{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:QYWPI2RB5YRRMJSVDNK27EK2CU","short_pith_number":"pith:QYWPI2RB","schema_version":"1.0","canonical_sha256":"862cf46a21ee231626551b55af915a151a1844b34bd69346775780274baad2ac","source":{"kind":"arxiv","id":"math/0612415","version":1},"attestation_state":"computed","paper":{"title":"The density of prime divisors in the arithmetic dynamics of quadratic polynomials","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Rafe Jones","submitted_at":"2006-12-14T20:25:02Z","abstract_excerpt":"We consider integer recurrences of the form a_n = f(a_{n-1}), where f is a quadratic polynomial with integer coefficients. We show, for four infinite families of f, that the set of primes dividing at least one term of such a sequence must have density zero, regardless of choice of a_0. The proof relies on tools from group theory and probability theory to develop a zero-density criterion in terms of arithmetic properties of the forward orbit of the critical point of f. This provides an analogy to results in real and complex dynamics, where analytic properties of the forward orbit of the critica"},"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":"math/0612415","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2006-12-14T20:25:02Z","cross_cats_sorted":[],"title_canon_sha256":"d229e997077f9ac07f1534a28d946ba313a22ccc20ff4c6fa383cdcc8593275e","abstract_canon_sha256":"74892e56d06a709926856a376b77cbc5a40d1839e60a246ccb583cc2138dad96"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:45.357370Z","signature_b64":"vsL5tivH/frNmLHPaPlG68CwcVQMBp0qWg0KqnVZQvs73og8D+kCkTvsHm/YKb4KKOE/hq5tbpEBuqFM5HuuCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"862cf46a21ee231626551b55af915a151a1844b34bd69346775780274baad2ac","last_reissued_at":"2026-05-18T02:57:45.356883Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:45.356883Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The density of prime divisors in the arithmetic dynamics of quadratic polynomials","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Rafe Jones","submitted_at":"2006-12-14T20:25:02Z","abstract_excerpt":"We consider integer recurrences of the form a_n = f(a_{n-1}), where f is a quadratic polynomial with integer coefficients. We show, for four infinite families of f, that the set of primes dividing at least one term of such a sequence must have density zero, regardless of choice of a_0. The proof relies on tools from group theory and probability theory to develop a zero-density criterion in terms of arithmetic properties of the forward orbit of the critical point of f. This provides an analogy to results in real and complex dynamics, where analytic properties of the forward orbit of the critica"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0612415","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":"math/0612415","created_at":"2026-05-18T02:57:45.356962+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0612415v1","created_at":"2026-05-18T02:57:45.356962+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0612415","created_at":"2026-05-18T02:57:45.356962+00:00"},{"alias_kind":"pith_short_12","alias_value":"QYWPI2RB5YRR","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_16","alias_value":"QYWPI2RB5YRRMJSV","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_8","alias_value":"QYWPI2RB","created_at":"2026-05-18T12:25:54.717736+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2605.21666","citing_title":"2007: An Arboreal Odyssey: A View of Arboreal Galois Representations and Applications, from Early in the Subject's History","ref_index":13,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QYWPI2RB5YRRMJSVDNK27EK2CU","json":"https://pith.science/pith/QYWPI2RB5YRRMJSVDNK27EK2CU.json","graph_json":"https://pith.science/api/pith-number/QYWPI2RB5YRRMJSVDNK27EK2CU/graph.json","events_json":"https://pith.science/api/pith-number/QYWPI2RB5YRRMJSVDNK27EK2CU/events.json","paper":"https://pith.science/paper/QYWPI2RB"},"agent_actions":{"view_html":"https://pith.science/pith/QYWPI2RB5YRRMJSVDNK27EK2CU","download_json":"https://pith.science/pith/QYWPI2RB5YRRMJSVDNK27EK2CU.json","view_paper":"https://pith.science/paper/QYWPI2RB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0612415&json=true","fetch_graph":"https://pith.science/api/pith-number/QYWPI2RB5YRRMJSVDNK27EK2CU/graph.json","fetch_events":"https://pith.science/api/pith-number/QYWPI2RB5YRRMJSVDNK27EK2CU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QYWPI2RB5YRRMJSVDNK27EK2CU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QYWPI2RB5YRRMJSVDNK27EK2CU/action/storage_attestation","attest_author":"https://pith.science/pith/QYWPI2RB5YRRMJSVDNK27EK2CU/action/author_attestation","sign_citation":"https://pith.science/pith/QYWPI2RB5YRRMJSVDNK27EK2CU/action/citation_signature","submit_replication":"https://pith.science/pith/QYWPI2RB5YRRMJSVDNK27EK2CU/action/replication_record"}},"created_at":"2026-05-18T02:57:45.356962+00:00","updated_at":"2026-05-18T02:57:45.356962+00:00"}