{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:3ZPJHRJBKN7SKV2626YBORX75R","short_pith_number":"pith:3ZPJHRJB","canonical_record":{"source":{"id":"math/0512071","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2005-12-05T09:48:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"df30dd6074df739f77a7620722cca176dd06f3f33aa0ba8b4d8d0ce65952b870","abstract_canon_sha256":"ce5a734dd75072c0a48a35bf77ae37d7706d896be02d4d55b13b1bfbbf9e7f03"},"schema_version":"1.0"},"canonical_sha256":"de5e93c521537f25575ed7b01746ffec43d3d55f178d95bf512d6b07afc71ac3","source":{"kind":"arxiv","id":"math/0512071","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0512071","created_at":"2026-07-04T14:59:41Z"},{"alias_kind":"arxiv_version","alias_value":"math/0512071v3","created_at":"2026-07-04T14:59:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0512071","created_at":"2026-07-04T14:59:41Z"},{"alias_kind":"pith_short_12","alias_value":"3ZPJHRJBKN7S","created_at":"2026-07-04T14:59:41Z"},{"alias_kind":"pith_short_16","alias_value":"3ZPJHRJBKN7SKV26","created_at":"2026-07-04T14:59:41Z"},{"alias_kind":"pith_short_8","alias_value":"3ZPJHRJB","created_at":"2026-07-04T14:59:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:3ZPJHRJBKN7SKV2626YBORX75R","target":"record","payload":{"canonical_record":{"source":{"id":"math/0512071","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2005-12-05T09:48:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"df30dd6074df739f77a7620722cca176dd06f3f33aa0ba8b4d8d0ce65952b870","abstract_canon_sha256":"ce5a734dd75072c0a48a35bf77ae37d7706d896be02d4d55b13b1bfbbf9e7f03"},"schema_version":"1.0"},"canonical_sha256":"de5e93c521537f25575ed7b01746ffec43d3d55f178d95bf512d6b07afc71ac3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:59:41.910747Z","signature_b64":"+AJciWM5WSqTfqXRnlmpeqilFLJCrNVZTjYyn2CZjIz20I/V9Uk3bwC/VBtyATm4L+Nr3/Hpoalp5VupKycLDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de5e93c521537f25575ed7b01746ffec43d3d55f178d95bf512d6b07afc71ac3","last_reissued_at":"2026-07-04T14:59:41.910404Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:59:41.910404Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0512071","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T14:59:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TjEY3vWjXGGLfIdXFa8K3O4LyuH5P1aO9Xgp8RAM6yxSFVsIwWFc5M79ZcUr0vkjwUyKqqBmQhVTSXk3QDKCAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T20:56:02.622031Z"},"content_sha256":"f7d52e63d9140f6d2eee6b8fc7db923344f86cae2bc8d5565ac89884b1a4eadd","schema_version":"1.0","event_id":"sha256:f7d52e63d9140f6d2eee6b8fc7db923344f86cae2bc8d5565ac89884b1a4eadd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:3ZPJHRJBKN7SKV2626YBORX75R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Combinatorial congruences and Stirling numbers","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2005-12-05T09:48:50Z","abstract_excerpt":"In this paper we obtain some sophisticated combinatorial congruences involving binomial coefficients and confirm two conjectures of the author and Davis. They are closely related to our investigation of the periodicity of the sequence $\\sum_{j=0}^l{l\\choose j}S(j,m)a^{l-j}(l=m,m+1,...)$ modulo a prime $p$, where $a$ and $m>0$ are integers, and those $S(j,m)$ are Stirling numbers of the second kind. We also give a new extension of Glaisher's congruence by showing that $(p-1)p^{[\\log_p m]}$ is a period of the sequence $\\sum_{j=r(mod p-1)}{l\\choose j}S(j,m)(l=m,m+1,...)$ modulo $p$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0512071","kind":"arxiv","version":3},"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/math/0512071/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":"2026-07-04T14:59:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zk85+2UmGZolseaRQ+jGvSm82mhVhci/4vno33D5GLwKZIfLTVaAd3ZombMLyl4cA3dxJxxwL7+Jzxreh2ZLDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T20:56:02.622529Z"},"content_sha256":"5f5ae1865e03eabcce87f4011b8b5cfbd1e2175c8a1d5050c9a72eee7902795c","schema_version":"1.0","event_id":"sha256:5f5ae1865e03eabcce87f4011b8b5cfbd1e2175c8a1d5050c9a72eee7902795c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3ZPJHRJBKN7SKV2626YBORX75R/bundle.json","state_url":"https://pith.science/pith/3ZPJHRJBKN7SKV2626YBORX75R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3ZPJHRJBKN7SKV2626YBORX75R/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":"2026-08-03T20:56:02Z","links":{"resolver":"https://pith.science/pith/3ZPJHRJBKN7SKV2626YBORX75R","bundle":"https://pith.science/pith/3ZPJHRJBKN7SKV2626YBORX75R/bundle.json","state":"https://pith.science/pith/3ZPJHRJBKN7SKV2626YBORX75R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3ZPJHRJBKN7SKV2626YBORX75R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:3ZPJHRJBKN7SKV2626YBORX75R","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":"ce5a734dd75072c0a48a35bf77ae37d7706d896be02d4d55b13b1bfbbf9e7f03","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.NT","submitted_at":"2005-12-05T09:48:50Z","title_canon_sha256":"df30dd6074df739f77a7620722cca176dd06f3f33aa0ba8b4d8d0ce65952b870"},"schema_version":"1.0","source":{"id":"math/0512071","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0512071","created_at":"2026-07-04T14:59:41Z"},{"alias_kind":"arxiv_version","alias_value":"math/0512071v3","created_at":"2026-07-04T14:59:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0512071","created_at":"2026-07-04T14:59:41Z"},{"alias_kind":"pith_short_12","alias_value":"3ZPJHRJBKN7S","created_at":"2026-07-04T14:59:41Z"},{"alias_kind":"pith_short_16","alias_value":"3ZPJHRJBKN7SKV26","created_at":"2026-07-04T14:59:41Z"},{"alias_kind":"pith_short_8","alias_value":"3ZPJHRJB","created_at":"2026-07-04T14:59:41Z"}],"graph_snapshots":[{"event_id":"sha256:5f5ae1865e03eabcce87f4011b8b5cfbd1e2175c8a1d5050c9a72eee7902795c","target":"graph","created_at":"2026-07-04T14:59:41Z","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/math/0512071/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we obtain some sophisticated combinatorial congruences involving binomial coefficients and confirm two conjectures of the author and Davis. They are closely related to our investigation of the periodicity of the sequence $\\sum_{j=0}^l{l\\choose j}S(j,m)a^{l-j}(l=m,m+1,...)$ modulo a prime $p$, where $a$ and $m>0$ are integers, and those $S(j,m)$ are Stirling numbers of the second kind. We also give a new extension of Glaisher's congruence by showing that $(p-1)p^{[\\log_p m]}$ is a period of the sequence $\\sum_{j=r(mod p-1)}{l\\choose j}S(j,m)(l=m,m+1,...)$ modulo $p$.","authors_text":"Zhi-Wei Sun","cross_cats":["math.CO"],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"2005-12-05T09:48:50Z","title":"Combinatorial congruences and Stirling numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0512071","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:f7d52e63d9140f6d2eee6b8fc7db923344f86cae2bc8d5565ac89884b1a4eadd","target":"record","created_at":"2026-07-04T14:59:41Z","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":"ce5a734dd75072c0a48a35bf77ae37d7706d896be02d4d55b13b1bfbbf9e7f03","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.NT","submitted_at":"2005-12-05T09:48:50Z","title_canon_sha256":"df30dd6074df739f77a7620722cca176dd06f3f33aa0ba8b4d8d0ce65952b870"},"schema_version":"1.0","source":{"id":"math/0512071","kind":"arxiv","version":3}},"canonical_sha256":"de5e93c521537f25575ed7b01746ffec43d3d55f178d95bf512d6b07afc71ac3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"de5e93c521537f25575ed7b01746ffec43d3d55f178d95bf512d6b07afc71ac3","first_computed_at":"2026-07-04T14:59:41.910404Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:59:41.910404Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+AJciWM5WSqTfqXRnlmpeqilFLJCrNVZTjYyn2CZjIz20I/V9Uk3bwC/VBtyATm4L+Nr3/Hpoalp5VupKycLDQ==","signature_status":"signed_v1","signed_at":"2026-07-04T14:59:41.910747Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0512071","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f7d52e63d9140f6d2eee6b8fc7db923344f86cae2bc8d5565ac89884b1a4eadd","sha256:5f5ae1865e03eabcce87f4011b8b5cfbd1e2175c8a1d5050c9a72eee7902795c"],"state_sha256":"33017ed7c178617dc6a959fec0ae1b8e1be90118aa1300a0d9e2fc3c4d1cdcc6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I8JbU/sR2nzJScUicUK96vpQ1XGzLuVCWC/jL5KiT2tFxluo9gKGUnStmDTsy3CxFQaptfbAuABe6FnGWKoMBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T20:56:02.628268Z","bundle_sha256":"a28b2db07b0e7cf782a0ec16bbe6d01326c499e9e75caec5efe65babab548efa"}}