{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:S6XGWTMWK5VLJQINJUUIES7VQC","short_pith_number":"pith:S6XGWTMW","canonical_record":{"source":{"id":"1701.02975","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-01-11T14:03:43Z","cross_cats_sorted":[],"title_canon_sha256":"c2c7a9dd67f960b29c9b8f8fc625ecbbdb754e2048d45757641501f264290800","abstract_canon_sha256":"dbd2ca9cb8291224ca0397642c16a9a860a714c495e4c772c952d538c273e1f5"},"schema_version":"1.0"},"canonical_sha256":"97ae6b4d96576ab4c10d4d28824bf5809f81a2edb30a9eef911844e40515f3e0","source":{"kind":"arxiv","id":"1701.02975","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.02975","created_at":"2026-05-18T00:52:59Z"},{"alias_kind":"arxiv_version","alias_value":"1701.02975v1","created_at":"2026-05-18T00:52:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.02975","created_at":"2026-05-18T00:52:59Z"},{"alias_kind":"pith_short_12","alias_value":"S6XGWTMWK5VL","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"S6XGWTMWK5VLJQIN","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"S6XGWTMW","created_at":"2026-05-18T12:31:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:S6XGWTMWK5VLJQINJUUIES7VQC","target":"record","payload":{"canonical_record":{"source":{"id":"1701.02975","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-01-11T14:03:43Z","cross_cats_sorted":[],"title_canon_sha256":"c2c7a9dd67f960b29c9b8f8fc625ecbbdb754e2048d45757641501f264290800","abstract_canon_sha256":"dbd2ca9cb8291224ca0397642c16a9a860a714c495e4c772c952d538c273e1f5"},"schema_version":"1.0"},"canonical_sha256":"97ae6b4d96576ab4c10d4d28824bf5809f81a2edb30a9eef911844e40515f3e0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:59.669192Z","signature_b64":"LdjHE2ks4MuVrbgGpcpWJDl5F4Ex3HMPxDkya/B9hl3qGrQ9NAGqHvaaSTuKX663yrG3gPWP29A5F79Wp6qmAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"97ae6b4d96576ab4c10d4d28824bf5809f81a2edb30a9eef911844e40515f3e0","last_reissued_at":"2026-05-18T00:52:59.668575Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:59.668575Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1701.02975","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":"2026-05-18T00:52:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"elBTfyVF2Xk+62iJ/m18E6mlWNuZEO8C6grCNi+am54erEJKLSLI2o9PQasXlUxNpyTc/4I+kPjLs8WsK5s8AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T11:25:21.279020Z"},"content_sha256":"2ef312ebd852a595280736d6d6fe96395706fc0347c33c8f51e22a81b76e594b","schema_version":"1.0","event_id":"sha256:2ef312ebd852a595280736d6d6fe96395706fc0347c33c8f51e22a81b76e594b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:S6XGWTMWK5VLJQINJUUIES7VQC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Structure and asymptotics for Catalan numbers modulo primes using automata","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Rob Burns","submitted_at":"2017-01-11T14:03:43Z","abstract_excerpt":"Let $C_n$ be the $n$th Catalan number. We show that the asymptotic density of the set $\\{n: C_n \\equiv 0 \\mod p \\}$ is $1$ for all primes $p$, We also show that if $n = p^k -1$ then $C_n \\equiv -1 \\mod p$. Finally we show that if $n \\equiv \\{ \\frac{p+1}{2}, \\frac{p+3}{2}, ..., p-2 \\} \\mod p$ then $p$ divides $C_n$. All results are obtained using the automata method of Rowland and Yassawi."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.02975","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":"2026-05-18T00:52:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5eWU9zBvzzspHftD2QAfy9X5/xgKfvV59GUvWNSN2uYsyHy+6q213CbfpMvoYJ7jSKRqjwZD2Fs3FbPvbqXTBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T11:25:21.279924Z"},"content_sha256":"5b10bedd502ca50c3253ba5fdbadfd3de4f995b2a7b919ebf6e98c847d22c2ac","schema_version":"1.0","event_id":"sha256:5b10bedd502ca50c3253ba5fdbadfd3de4f995b2a7b919ebf6e98c847d22c2ac"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/S6XGWTMWK5VLJQINJUUIES7VQC/bundle.json","state_