{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:HW27QSY6KGXM6GQD4W47SW3D4F","short_pith_number":"pith:HW27QSY6","canonical_record":{"source":{"id":"1407.8465","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-31T15:53:32Z","cross_cats_sorted":[],"title_canon_sha256":"09589902e164767c41a4558d2917079b1d8838421c384c06e2e6fb8a97de5498","abstract_canon_sha256":"398a2aba4cc3dd76c8e5eb7c09780192aaec2322be8a4a2999f9bd8a74909c46"},"schema_version":"1.0"},"canonical_sha256":"3db5f84b1e51aecf1a03e5b9f95b63e166ebffed01d3c9964907bb1a9f2fecff","source":{"kind":"arxiv","id":"1407.8465","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.8465","created_at":"2026-07-05T07:31:49Z"},{"alias_kind":"arxiv_version","alias_value":"1407.8465v5","created_at":"2026-07-05T07:31:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.8465","created_at":"2026-07-05T07:31:49Z"},{"alias_kind":"pith_short_12","alias_value":"HW27QSY6KGXM","created_at":"2026-07-05T07:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"HW27QSY6KGXM6GQD","created_at":"2026-07-05T07:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"HW27QSY6","created_at":"2026-07-05T07:31:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:HW27QSY6KGXM6GQD4W47SW3D4F","target":"record","payload":{"canonical_record":{"source":{"id":"1407.8465","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-31T15:53:32Z","cross_cats_sorted":[],"title_canon_sha256":"09589902e164767c41a4558d2917079b1d8838421c384c06e2e6fb8a97de5498","abstract_canon_sha256":"398a2aba4cc3dd76c8e5eb7c09780192aaec2322be8a4a2999f9bd8a74909c46"},"schema_version":"1.0"},"canonical_sha256":"3db5f84b1e51aecf1a03e5b9f95b63e166ebffed01d3c9964907bb1a9f2fecff","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:31:49.741580Z","signature_b64":"YlkG4sosnyQKSqAZ+KPcO5Iruii5SWugxXGKI67cSXHIiS4vIAP7KoRLv2jJg4jGzXjt8DcAfi0us6Z7WXCcDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3db5f84b1e51aecf1a03e5b9f95b63e166ebffed01d3c9964907bb1a9f2fecff","last_reissued_at":"2026-07-05T07:31:49.741043Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:31:49.741043Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1407.8465","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T07:31:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QD2SV11E9+b07dAXf5vSvukOXyYQzZ4CjRBAF86Cfj/467Ub2TcQY9ETtzTwi37eVls8Q7DHm3M5RkhBWILtCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T07:42:16.670865Z"},"content_sha256":"0a27c5b162e826790fb0ca73d58fc9e8dd90c45708857322f80e6cf2f8d1b100","schema_version":"1.0","event_id":"sha256:0a27c5b162e826790fb0ca73d58fc9e8dd90c45708857322f80e6cf2f8d1b100"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:HW27QSY6KGXM6GQD4W47SW3D4F","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"New congruences involving harmonic numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2014-07-31T15:53:32Z","abstract_excerpt":"Let $p>3$ be a prime. For any $p$-adic integer $a$, we determine $$\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k,\\ \\ \\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k^{(2)},\\ \\ \\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}k\\frac{H_k^{(2)}}{2k+1}$$ modulo $p^2$, where $H_k=\\sum_{0<j\\le k}1/j$ and $H_k^{(2)}=\\sum_{0<j\\le k}1/j^2$. In particular, we show that \\begin{gather*}\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k\\equiv(-1)^{\\langle a\\rangle_p}\\,2\\left(B_{p-1}(a)-B_{p-1}\\right)\\pmod p, \\\\\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k^{(2)}\\equiv -E_{p-3}(a)\\pmod p, \\\\(2a-1)\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}k\\fr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.8465","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1407.8465/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T07:31:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+QcrCJvSnIm1/Q71OMYCHotmWuYfhUbcSQXbwI5SI7h9kq9NrZq0FLD7aInDqkQ+wCvdgnvDK+XCwBpciU9FDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T07:42:16.671696Z"},"content_sha256":"0cd9dedcfb178af930da5c46bfacc06cb9c21149f607316138661ca41e2d53b5","schema_version":"1.0","event_id":"sha256:0cd9dedcfb178af930da5c46bfacc06cb9c21149f607316138661ca41e2d53b5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HW27QSY6KGXM6GQD4W47SW3D4F/bundle.json","state_url":"https://pith.science/pith/HW27QSY6KGXM6GQD4W47SW3D4