{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:BDDXTJWP6WSSU3F5GUB3T3U2QO","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":"8c2f5e33664aa74af364a2ea98cb6ae39636f5e1da8a09c92c86f86f9e4d355c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-03-08T05:09:43Z","title_canon_sha256":"86b4d63ca183392a21277cc7c6d8c8b5f5ffc9114f9e446c09ebf1d7bdf99e2d"},"schema_version":"1.0","source":{"id":"1703.02705","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.02705","created_at":"2026-05-18T00:49:05Z"},{"alias_kind":"arxiv_version","alias_value":"1703.02705v1","created_at":"2026-05-18T00:49:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.02705","created_at":"2026-05-18T00:49:05Z"},{"alias_kind":"pith_short_12","alias_value":"BDDXTJWP6WSS","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BDDXTJWP6WSSU3F5","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BDDXTJWP","created_at":"2026-05-18T12:31:08Z"}],"graph_snapshots":[{"event_id":"sha256:4cce0dac29d19924a7de359462399456bd7aa8ac8fe6be33787d35a9a5c50587","target":"graph","created_at":"2026-05-18T00:49:05Z","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":"Let $C_n$ be the $n$th Catalan number. For any prime $p \\geq 5$ we show that the set $\\{C_n : n \\in \\mathbb{N} \\}$ contains all residues mod $p$. In addition all residues are attained infinitely often. Any positive integer can be expressed as the product of central binomial coefficients modulo $p$. The directed sub-graph of the automata for $C_n \\mod p$ consisting of the constant states and transitions between them has a cycle which visits all vertices.","authors_text":"Rob Burns","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-03-08T05:09:43Z","title":"The Catalan numbers have no forbidden residue modulo primes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.02705","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":"sha256:25725833b37172f2882dc47201330b21b14ca9bf1b74b25aaadc123787120b82","target":"record","created_at":"2026-05-18T00:49:05Z","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":"8c2f5e33664aa74af364a2ea98cb6ae39636f5e1da8a09c92c86f86f9e4d355c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-03-08T05:09:43Z","title_canon_sha256":"86b4d63ca183392a21277cc7c6d8c8b5f5ffc9114f9e446c09ebf1d7bdf99e2d"},"schema_version":"1.0","source":{"id":"1703.02705","kind":"arxiv","version":1}},"canonical_sha256":"08c779a6cff5a52a6cbd3503b9ee9a83924c7ac0957f163337c59bb4a3d389bc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"08c779a6cff5a52a6cbd3503b9ee9a83924c7ac0957f163337c59bb4a3d389bc","first_computed_at":"2026-05-18T00:49:05.826082Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:49:05.826082Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qsnl5bqCzzyern/MWDjs77YamICHjAOtrCM6dfD0iSQiiRVPESH/UP0G8kJb3/IPwx2kI5323MCA9ZAUDantBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:49:05.826648Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.02705","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:25725833b37172f2882dc47201330b21b14ca9bf1b74b25aaadc123787120b82","sha256:4cce0dac29d19924a7de359462399456bd7aa8ac8fe6be33787d35a9a5c50587"],"state_sha256":"10b0f7a2eff6c78ba5c38713446e0489ec38a6e9eb4313ac1b7ba5f1936f485b"}