{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:TE2X34FUMGRSH4M3353B55AXUM","short_pith_number":"pith:TE2X34FU","canonical_record":{"source":{"id":"math/0508087","kind":"arxiv","version":5},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2005-08-04T07:28:54Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"9d03123a8c5ce4cc41a6e95efd7c23dbe5e8059ce2ddebf432235690a1af0988","abstract_canon_sha256":"d6d525d6a9cc9de997067dde1cc5d09f7b68e9630458fa9f6574b780b53eb7a1"},"schema_version":"1.0"},"canonical_sha256":"99357df0b461a323f19bdf761ef417a31add71ac4eb76d366f75be5bc14ed383","source":{"kind":"arxiv","id":"math/0508087","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0508087","created_at":"2026-07-04T15:03:15Z"},{"alias_kind":"arxiv_version","alias_value":"math/0508087v5","created_at":"2026-07-04T15:03:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0508087","created_at":"2026-07-04T15:03:15Z"},{"alias_kind":"pith_short_12","alias_value":"TE2X34FUMGRS","created_at":"2026-07-04T15:03:15Z"},{"alias_kind":"pith_short_16","alias_value":"TE2X34FUMGRSH4M3","created_at":"2026-07-04T15:03:15Z"},{"alias_kind":"pith_short_8","alias_value":"TE2X34FU","created_at":"2026-07-04T15:03:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:TE2X34FUMGRSH4M3353B55AXUM","target":"record","payload":{"canonical_record":{"source":{"id":"math/0508087","kind":"arxiv","version":5},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2005-08-04T07:28:54Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"9d03123a8c5ce4cc41a6e95efd7c23dbe5e8059ce2ddebf432235690a1af0988","abstract_canon_sha256":"d6d525d6a9cc9de997067dde1cc5d09f7b68e9630458fa9f6574b780b53eb7a1"},"schema_version":"1.0"},"canonical_sha256":"99357df0b461a323f19bdf761ef417a31add71ac4eb76d366f75be5bc14ed383","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:03:15.418664Z","signature_b64":"/xnGaB8kzbV11KaLSijuxmo4TLNnREDdX8l9VzSB/sPLvk/RuLFYm6DsuReQWw8UJkOXvsEsbzT6Fomcv2PiDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99357df0b461a323f19bdf761ef417a31add71ac4eb76d366f75be5bc14ed383","last_reissued_at":"2026-07-04T15:03:15.418278Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:03:15.418278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0508087","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":"2026-07-04T15:03:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y7AgVCFQTLssxz1kpor2kP08p3053gxs1aH1oAU4KXMY03N672RxuwRxt7ldTrgaAk4uFlgTHvjcNVLI7zeJCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T01:02:25.007025Z"},"content_sha256":"ed93ceec0109292d9ef932ae7b83fd9f2f4eb4f92d1783549d6f24c90cb5ce42","schema_version":"1.0","event_id":"sha256:ed93ceec0109292d9ef932ae7b83fd9f2f4eb4f92d1783549d6f24c90cb5ce42"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:TE2X34FUMGRSH4M3353B55AXUM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Combinatorial congruences modulo prime powers","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Donald M. Davis, Zhi-Wei Sun","submitted_at":"2005-08-04T07:28:54Z","abstract_excerpt":"Let p be any prime, and let a and n be nonnegative integers. Let $r\\in Z$ and $f(x)\\in Z[x]$. We establish the congruence\n  $$p^{\\deg f}\\sum_{k=r(mod p^a)}\\binom{n}{k}(-1)^k f((k-r)/p^a) =0 (mod p^{\\sum_{i=a}^{\\infty}[n/p^i]})$$ (motivated by a conjecture arising from algebraic topology), and obtain the following vast generalization of Lucas' theorem: If a is greater than one, and $l,s,t$ are nonnegative integers with $s,t<p$, then\n  $$\\frac{1}{[n/p^{a-1}]!} \\sum_{k=r(mod p^a)} \\binom{pn+s}{pk+t}(-1)^{pk}((k-r)/p^{a-1})^l =\\frac {1}{[n/p^{a-1}]!} \\sum_{k=r(mod p^a)}\\binom{n}{k}\\binom{s}{t}(-1)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0508087","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/math/0508087/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":"2026-07-04T15:03:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aU4pHDBcCWeR+1MnEjROhIVrLnGSFFAVYlWPOOds8NeTp9+A1hJlMBUHtC0RO6dYYHViXJC8kxtEtcTsUMbRDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T01:02:25.007836Z"},"content_sha256":"3b109bb9ee157ff575b0be85d3828061338f188c6dedc4e1d31c9693500977f8","schema_version":"1.0","event_id":"sha256:3b109bb9ee157ff575b0be85d3828061338f188c6dedc4e1d31c9693500977f8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TE2X34FUMGRSH4M3353B55AXUM/bundle.json","state_url":"https://pith.science