{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:GURGUZFQYTSSI5YLD4ENZOV3KR","short_pith_number":"pith:GURGUZFQ","schema_version":"1.0","canonical_sha256":"35226a64b0c4e524770b1f08dcbabb54554f42bc8bd0c7dddcfdef2b972ee221","source":{"kind":"arxiv","id":"1608.02177","version":1},"attestation_state":"computed","paper":{"title":"Index Divisibility in Dynamical Sequences and Cyclic Orbits Modulo $p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Annie S. Chen, Katherine E. Stange, T. Alden Gassert","submitted_at":"2016-08-07T04:38:34Z","abstract_excerpt":"Let $\\phi(x) = x^d + c$ be an integral polynomial of degree at least 2, and consider the sequence $(\\phi^n(0))_{n=0}^\\infty$, which is the orbit of $0$ under iteration by $\\phi$. Let $D_{d,c}$ denote the set of positive integers $n$ for which $n \\mid \\phi^n(0)$. We give a characterization of $D_{d,c}$ in terms of a directed graph and describe a number of its properties, including its cardinality and the primes contained therein. In particular, we study the question of which primes $p$ have the property that the orbit of $0$ is a single $p$-cycle modulo $p$. We show that the set of such primes "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1608.02177","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-08-07T04:38:34Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"a4292395ec7c3fd286f867ecfe5c4594955c0cf70d83d8874f662ec157458b04","abstract_canon_sha256":"0a551f3a9cabeffd0ac4fbe017c8c395810d16647555acc7278533e66d79d809"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:40.822029Z","signature_b64":"nVf6prYDgpVEZdzPJURHEBbaqBM69c40U9b7/jTKNCuihF86j0kiDSNSFlRyMyk3bE1rMBgbYJ7uRekp4AhPCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"35226a64b0c4e524770b1f08dcbabb54554f42bc8bd0c7dddcfdef2b972ee221","last_reissued_at":"2026-05-18T01:09:40.821470Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:40.821470Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Index Divisibility in Dynamical Sequences and Cyclic Orbits Modulo $p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Annie S. Chen, Katherine E. Stange, T. Alden Gassert","submitted_at":"2016-08-07T04:38:34Z","abstract_excerpt":"Let $\\phi(x) = x^d + c$ be an integral polynomial of degree at least 2, and consider the sequence $(\\phi^n(0))_{n=0}^\\infty$, which is the orbit of $0$ under iteration by $\\phi$. Let $D_{d,c}$ denote the set of positive integers $n$ for which $n \\mid \\phi^n(0)$. We give a characterization of $D_{d,c}$ in terms of a directed graph and describe a number of its properties, including its cardinality and the primes contained therein. In particular, we study the question of which primes $p$ have the property that the orbit of $0$ is a single $p$-cycle modulo $p$. We show that the set of such primes "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.02177","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1608.02177","created_at":"2026-05-18T01:09:40.821560+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.02177v1","created_at":"2026-05-18T01:09:40.821560+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.02177","created_at":"2026-05-18T01:09:40.821560+00:00"},{"alias_kind":"pith_short_12","alias_value":"GURGUZFQYTSS","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_16","alias_value":"GURGUZFQYTSSI5YL","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_8","alias_value":"GURGUZFQ","created_at":"2026-05-18T12:30:19.053100+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GURGUZFQYTSSI5YLD4ENZOV3KR","json":"https://pith.science/pith/GURGUZFQYTSSI5YLD4ENZOV3KR.json","graph_json":"https://pith.science/api/pith-number/GURGUZFQYTSSI5YLD4ENZOV3KR/graph.json","events_json":"https://pith.science/api/pith-number/GURGUZFQYTSSI5YLD4ENZOV3KR/events.json","paper":"https://pith.science/paper/GURGUZFQ"},"agent_actions":{"view_html":"https://pith.science/pith/GURGUZFQYTSSI5YLD4ENZOV3KR","download_json":"https://pith.science/pith/GURGUZFQYTSSI5YLD4ENZOV3KR.json","view_paper":"https://pith.science/paper/GURGUZFQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.02177&json=true","fetch_graph":"https://pith.science/api/pith-number/GURGUZFQYTSSI5YLD4ENZOV3KR/graph.json","fetch_events":"https://pith.science/api/pith-number/GURGUZFQYTSSI5YLD4ENZOV3KR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GURGUZFQYTSSI5YLD4ENZOV3KR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GURGUZFQYTSSI5YLD4ENZOV3KR/action/storage_attestation","attest_author":"https://pith.science/pith/GURGUZFQYTSSI5YLD4ENZOV3KR/action/author_attestation","sign_citation":"https://pith.science/pith/GURGUZFQYTSSI5YLD4ENZOV3KR/action/citation_signature","submit_replication":"https://pith.science/pith/GURGUZFQYTSSI5YLD4ENZOV3KR/action/replication_record"}},"created_at":"2026-05-18T01:09:40.821560+00:00","updated_at":"2026-05-18T01:09:40.821560+00:00"}