{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:55VYV45RAYL63XT7QQMA2YBINR","short_pith_number":"pith:55VYV45R","schema_version":"1.0","canonical_sha256":"ef6b8af3b10617edde7f84180d60286c445df6b1e0f214f62969b7cbd19e5ea8","source":{"kind":"arxiv","id":"1410.1712","version":2},"attestation_state":"computed","paper":{"title":"A New Super Congruence Involving Multiple Harmonic Sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2014-10-07T13:01:48Z","abstract_excerpt":"Let ${\\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p\\ge 5$ and $r\\ge 2$, we prove that \\begin{equation}\n  \\sum\\limits_{\\begin{smallmatrix}\n  {{l}_{1}}+{{l}_{2}}+\\cdots +{{l}_{5}}={{p}^{r}}\n  {{l}_{1}},\\cdots ,{{l}_{5}}\\in {\\mathcal{P}_{p}} \\end{smallmatrix}}{\\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}{{l}_{5}}}}\\equiv -\\frac{5!}{6}{{B}_{p-5}}{{p}^{r-1}} \\pmod{{{p}^{r}}}. \\end{equation}\n  This gives an extension of a family of super congruences found by Wang, Cai and Zhao."},"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":"1410.1712","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-07T13:01:48Z","cross_cats_sorted":[],"title_canon_sha256":"8ccfd48d6fee0ef26eb263093684f6616f44eae840da77619bbb5bb2f42c309d","abstract_canon_sha256":"9487330185aa66562ce02c109f158a777403ffa8f78b9855678682615f26f004"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:40:17.586920Z","signature_b64":"+4iRA6gEGUpT6/00xVAyZaleUHOMJ3OHapx47JIBMDjB1ivknqf5L29g37P39Ip5gHjxHEbcIpIA4cA8UGMfAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ef6b8af3b10617edde7f84180d60286c445df6b1e0f214f62969b7cbd19e5ea8","last_reissued_at":"2026-05-18T02:40:17.586260Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:40:17.586260Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A New Super Congruence Involving Multiple Harmonic Sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2014-10-07T13:01:48Z","abstract_excerpt":"Let ${\\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p\\ge 5$ and $r\\ge 2$, we prove that \\begin{equation}\n  \\sum\\limits_{\\begin{smallmatrix}\n  {{l}_{1}}+{{l}_{2}}+\\cdots +{{l}_{5}}={{p}^{r}}\n  {{l}_{1}},\\cdots ,{{l}_{5}}\\in {\\mathcal{P}_{p}} \\end{smallmatrix}}{\\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}{{l}_{5}}}}\\equiv -\\frac{5!}{6}{{B}_{p-5}}{{p}^{r-1}} \\pmod{{{p}^{r}}}. \\end{equation}\n  This gives an extension of a family of super congruences found by Wang, Cai and Zhao."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.1712","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":"1410.1712","created_at":"2026-05-18T02:40:17.586365+00:00"},{"alias_kind":"arxiv_version","alias_value":"1410.1712v2","created_at":"2026-05-18T02:40:17.586365+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.1712","created_at":"2026-05-18T02:40:17.586365+00:00"},{"alias_kind":"pith_short_12","alias_value":"55VYV45RAYL6","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_16","alias_value":"55VYV45RAYL63XT7","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_8","alias_value":"55VYV45R","created_at":"2026-05-18T12:28:14.216126+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/55VYV45RAYL63XT7QQMA2YBINR","json":"https://pith.science/pith/55VYV45RAYL63XT7QQMA2YBINR.json","graph_json":"https://pith.science/api/pith-number/55VYV45RAYL63XT7QQMA2YBINR/graph.json","events_json":"https://pith.science/api/pith-number/55VYV45RAYL63XT7QQMA2YBINR/events.json","paper":"https://pith.science/paper/55VYV45R"},"agent_actions":{"view_html":"https://pith.science/pith/55VYV45RAYL63XT7QQMA2YBINR","download_json":"https://pith.science/pith/55VYV45RAYL63XT7QQMA2YBINR.json","view_paper":"https://pith.science/paper/55VYV45R","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1410.1712&json=true","fetch_graph":"https://pith.science/api/pith-number/55VYV45RAYL63XT7QQMA2YBINR/graph.json","fetch_events":"https://pith.science/api/pith-number/55VYV45RAYL63XT7QQMA2YBINR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/55VYV45RAYL63XT7QQMA2YBINR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/55VYV45RAYL63XT7QQMA2YBINR/action/storage_attestation","attest_author":"https://pith.science/pith/55VYV45RAYL63XT7QQMA2YBINR/action/author_attestation","sign_citation":"https://pith.science/pith/55VYV45RAYL63XT7QQMA2YBINR/action/citation_signature","submit_replication":"https://pith.science/pith/55VYV45RAYL63XT7QQMA2YBINR/action/replication_record"}},"created_at":"2026-05-18T02:40:17.586365+00:00","updated_at":"2026-05-18T02:40:17.586365+00:00"}