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For any prime $p \\ge 11$ and integer $r\\ge 2$, we prove that $$ \\sum\\limits_{\\begin{smallmatrix}\n  {{l}_{1}}+{{l}_{2}}+\\cdots +{{l}_{6}}={{p}^{r}}\n  {{l}_{1}},\\cdots ,{{l}_{6}}\\in {\\mathcal{P}_{p}} \\end{smallmatrix}}{\\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}{{l}_{5}}{l}_{6}}}\\equiv - \\frac{{5!}}{18}p^{r-1}B_{p-3}^{2} \\pmod{{{p}^{r}}}. $$ This extends a family of curious congruences. We also obtain other interesting congruences involving multiple harmonic sums and "},"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":"1504.03227","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-04-13T15:49:16Z","cross_cats_sorted":[],"title_canon_sha256":"39f868b7ba09bcca9e13b2bcac762c7cd8d6afa878f35630918c466c3eb8355d","abstract_canon_sha256":"ece822046272d4361697851a1e3d5987d68c4703e8617d3090b2077a3c257240"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:47.463359Z","signature_b64":"mYnZOEPTCZSgpd6RLGQ4Vn8pwrdkeK6jI8b1wJlLtsbDK6caLIy8lHe40CShDJZUgCJMutK3XyZl0niYMvRjAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7a2a8883a68d2b18f6e196c169014754ec23147981ea9e93305664d22a008b39","last_reissued_at":"2026-05-18T01:21:47.462695Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:47.462695Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New Congruences on Multiple Harmonic Sums and Bernoulli Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2015-04-13T15:49:16Z","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 $p \\ge 11$ and integer $r\\ge 2$, we prove that $$ \\sum\\limits_{\\begin{smallmatrix}\n  {{l}_{1}}+{{l}_{2}}+\\cdots +{{l}_{6}}={{p}^{r}}\n  {{l}_{1}},\\cdots ,{{l}_{6}}\\in {\\mathcal{P}_{p}} \\end{smallmatrix}}{\\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}{{l}_{5}}{l}_{6}}}\\equiv - \\frac{{5!}}{18}p^{r-1}B_{p-3}^{2} \\pmod{{{p}^{r}}}. $$ This extends a family of curious congruences. 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