F/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HW27QSY6KGXM6GQD4W47SW3D4F/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":"2026-08-04T07:42:16Z","links":{"resolver":"https://pith.science/pith/HW27QSY6KGXM6GQD4W47SW3D4F","bundle":"https://pith.science/pith/HW27QSY6KGXM6GQD4W47SW3D4F/bundle.json","state":"https://pith.science/pith/HW27QSY6KGXM6GQD4W47SW3D4F/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HW27QSY6KGXM6GQD4W47SW3D4F/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:HW27QSY6KGXM6GQD4W47SW3D4F","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":"398a2aba4cc3dd76c8e5eb7c09780192aaec2322be8a4a2999f9bd8a74909c46","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-31T15:53:32Z","title_canon_sha256":"09589902e164767c41a4558d2917079b1d8838421c384c06e2e6fb8a97de5498"},"schema_version":"1.0","source":{"id":"1407.8465","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.8465","created_at":"2026-07-05T07:31:49Z"},{"alias_kind":"arxiv_version","alias_value":"1407.8465v5","created_at":"2026-07-05T07:31:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.8465","created_at":"2026-07-05T07:31:49Z"},{"alias_kind":"pith_short_12","alias_value":"HW27QSY6KGXM","created_at":"2026-07-05T07:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"HW27QSY6KGXM6GQD","created_at":"2026-07-05T07:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"HW27QSY6","created_at":"2026-07-05T07:31:49Z"}],"graph_snapshots":[{"event_id":"sha256:0cd9dedcfb178af930da5c46bfacc06cb9c21149f607316138661ca41e2d53b5","target":"graph","created_at":"2026-07-05T07:31:49Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1407.8465/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $p>3$ be a prime. For any $p$-adic integer $a$, we determine $$\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k,\\ \\ \\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k^{(2)},\\ \\ \\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}k\\frac{H_k^{(2)}}{2k+1}$$ modulo $p^2$, where $H_k=\\sum_{0<j\\le k}1/j$ and $H_k^{(2)}=\\sum_{0<j\\le k}1/j^2$. In particular, we show that \\begin{gather*}\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k\\equiv(-1)^{\\langle a\\rangle_p}\\,2\\left(B_{p-1}(a)-B_{p-1}\\right)\\pmod p, \\\\\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k^{(2)}\\equiv -E_{p-3}(a)\\pmod p, \\\\(2a-1)\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}k\\fr","authors_text":"Zhi-Wei Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-31T15:53:32Z","title":"New congruences involving harmonic numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.8465","kind":"arxiv","version":5},"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":"sha256:0a27c5b162e826790fb0ca73d58fc9e8dd90c45708857322f80e6cf2f8d1b100","target":"record","created_at":"2026-07-05T07:31:49Z","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":"398a2aba4cc3dd76c8e5eb7c09780192aaec2322be8a4a2999f9bd8a74909c46","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-31T15:53:32Z","title_canon_sha256":"09589902e164767c41a4558d2917079b1d8838421c384c06e2e6fb8a97de5498"},"schema_version":"1.0","source":{"id":"1407.8465","kind":"arxiv","version":5}},"canonical_sha256":"3db5f84b1e51aecf1a03e5b9f95b63e166ebffed01d3c9964907bb1a9f2fecff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3db5f84b1e51aecf1a03e5b9f95b63e166ebffed01d3c9964907bb1a9f2fecff","first_computed_at":"2026-07-05T07:31:49.741043Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:31:49.741043Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YlkG4sosnyQKSqAZ+KPcO5Iruii5SWugxXGKI67cSXHIiS4vIAP7KoRLv2jJg4jGzXjt8DcAfi0us6Z7WXCcDg==","signature_status":"signed_v1","signed_at":"2026-07-05T07:31:49.741580Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.8465","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0a27c5b162e826790fb0ca73d58fc9e8dd90c45708857322f80e6cf2f8d1b100","sha256:0cd9dedcfb178af930da5c46bfacc06cb9c21149f607316138661ca41e2d53b5"],"state_sha256":"2b5a41d6343d4ecaefa696c7b52a36eaac5092ab1be2073acd5941935e7b0550"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5abg7nmWiq4XP+EG8oufalzJ3AQNBZ8NYMImNdziSSaF4uSH2WhRxVDJEaHvcmuu7QAO324Jta6nsLwaoZwCDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T07:42:16.678452Z","bundle_sha256":"94d58268c6a91bdd02031a759996b3926d644fc3d42dd47ef6ac50c8572404e1"}}