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In this paper we prove that for every sufficiently large prime $p$ $$ \\sum_{\\substack{l_1+l_2+\\cdots+l_n=mp^r p\\nmid l_1 l_2 \\cdots l_n }} \\frac1{l_1l_2\\cdots l_n} \\equiv p^{r-1} \\sum_{{\\bf k}\\vdash n} C_{m,{\\bf k}} B_{p-{\\bf k}} \\pmod{p^r} $$ where $B_{p-{\\bf k}}=B_{p-k_1}B_{p-k_2}\\cdots B_{p-k_t}$ are products of Bernoulli numbers and the coefficients $C_{m,{\\bf k}}$ are polynomials of $m$ independent of"},"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":"1702.08401","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-02-27T17:48:25Z","cross_cats_sorted":[],"title_canon_sha256":"b164031a0e825b8a6e271c01aa0daaf9e2f1d0d7a34477ed9932da4fc06372de","abstract_canon_sha256":"7c1dfe0f738f1637c5c60e19968420197cb9fa891e3763d5c8051ef652c24795"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:19.461730Z","signature_b64":"rf78PPqDLoseyY97BEyzmh1aK2EVL/KGyAYredok8kLyidxCxB5rLNT9qrQuMYzfR1yFzbI35TQJiWSvGx4FAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3acf3b8442473e7bada6f2aa71a486b84bf7ce5621764d5de7a5f4e80978dcf2","last_reissued_at":"2026-05-18T00:19:19.461158Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:19.461158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Super Congruences Involving Multiple Harmonic Sums and Bernoulli Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jianqiang Zhao, Kevin Chen","submitted_at":"2017-02-27T17:48:25Z","abstract_excerpt":"Let $m$, $r$ and $n$ be positive integers. We denote by ${\\bf k}\\vdash n$ any tuple of odd positive integers ${\\bf k}=(k_1,\\dots,k_t)$ such that $k_1+\\dots+k_t=n$ and $k_j\\ge 3$ for all $j$. In this paper we prove that for every sufficiently large prime $p$ $$ \\sum_{\\substack{l_1+l_2+\\cdots+l_n=mp^r p\\nmid l_1 l_2 \\cdots l_n }} \\frac1{l_1l_2\\cdots l_n} \\equiv p^{r-1} \\sum_{{\\bf k}\\vdash n} C_{m,{\\bf k}} B_{p-{\\bf k}} \\pmod{p^r} $$ where $B_{p-{\\bf k}}=B_{p-k_1}B_{p-k_2}\\cdots B_{p-k_t}$ are products of Bernoulli numbers and the coefficients $C_{m,{\\bf k}}$ are polynomials of $m$ independent of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.08401","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":"1702.08401","created_at":"2026-05-18T00:19:19.461256+00:00"},{"alias_kind":"arxiv_version","alias_value":"1702.08401v2","created_at":"2026-05-18T00:19:19.461256+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.08401","created_at":"2026-05-18T00:19:19.461256+00:00"},{"alias_kind":"pith_short_12","alias_value":"HLHTXBCCI47H","created_at":"2026-05-18T12:31:18.294218+00:00"},{"alias_kind":"pith_short_16","alias_value":"HLHTXBCCI47HXLNG","created_at":"2026-05-18T12:31:18.294218+00:00"},{"alias_kind":"pith_short_8","alias_value":"HLHTXBCC","created_at":"2026-05-18T12:31:18.294218+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/HLHTXBCCI47HXLNG6KVHDJEGXB","json":"https://pith.science/pith/HLHTXBCCI47HXLNG6KVHDJEGXB.json","graph_json":"https://pith.science/api/pith-number/HLHTXBCCI47HXLNG6KVHDJEGXB/graph.json","events_json":"https://pith.science/api/pith-number/HLHTXBCCI47HXLNG6KVHDJEGXB/events.json","paper":"https://pith.science/paper/HLHTXBCC"},"agent_actions":{"view_html":"https://pith.science/pith/HLHTXBCCI47HXLNG6KVHDJEGXB","download_json":"https://pith.science/pith/HLHTXBCCI47HXLNG6KVHDJEGXB.json","view_paper":"https://pith.science/paper/HLHTXBCC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1702.08401&json=true","fetch_graph":"https://pith.science/api/pith-number/HLHTXBCCI47HXLNG6KVHDJEGXB/graph.json","fetch_events":"https://pith.science/api/pith-number/HLHTXBCCI47HXLNG6KVHDJEGXB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HLHTXBCCI47HXLNG6KVHDJEGXB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HLHTXBCCI47HXLNG6KVHDJEGXB/action/storage_attestation","attest_author":"https://pith.science/pith/HLHTXBCCI47HXLNG6KVHDJEGXB/action/author_attestation","sign_citation":"https://pith.science/pith/HLHTXBCCI47HXLNG6KVHDJEGXB/action/citation_signature","submit_replication":"https://pith.science/pith/HLHTXBCCI47HXLNG6KVHDJEGXB/action/replication_record"}},"created_at":"2026-05-18T00:19:19.461256+00:00","updated_at":"2026-05-18T00:19:19.461256+00:00"}