCMOL7PEX","json":"https://pith.science/pith/OMEKICKVF47ACU32XQCMOL7PEX.json","graph_json":"https://pith.science/api/pith-number/OMEKICKVF47ACU32XQCMOL7PEX/graph.json","events_json":"https://pith.science/api/pith-number/OMEKICKVF47ACU32XQCMOL7PEX/events.json","paper":"https://pith.science/paper/OMEKICKV"},"agent_actions":{"view_html":"https://pith.science/pith/OMEKICKVF47ACU32XQCMOL7PEX","download_json":"https://pith.science/pith/OMEKICKVF47ACU32XQCMOL7PEX.json","view_paper":"https://pith.science/paper/OMEKICKV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.0523&json=true","fetch_graph":"https://pith.science/api/pith-number/OMEKICKVF47ACU32XQCMOL7PEX/graph.json","fetch_events":"https://pith.science/api/pith-number/OMEKICKVF47ACU32XQCMOL7PEX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OMEKICKVF47ACU32XQCMOL7PEX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OMEKICKVF47ACU32XQCMOL7PEX/action/storage_attestation","attest_author":"https://pith.science/pith/OMEKICKVF47ACU32XQCMOL7PEX/action/author_attestation","sign_citation":"https://pith.science/pith/OMEKICKVF47ACU32XQCMOL7PEX/action/citation_signature","submit_replication":"https://pith.science/pith/OMEKICKVF47ACU32XQCMOL7PEX/action/replication_record"}},"created_at":"2026-05-18T01:20:03.779095+00:00","updated_at":"2026-05-18T01:20:03.779095+00:00"}