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Recently, Wang and Cai proved that for every positive integer $r$ and prime $p>2$ $$ \\sum_{\\substack{i+j+k=p^r\\\\ i,j,k\\in{\\mathfrak P}_p}} \\frac1{ijk} \\equiv\n  -2p^{r-1} B_{p-3} \\pmod{p^r}, $$ where $B_{p-3}$ is the $(p-3)$-rd Bernoulli number. In this paper we prove the following analogous result: Let $n=2$ or $4$. Then for every positive integer $r\\ge n/2$ and prime $p>4$ $$ \\sum_{\\substack{i_1+\\cdots+i_n=p^r\\\\ i_1,\\dots,i_n\\in{\\mathfrak P}_p}} \\frac1{i_1i_2\\cdots i_n} \\equiv\n  -\\frac{n!}{n+1} p^{r}"},"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":"1404.3549","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-14T12:02:16Z","cross_cats_sorted":[],"title_canon_sha256":"470c0e80283d28bb8fba11c6d94598b79010a60c81344b128d87b127b1233a72","abstract_canon_sha256":"a346208bd47c9dfc15dd6e62f4604ba790980492d5cb43d4cc1d0243dcee0968"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:14.707217Z","signature_b64":"HJT9imd3+doXarlekw81TQ/FRzJ9xIDt1RJNSIoEGLOTABf5bwZyloHD2jRMEctf05tnUd5gVcqAtiQWG05BBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5874341254a14b7fa6ab3afc27788c7033ebf6cc9ff2d70c167437f4db2e2739","last_reissued_at":"2026-05-18T00:19:14.706490Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:14.706490Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruences Involving Multiple Harmonic Sums and Finite Multiple Zeta Values","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jianqiang Zhao","submitted_at":"2014-04-14T12:02:16Z","abstract_excerpt":"Let $p$ be a prime and ${\\mathfrak P}_p$ the set of positive integers which are prime to $p$. Recently, Wang and Cai proved that for every positive integer $r$ and prime $p>2$ $$ \\sum_{\\substack{i+j+k=p^r\\\\ i,j,k\\in{\\mathfrak P}_p}} \\frac1{ijk} \\equiv\n  -2p^{r-1} B_{p-3} \\pmod{p^r}, $$ where $B_{p-3}$ is the $(p-3)$-rd Bernoulli number. In this paper we prove the following analogous result: Let $n=2$ or $4$. Then for every positive integer $r\\ge n/2$ and prime $p>4$ $$ \\sum_{\\substack{i_1+\\cdots+i_n=p^r\\\\ i_1,\\dots,i_n\\in{\\mathfrak P}_p}} \\frac1{i_1i_2\\cdots i_n} \\equiv\n  -\\frac{n!}{n+1} p^{r}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.3549","kind":"arxiv","version":3},"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":"1404.3549","created_at":"2026-05-18T00:19:14.706611+00:00"},{"alias_kind":"arxiv_version","alias_value":"1404.3549v3","created_at":"2026-05-18T00:19:14.706611+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.3549","created_at":"2026-05-18T00:19:14.706611+00:00"},{"alias_kind":"pith_short_12","alias_value":"LB2DIESUUFFX","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_16","alias_value":"LB2DIESUUFFX7JVL","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_8","alias_value":"LB2DIESU","created_at":"2026-05-18T12:28:35.611951+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/LB2DIESUUFFX7JVLHL6CO6EMOA","json":"https://pith.science/pith/LB2DIESUUFFX7JVLHL6CO6EMOA.json","graph_json":"https://pith.science/api/pith-number/LB2DIESUUFFX7JVLHL6CO6EMOA/graph.json","events_json":"https://pith.science/api/pith-number/LB2DIESUUFFX7JVLHL6CO6EMOA/events.json","paper":"https://pith.science/paper/LB2DIESU"},"agent_actions":{"view_html":"https://pith.science/pith/LB2DIESUUFFX7JVLHL6CO6EMOA","download_json":"https://pith.science/pith/LB2DIESUUFFX7JVLHL6CO6EMOA.json","view_paper":"https://pith.science/paper/LB2DIESU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1404.3549&json=true","fetch_graph":"https://pith.science/api/pith-number/LB2DIESUUFFX7JVLHL6CO6EMOA/graph.json","fetch_events":"https://pith.science/api/pith-number/LB2DIESUUFFX7JVLHL6CO6EMOA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LB2DIESUUFFX7JVLHL6CO6EMOA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LB2DIESUUFFX7JVLHL6CO6EMOA/action/storage_attestation","attest_author":"https://pith.science/pith/LB2DIESUUFFX7JVLHL6CO6EMOA/action/author_attestation","sign_citation":"https://pith.science/pith/LB2DIESUUFFX7JVLHL6CO6EMOA/action/citation_signature","submit_replication":"https://pith.science/pith/LB2DIESUUFFX7JVLHL6CO6EMOA/action/replication_record"}},"created_at":"2026-05-18T00:19:14.706611+00:00","updated_at":"2026-05-18T00:19:14.706611+00:00"}