/pith/TE2X34FUMGRSH4M3353B55AXUM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TE2X34FUMGRSH4M3353B55AXUM/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":"2026-08-04T01:02:25Z","links":{"resolver":"https://pith.science/pith/TE2X34FUMGRSH4M3353B55AXUM","bundle":"https://pith.science/pith/TE2X34FUMGRSH4M3353B55AXUM/bundle.json","state":"https://pith.science/pith/TE2X34FUMGRSH4M3353B55AXUM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TE2X34FUMGRSH4M3353B55AXUM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:TE2X34FUMGRSH4M3353B55AXUM","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":"d6d525d6a9cc9de997067dde1cc5d09f7b68e9630458fa9f6574b780b53eb7a1","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.NT","submitted_at":"2005-08-04T07:28:54Z","title_canon_sha256":"9d03123a8c5ce4cc41a6e95efd7c23dbe5e8059ce2ddebf432235690a1af0988"},"schema_version":"1.0","source":{"id":"math/0508087","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0508087","created_at":"2026-07-04T15:03:15Z"},{"alias_kind":"arxiv_version","alias_value":"math/0508087v5","created_at":"2026-07-04T15:03:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0508087","created_at":"2026-07-04T15:03:15Z"},{"alias_kind":"pith_short_12","alias_value":"TE2X34FUMGRS","created_at":"2026-07-04T15:03:15Z"},{"alias_kind":"pith_short_16","alias_value":"TE2X34FUMGRSH4M3","created_at":"2026-07-04T15:03:15Z"},{"alias_kind":"pith_short_8","alias_value":"TE2X34FU","created_at":"2026-07-04T15:03:15Z"}],"graph_snapshots":[{"event_id":"sha256:3b109bb9ee157ff575b0be85d3828061338f188c6dedc4e1d31c9693500977f8","target":"graph","created_at":"2026-07-04T15:03:15Z","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/0508087/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let p be any prime, and let a and n be nonnegative integers. Let $r\\in Z$ and $f(x)\\in Z[x]$. We establish the congruence\n  $$p^{\\deg f}\\sum_{k=r(mod p^a)}\\binom{n}{k}(-1)^k f((k-r)/p^a) =0 (mod p^{\\sum_{i=a}^{\\infty}[n/p^i]})$$ (motivated by a conjecture arising from algebraic topology), and obtain the following vast generalization of Lucas' theorem: If a is greater than one, and $l,s,t$ are nonnegative integers with $s,t<p$, then\n  $$\\frac{1}{[n/p^{a-1}]!} \\sum_{k=r(mod p^a)} \\binom{pn+s}{pk+t}(-1)^{pk}((k-r)/p^{a-1})^l =\\frac {1}{[n/p^{a-1}]!} \\sum_{k=r(mod p^a)}\\binom{n}{k}\\binom{s}{t}(-1)","authors_text":"Donald M. Davis, Zhi-Wei Sun","cross_cats":["math.CO"],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"2005-08-04T07:28:54Z","title":"Combinatorial congruences modulo prime powers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0508087","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:ed93ceec0109292d9ef932ae7b83fd9f2f4eb4f92d1783549d6f24c90cb5ce42","target":"record","created_at":"2026-07-04T15:03:15Z","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":"d6d525d6a9cc9de997067dde1cc5d09f7b68e9630458fa9f6574b780b53eb7a1","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.NT","submitted_at":"2005-08-04T07:28:54Z","title_canon_sha256":"9d03123a8c5ce4cc41a6e95efd7c23dbe5e8059ce2ddebf432235690a1af0988"},"schema_version":"1.0","source":{"id":"math/0508087","kind":"arxiv","version":5}},"canonical_sha256":"99357df0b461a323f19bdf761ef417a31add71ac4eb76d366f75be5bc14ed383","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"99357df0b461a323f19bdf761ef417a31add71ac4eb76d366f75be5bc14ed383","first_computed_at":"2026-07-04T15:03:15.418278Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:03:15.418278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/xnGaB8kzbV11KaLSijuxmo4TLNnREDdX8l9VzSB/sPLvk/RuLFYm6DsuReQWw8UJkOXvsEsbzT6Fomcv2PiDQ==","signature_status":"signed_v1","signed_at":"2026-07-04T15:03:15.418664Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0508087","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ed93ceec0109292d9ef932ae7b83fd9f2f4eb4f92d1783549d6f24c90cb5ce42","sha256:3b109bb9ee157ff575b0be85d3828061338f188c6dedc4e1d31c9693500977f8"],"state_sha256":"32106792c102cc7614c11cb86ae0ff4630038ecb27a062737f9cd9a4548b9721"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3MdLLrCsG7tQrYhhNQG1mdYNdrASefocDalaSv7aTdVCZybk+IMEO9hFFq/cpcsMkR/Bx00LKkfKTriVlEd6Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T01:02:25.013213Z","bundle_sha256":"b1042fbf4dc93f781351ec872851a636a15e038ca78d654d98572f09ee614cc8"}}