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For a set $\\calW_K$ of primes of $K$, let $O_{K,\\calW_K}=\\{x\\in K: \\ord_{\\pp}x \\geq 0, \\forall \\pp \\not \\in \\calW_K\\}$. Let $P \\in E(K)$ be a generator of $E(K)$ modulo the torsion subgroup. Let $(x_n(P),y_n(P))$ be the affine coordinates of $[n]P$ with respect to a fixed Weierstrass equation of $E$. We show that there exists a set $\\calW_K$ of primes of $K$ of natural density one such that in $O_{K,\\calW_K}$ multiplication of indices (with respect to some fixed multiple of $P$) is existentially defina"},"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":"0901.4168","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-01-27T03:53:06Z","cross_cats_sorted":["math.AG","math.LO"],"title_canon_sha256":"575e344c2bef3129377260b730cbf5735b4e1b0386004f08a5dd57a20f2763f6","abstract_canon_sha256":"3285d3e788c073534ced41bb96d7d35bd6736538bb51d02317bf375c8191c4d3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:34.988736Z","signature_b64":"wQGK2AwA4AfB8sL7j1Or4qrG/RRITA5gxXl1zcQ1ukEdP9zybM+FprtYzkDYaZXC42svxFU3DKR570NVOm6vBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"88fc35ef4a5a605ccd0992e3a490eff84d79322c61e545ed94655ed9d82630b2","last_reissued_at":"2026-05-18T04:18:34.988118Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:34.988118Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Using Indices of Points on an Elliptic Curve to Construct A Diophantine Model of $\\Z$ and Define $\\Z$ Using One Universal Quantifier in Very Large Subrings of Number Fields, Including $\\Q$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.LO"],"primary_cat":"math.NT","authors_text":"Alexandra Shlapentokh","submitted_at":"2009-01-27T03:53:06Z","abstract_excerpt":"Let $K$ be a number field and let $E$ be an elliptic curve defined and of rank one over $K$. For a set $\\calW_K$ of primes of $K$, let $O_{K,\\calW_K}=\\{x\\in K: \\ord_{\\pp}x \\geq 0, \\forall \\pp \\not \\in \\calW_K\\}$. Let $P \\in E(K)$ be a generator of $E(K)$ modulo the torsion subgroup. Let $(x_n(P),y_n(P))$ be the affine coordinates of $[n]P$ with respect to a fixed Weierstrass equation of $E$. We show that there exists a set $\\calW_K$ of primes of $K$ of natural density one such that in $O_{K,\\calW_K}$ multiplication of indices (with respect to some fixed multiple of $P$) is existentially defina"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.4168","kind":"arxiv","version":3},"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":"0901.4168","created_at":"2026-05-18T04:18:34.988231+00:00"},{"alias_kind":"arxiv_version","alias_value":"0901.4168v3","created_at":"2026-05-18T04:18:34.988231+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.4168","created_at":"2026-05-18T04:18:34.988231+00:00"},{"alias_kind":"pith_short_12","alias_value":"RD6DL32KLJQF","created_at":"2026-05-18T12:26:01.383474+00:00"},{"alias_kind":"pith_short_16","alias_value":"RD6DL32KLJQFZTIJ","created_at":"2026-05-18T12:26:01.383474+00:00"},{"alias_kind":"pith_short_8","alias_value":"RD6DL32K","created_at":"2026-05-18T12:26:01.383474+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/RD6DL32KLJQFZTIJSLR2JEHP7B","json":"https://pith.science/pith/RD6DL32KLJQFZTIJSLR2JEHP7B.json","graph_json":"https://pith.science/api/pith-number/RD6DL32KLJQFZTIJSLR2JEHP7B/graph.json","events_json":"https://pith.science/api/pith-number/RD6DL32KLJQFZTIJSLR2JEHP7B/events.json","paper":"https://pith.science/paper/RD6DL32K"},"agent_actions":{"view_html":"https://pith.science/pith/RD6DL32KLJQFZTIJSLR2JEHP7B","download_json":"https://pith.science/pith/RD6DL32KLJQFZTIJSLR2JEHP7B.json","view_paper":"https://pith.science/paper/RD6DL32K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0901.4168&json=true","fetch_graph":"https://pith.science/api/pith-number/RD6DL32KLJQFZTIJSLR2JEHP7B/graph.json","fetch_events":"https://pith.science/api/pith-number/RD6DL32KLJQFZTIJSLR2JEHP7B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RD6DL32KLJQFZTIJSLR2JEHP7B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RD6DL32KLJQFZTIJSLR2JEHP7B/action/storage_attestation","attest_author":"https://pith.science/pith/RD6DL32KLJQFZTIJSLR2JEHP7B/action/author_attestation","sign_citation":"https://pith.science/pith/RD6DL32KLJQFZTIJSLR2JEHP7B/action/citation_signature","submit_replication":"https://pith.science/pith/RD6DL32KLJQFZTIJSLR2JEHP7B/action/replication_record"}},"created_at":"2026-05-18T04:18:34.988231+00:00","updated_at":"2026-05-18T04:18:34.988231+00:00"}