url":"https://pith.science/pith/S6XGWTMWK5VLJQINJUUIES7VQC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/S6XGWTMWK5VLJQINJUUIES7VQC/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":"2026-06-09T11:25:21Z","links":{"resolver":"https://pith.science/pith/S6XGWTMWK5VLJQINJUUIES7VQC","bundle":"https://pith.science/pith/S6XGWTMWK5VLJQINJUUIES7VQC/bundle.json","state":"https://pith.science/pith/S6XGWTMWK5VLJQINJUUIES7VQC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/S6XGWTMWK5VLJQINJUUIES7VQC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:S6XGWTMWK5VLJQINJUUIES7VQC","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":"dbd2ca9cb8291224ca0397642c16a9a860a714c495e4c772c952d538c273e1f5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-01-11T14:03:43Z","title_canon_sha256":"c2c7a9dd67f960b29c9b8f8fc625ecbbdb754e2048d45757641501f264290800"},"schema_version":"1.0","source":{"id":"1701.02975","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.02975","created_at":"2026-05-18T00:52:59Z"},{"alias_kind":"arxiv_version","alias_value":"1701.02975v1","created_at":"2026-05-18T00:52:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.02975","created_at":"2026-05-18T00:52:59Z"},{"alias_kind":"pith_short_12","alias_value":"S6XGWTMWK5VL","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"S6XGWTMWK5VLJQIN","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"S6XGWTMW","created_at":"2026-05-18T12:31:43Z"}],"graph_snapshots":[{"event_id":"sha256:5b10bedd502ca50c3253ba5fdbadfd3de4f995b2a7b919ebf6e98c847d22c2ac","target":"graph","created_at":"2026-05-18T00:52:59Z","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. We show that the asymptotic density of the set $\\{n: C_n \\equiv 0 \\mod p \\}$ is $1$ for all primes $p$, We also show that if $n = p^k -1$ then $C_n \\equiv -1 \\mod p$. Finally we show that if $n \\equiv \\{ \\frac{p+1}{2}, \\frac{p+3}{2}, ..., p-2 \\} \\mod p$ then $p$ divides $C_n$. All results are obtained using the automata method of Rowland and Yassawi.","authors_text":"Rob Burns","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-01-11T14:03:43Z","title":"Structure and asymptotics for Catalan numbers modulo primes using automata"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.02975","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":"sha256:2ef312ebd852a595280736d6d6fe96395706fc0347c33c8f51e22a81b76e594b","target":"record","created_at":"2026-05-18T00:52:59Z","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":"dbd2ca9cb8291224ca0397642c16a9a860a714c495e4c772c952d538c273e1f5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-01-11T14:03:43Z","title_canon_sha256":"c2c7a9dd67f960b29c9b8f8fc625ecbbdb754e2048d45757641501f264290800"},"schema_version":"1.0","source":{"id":"1701.02975","kind":"arxiv","version":1}},"canonical_sha256":"97ae6b4d96576ab4c10d4d28824bf5809f81a2edb30a9eef911844e40515f3e0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"97ae6b4d96576ab4c10d4d28824bf5809f81a2edb30a9eef911844e40515f3e0","first_computed_at":"2026-05-18T00:52:59.668575Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:52:59.668575Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LdjHE2ks4MuVrbgGpcpWJDl5F4Ex3HMPxDkya/B9hl3qGrQ9NAGqHvaaSTuKX663yrG3gPWP29A5F79Wp6qmAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:52:59.669192Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.02975","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2ef312ebd852a595280736d6d6fe96395706fc0347c33c8f51e22a81b76e594b","sha256:5b10bedd502ca50c3253ba5fdbadfd3de4f995b2a7b919ebf6e98c847d22c2ac"],"state_sha256":"89d2e9597b550d11ee2c231aa237dcf0dec7b8bfd540c7cedeca8706360b1a7e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0tFHU+ADWKzjhS2aKWHONsOZFBGxZA41CHYzK86G8kTSLuIZu/SnQqKGe2jwyWgQcWmABz4OsK7pGv/iXGLCDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T11:25:21.283971Z","bundle_sha256":"d42d31b79c308c7071561e4a9a46de0bfff0bbad2bff9d8d4c90ea1b98a43c93"}}