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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}","authors_text":"Jianqiang Zhao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-14T12:02:16Z","title":"Congruences Involving Multiple Harmonic Sums and Finite Multiple Zeta Values"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.3549","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:533a682c50f69956c3758fc711a5f56d302fc4f37ae8640823a4ffce740bc2de","target":"record","created_at":"2026-05-18T00:19:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a346208bd47c9dfc15dd6e62f4604ba790980492d5cb43d4cc1d0243dcee0968","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-04-14T12:02:16Z","title_canon_sha256":"470c0e80283d28bb8fba11c6d94598b79010a60c81344b128d87b127b1233a72"},"schema_version":"1.0","source":{"id":"1404.3549","kind":"arxiv","version":3}},"canonical_sha256":"5874341254a14b7fa6ab3afc27788c7033ebf6cc9ff2d70c167437f4db2e2739","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5874341254a14b7fa6ab3afc27788c7033ebf6cc9ff2d70c167437f4db2e2739","first_computed_at":"2026-05-18T00:19:14.706490Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:19:14.706490Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HJT9imd3+doXarlekw81TQ/FRzJ9xIDt1RJNSIoEGLOTABf5bwZyloHD2jRMEctf05tnUd5gVcqAtiQWG05BBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:19:14.707217Z","signed_message":"canonical_sha256_bytes"},"source_id":"1404.3549","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:533a682c50f69956c3758fc711a5f56d302fc4f37ae8640823a4ffce740bc2de","sha256:07850787b1ffc5b03f842f5db1bef5b0c8d56f394c9c1272210cacdb46b583f4"],"state_sha256":"16431030f673ff674aaaf7a8bc1dfb01d04257fc1bd2f1b01e17a5aecd24ad9b"}