{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:LBOKA7IXUFHLDTUKJ4NRUCZ6R2","short_pith_number":"pith:LBOKA7IX","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"},"canonical_sha256":"585ca07d17a14eb1ce8a4f1b1a0b3e8ea4f3a3d0c33d132e16e17993d48f4188","source":{"kind":"arxiv","id":"1902.05258","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1902.05258","created_at":"2026-05-17T23:54:01Z"},{"alias_kind":"arxiv_version","alias_value":"1902.05258v1","created_at":"2026-05-17T23:54:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.05258","created_at":"2026-05-17T23:54:01Z"},{"alias_kind":"pith_short_12","alias_value":"LBOKA7IXUFHL","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"LBOKA7IXUFHLDTUK","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"LBOKA7IX","created_at":"2026-05-18T12:33:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:LBOKA7IXUFHLDTUKJ4NRUCZ6R2","target":"record","payload":{"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"},"canonical_sha256":"585ca07d17a14eb1ce8a4f1b1a0b3e8ea4f3a3d0c33d132e16e17993d48f4188","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"},"source_kind":"arxiv","source_id":"1902.05258","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:54:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qhlhf8caQijU2OKx6hMOhoV29sVjv7Q/SwDRzmq9tiZOMy9+UNwzB/yxkuAyoEQMKH3VhGs2MKYseZVsa0d1BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T10:23:05.684612Z"},"content_sha256":"88a3b3f413278a45be139c1610f8bae742906cf6047c34f36deb250eb645528f","schema_version":"1.0","event_id":"sha256:88a3b3f413278a45be139c1610f8bae742906cf6047c34f36deb250eb645528f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:LBOKA7IXUFHLDTUKJ4NRUCZ6R2","target":"graph","payload":{"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. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.05258","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:54:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oEKuL29L13/3rjZDnqWU5464F41DzwhAzRGVnuVEYFulWKxUYBsBg0wIdhsjbleSAXRFDXTF/J61oz811vlgAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T10:23:05.685066Z"},"content_sha256":"0aaadd251b3229170a194b1a9facfc3d3f42fc37c43e0551532e535498e78cde","schema_version":"1.0","event_id":"sha256:0aaadd251b3229170a194b1a9facfc3d3f42fc37c43e0551532e535498e78cde"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LBOKA7IXUFHLDTUKJ4NRUCZ6R2/bundle.json","state_url":"https://pith.science/pith/LBOKA7IXUFHLDTUKJ4NRUCZ6R2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LBOKA7IXUFHLDTUKJ4NRUCZ6R2/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-12T10:23:05Z","links":{"resolver":"https://pith.science/pith/LBOKA7IXUFHLDTUKJ4NRUCZ6R2","bundle":"https://pith.science/pith/LBOKA7IXUFHLDTUKJ4NRUCZ6R2/bundle.json","state":"https://pith.science/pith/LBOKA7IXUFHLDTUKJ4NRUCZ6R2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LBOKA7IXUFHLDTUKJ4NRUCZ6R2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:LBOKA7IXUFHLDTUKJ4NRUCZ6R2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e4353663b302168fc6d9f4055fd159e7e7f736565ea7d5fce6691a3fe9c87791","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-02-14T08:31:33Z","title_canon_sha256":"f67a3559f5932b55bcca9fa2e82f1daf81e7e7e293271fab3cc9c7a3d7e7eaa7"},"schema_version":"1.0","source":{"id":"1902.05258","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1902.05258","created_at":"2026-05-17T23:54:01Z"},{"alias_kind":"arxiv_version","alias_value":"1902.05258v1","created_at":"2026-05-17T23:54:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.05258","created_at":"2026-05-17T23:54:01Z"},{"alias_kind":"pith_short_12","alias_value":"LBOKA7IXUFHL","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"LBOKA7IXUFHLDTUK","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"LBOKA7IX","created_at":"2026-05-18T12:33:21Z"}],"graph_snapshots":[{"event_id":"sha256:0aaadd251b3229170a194b1a9facfc3d3f42fc37c43e0551532e535498e78cde","target":"graph","created_at":"2026-05-17T23:54:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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. 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","authors_text":"Ren\\'e Gy","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-02-14T08:31:33Z","title":"Extended Congruences for Harmonic Numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.05258","kind":"arxiv","version":1},"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":"sha256:88a3b3f413278a45be139c1610f8bae742906cf6047c34f36deb250eb645528f","target":"record","created_at":"2026-05-17T23:54:01Z","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":"e4353663b302168fc6d9f4055fd159e7e7f736565ea7d5fce6691a3fe9c87791","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-02-14T08:31:33Z","title_canon_sha256":"f67a3559f5932b55bcca9fa2e82f1daf81e7e7e293271fab3cc9c7a3d7e7eaa7"},"schema_version":"1.0","source":{"id":"1902.05258","kind":"arxiv","version":1}},"canonical_sha256":"585ca07d17a14eb1ce8a4f1b1a0b3e8ea4f3a3d0c33d132e16e17993d48f4188","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"585ca07d17a14eb1ce8a4f1b1a0b3e8ea4f3a3d0c33d132e16e17993d48f4188","first_computed_at":"2026-05-17T23:54:01.440052Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:54:01.440052Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CHRUgLb4je5FdPENte4ebFppUu66O8g6ZuwMdGo5m2bAhxnPIfW8dDzhZysqS8Uwh3dS4k/Zt3XwFKnQL4s4AQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:54:01.440902Z","signed_message":"canonical_sha256_bytes"},"source_id":"1902.05258","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:88a3b3f413278a45be139c1610f8bae742906cf6047c34f36deb250eb645528f","sha256:0aaadd251b3229170a194b1a9facfc3d3f42fc37c43e0551532e535498e78cde"],"state_sha256":"cedb4366581ffe443fb08c1ea1f4978f8f502c3c242b8d6835da7dcf21c3aa85"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cOmvbpN5DZUau/8X5oU98TGSCQUzQ+VjnP81bYo6vosWShFVbdMl56BD3vdQnFZqdYtTYz0IOpVQbNMVXB1aDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T10:23:05.689578Z","bundle_sha256":"124013fc2df65d07de69f02853ae9ccc0a1a762701795d1b41a85eff1e6e272e"}}