{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:HW27QSY6KGXM6GQD4W47SW3D4F","short_pith_number":"pith:HW27QSY6","schema_version":"1.0","canonical_sha256":"3db5f84b1e51aecf1a03e5b9f95b63e166ebffed01d3c9964907bb1a9f2fecff","source":{"kind":"arxiv","id":"1407.8465","version":5},"attestation_state":"computed","paper":{"title":"New congruences involving harmonic numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2014-07-31T15:53:32Z","abstract_excerpt":"Let $p>3$ be a prime. For any $p$-adic integer $a$, we determine $$\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k,\\ \\ \\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k^{(2)},\\ \\ \\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}k\\frac{H_k^{(2)}}{2k+1}$$ modulo $p^2$, where $H_k=\\sum_{0<j\\le k}1/j$ and $H_k^{(2)}=\\sum_{0<j\\le k}1/j^2$. In particular, we show that \\begin{gather*}\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k\\equiv(-1)^{\\langle a\\rangle_p}\\,2\\left(B_{p-1}(a)-B_{p-1}\\right)\\pmod p, \\\\\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k^{(2)}\\equiv -E_{p-3}(a)\\pmod p, \\\\(2a-1)\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}k\\fr"},"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":"1407.8465","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-31T15:53:32Z","cross_cats_sorted":[],"title_canon_sha256":"09589902e164767c41a4558d2917079b1d8838421c384c06e2e6fb8a97de5498","abstract_canon_sha256":"398a2aba4cc3dd76c8e5eb7c09780192aaec2322be8a4a2999f9bd8a74909c46"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:31:49.741580Z","signature_b64":"YlkG4sosnyQKSqAZ+KPcO5Iruii5SWugxXGKI67cSXHIiS4vIAP7KoRLv2jJg4jGzXjt8DcAfi0us6Z7WXCcDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3db5f84b1e51aecf1a03e5b9f95b63e166ebffed01d3c9964907bb1a9f2fecff","last_reissued_at":"2026-07-05T07:31:49.741043Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:31:49.741043Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New congruences involving harmonic numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2014-07-31T15:53:32Z","abstract_excerpt":"Let $p>3$ be a prime. For any $p$-adic integer $a$, we determine $$\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k,\\ \\ \\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k^{(2)},\\ \\ \\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}k\\frac{H_k^{(2)}}{2k+1}$$ modulo $p^2$, where $H_k=\\sum_{0<j\\le k}1/j$ and $H_k^{(2)}=\\sum_{0<j\\le k}1/j^2$. In particular, we show that \\begin{gather*}\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k\\equiv(-1)^{\\langle a\\rangle_p}\\,2\\left(B_{p-1}(a)-B_{p-1}\\right)\\pmod p, \\\\\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k^{(2)}\\equiv -E_{p-3}(a)\\pmod p, \\\\(2a-1)\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}k\\fr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.8465","kind":"arxiv","version":5},"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/1407.8465/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":"1407.8465","created_at":"2026-07-05T07:31:49.741103+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.8465v5","created_at":"2026-07-05T07:31:49.741103+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.8465","created_at":"2026-07-05T07:31:49.741103+00:00"},{"alias_kind":"pith_short_12","alias_value":"HW27QSY6KGXM","created_at":"2026-07-05T07:31:49.741103+00:00"},{"alias_kind":"pith_short_16","alias_value":"HW27QSY6KGXM6GQD","created_at":"2026-07-05T07:31:49.741103+00:00"},{"alias_kind":"pith_short_8","alias_value":"HW27QSY6","created_at":"2026-07-05T07:31:49.741103+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/HW27QSY6KGXM6GQD4W47SW3D4F","json":"https://pith.science/pith/HW27QSY6KGXM6GQD4W47SW3D4F.json","graph_json":"https://pith.science/api/pith-number/HW27QSY6KGXM6GQD4W47SW3D4F/graph.json","events_json":"https://pith.science/api/pith-number/HW27QSY6KGXM6GQD4W47SW3D4F/events.json","paper":"https://pith.science/paper/HW27QSY6"},"agent_actions":{"view_html":"https://pith.science/pith/HW27QSY6KGXM6GQD4W47SW3D4F","download_json":"https://pith.science/pith/HW27QSY6KGXM6GQD4W47SW3D4F.json","view_paper":"https://pith.science/paper/HW27QSY6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.8465&json=true","fetch_graph":"https://pith.science/api/pith-number/HW27QSY6KGXM6GQD4W47SW3D4F/graph.json","fetch_events":"https://pith.science/api/pith-number/HW27QSY6KGXM6GQD4W47SW3D4F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HW27QSY6KGXM6GQD4W47SW3D4F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HW27QSY6KGXM6GQD4W47SW3D4F/action/storage_attestation","attest_author":"https://pith.science/pith/HW27QSY6KGXM6GQD4W47SW3D4F/action/author_attestation","sign_citation":"https://pith.science/pith/HW27QSY6KGXM6GQD4W47SW3D4F/action/citation_signature","submit_replication":"https://pith.science/pith/HW27QSY6KGXM6GQD4W47SW3D4F/action/replication_record"}},"created_at":"2026-07-05T07:31:49.741103+00:00","updated_at":"2026-07-05T07:31:49.741103+00:00"}