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Moreover, we show that \\begin{equation*}\\sum_{j=0}^{n-1}\\left(\\frac{(2^{j+1}-1)}{(j+1)}\\frac{(2^{j+2}-1)}{(j+2)}\\frac{B_{j+2}}{2^{j}}H^{(j+1)}_{\\frac{p-1}{2}}+2(-1)^j\\frac{q_p^{j+1}}{j+1}\\right)p^j\\equiv 0 \\pmod {p^n} \\end{equation*} holds under the condi"},"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":"1902.05258","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-02-14T08:31:33Z","cross_cats_sorted":[],"title_canon_sha256":"f67a3559f5932b55bcca9fa2e82f1daf81e7e7e293271fab3cc9c7a3d7e7eaa7","abstract_canon_sha256":"e4353663b302168fc6d9f4055fd159e7e7f736565ea7d5fce6691a3fe9c87791"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:01.440902Z","signature_b64":"CHRUgLb4je5FdPENte4ebFppUu66O8g6ZuwMdGo5m2bAhxnPIfW8dDzhZysqS8Uwh3dS4k/Zt3XwFKnQL4s4AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"585ca07d17a14eb1ce8a4f1b1a0b3e8ea4f3a3d0c33d132e16e17993d48f4188","last_reissued_at":"2026-05-17T23:54:01.440052Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:01.440052Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extended Congruences for Harmonic Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ren\\'e Gy","submitted_at":"2019-02-14T08:31:33Z","abstract_excerpt":"We derive $p$-adic expansions for the generalized Harmonic numbers $H^{(j)}_{p-1}$ and $H^{(j)}_{\\frac{p-1}{2}}$ involving the Bernoulli numbers $B_j$ and the the base-2 Fermat quotient $q_p$. While most of our results are not new, we obtain them elementarily, without resorting to the theory of $p$-adic L-functions as was the case previously. 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