{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2007:TB4LZNEXPIAGPZWUQDQXD2YPEG","short_pith_number":"pith:TB4LZNEX","schema_version":"1.0","canonical_sha256":"9878bcb4977a0067e6d480e171eb0f2185c538aa29f3f63c41bb233aaaf8b472","source":{"kind":"arxiv","id":"0707.2505","version":2},"attestation_state":"computed","paper":{"title":"Primitive Divisors in Arithmetic Dynamics","license":"","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Joseph H. Silverman, Patrick Ingram","submitted_at":"2007-07-17T12:47:12Z","abstract_excerpt":"Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in lowest terms. We prove that for all but finitely many n > 0, the numerator A_n has a primitive divisor, i.e., there is a prime p such that p divides A_n and p does not divide A_i for all i < n. More generally, we prove an analogous result when F is defined over a number field and 0 is a periodic point for F."},"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":"0707.2505","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2007-07-17T12:47:12Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"6d0c746bcea6e7113cf10bd37ea9ff433b0b7a8ca9c8ae47dd5519c308407252","abstract_canon_sha256":"75522aa4ac3874d15955ae20793cb1893c8d449e29b2a30f84fbee0356df5496"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:29.324633Z","signature_b64":"5tZMjhFUbHU5bI6r6u+kFu78HocOZ5OR78/xJKojfdblSog10EYWG0pmx4XYIwtuGoJBTbOll/g49mRM/jewBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9878bcb4977a0067e6d480e171eb0f2185c538aa29f3f63c41bb233aaaf8b472","last_reissued_at":"2026-05-18T02:16:29.323896Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:29.323896Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Primitive Divisors in Arithmetic Dynamics","license":"","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Joseph H. Silverman, Patrick Ingram","submitted_at":"2007-07-17T12:47:12Z","abstract_excerpt":"Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in lowest terms. We prove that for all but finitely many n > 0, the numerator A_n has a primitive divisor, i.e., there is a prime p such that p divides A_n and p does not divide A_i for all i < n. More generally, we prove an analogous result when F is defined over a number field and 0 is a periodic point for F."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0707.2505","kind":"arxiv","version":2},"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":"0707.2505","created_at":"2026-05-18T02:16:29.324015+00:00"},{"alias_kind":"arxiv_version","alias_value":"0707.2505v2","created_at":"2026-05-18T02:16:29.324015+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0707.2505","created_at":"2026-05-18T02:16:29.324015+00:00"},{"alias_kind":"pith_short_12","alias_value":"TB4LZNEXPIAG","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_16","alias_value":"TB4LZNEXPIAGPZWU","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_8","alias_value":"TB4LZNEX","created_at":"2026-05-18T12:25:56.245647+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/TB4LZNEXPIAGPZWUQDQXD2YPEG","json":"https://pith.science/pith/TB4LZNEXPIAGPZWUQDQXD2YPEG.json","graph_json":"https://pith.science/api/pith-number/TB4LZNEXPIAGPZWUQDQXD2YPEG/graph.json","events_json":"https://pith.science/api/pith-number/TB4LZNEXPIAGPZWUQDQXD2YPEG/events.json","paper":"https://pith.science/paper/TB4LZNEX"},"agent_actions":{"view_html":"https://pith.science/pith/TB4LZNEXPIAGPZWUQDQXD2YPEG","download_json":"https://pith.science/pith/TB4LZNEXPIAGPZWUQDQXD2YPEG.json","view_paper":"https://pith.science/paper/TB4LZNEX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0707.2505&json=true","fetch_graph":"https://pith.science/api/pith-number/TB4LZNEXPIAGPZWUQDQXD2YPEG/graph.json","fetch_events":"https://pith.science/api/pith-number/TB4LZNEXPIAGPZWUQDQXD2YPEG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TB4LZNEXPIAGPZWUQDQXD2YPEG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TB4LZNEXPIAGPZWUQDQXD2YPEG/action/storage_attestation","attest_author":"https://pith.science/pith/TB4LZNEXPIAGPZWUQDQXD2YPEG/action/author_attestation","sign_citation":"https://pith.science/pith/TB4LZNEXPIAGPZWUQDQXD2YPEG/action/citation_signature","submit_replication":"https://pith.science/pith/TB4LZNEXPIAGPZWUQDQXD2YPEG/action/replication_record"}},"created_at":"2026-05-18T02:16:29.324015+00:00","updated_at":"2026-05-18T02:16:29.324015+00:00"}