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Thus we can recover $p$ and $q$ from the data $D$ and $ x([k]Q))$. Furthermore, under the Generalized Riemann hypothesis, we prove that one can take $r<c\\log^4D$ such that the elliptic curve $E_{2rD}$"},"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":"1207.0274","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-07-02T03:44:22Z","cross_cats_sorted":[],"title_canon_sha256":"494761280e3ceefae96e41c4aa9693604747668e26c094d7df6234cd068eb6c2","abstract_canon_sha256":"37410db386e6d7ec6ae2f52f7765a3012bd2dd76a0e9364be85522d37e553e54"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:10.717485Z","signature_b64":"t+iBVIFFKqXn2di5zr6U+8ej5eedC9WlIvmbdMjUUJCcxyhMXJ5FMeMTe9Aqy8Jw0NJtUzCjDeOdeNbXTUyDCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d609b9d2392a069ca97433e7fce726c78027afa194d8323f271d843792d56b6","last_reissued_at":"2026-05-18T02:25:10.717008Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:10.717008Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Factoring integer using elliptic curves over rational number field $\\mathbb{Q}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jinxiang Zeng, Xiumei Li","submitted_at":"2012-07-02T03:44:22Z","abstract_excerpt":"For the integer $ D=pq$ of the product of two distinct odd primes, we construct an elliptic curve $E_{2rD}:y^2=x^3-2rDx$ over $\\mathbb Q$, where $r$ is a parameter dependent on the classes of $p$ and $q$ modulo 8, and show, under the parity conjecture, that the elliptic curve has rank one and $v_p(x([k]Q))\\not=v_q(x([k]Q))$ for odd $k$ and a generator $Q$ of the free part of $E_{2rD}(\\mathbb Q)$. Thus we can recover $p$ and $q$ from the data $D$ and $ x([k]Q))$. Furthermore, under the Generalized Riemann hypothesis, we prove that one can take $r<c\\log^4D$ such that the elliptic curve $E_{2rD}$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.0274","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":"1207.0274","created_at":"2026-05-18T02:25:10.717070+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.0274v3","created_at":"2026-05-18T02:25:10.717070+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.0274","created_at":"2026-05-18T02:25:10.717070+00:00"},{"alias_kind":"pith_short_12","alias_value":"LVQJXHJDSKQG","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_16","alias_value":"LVQJXHJDSKQGTSUX","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_8","alias_value":"LVQJXHJD","created_at":"2026-05-18T12:27:14.488303+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/LVQJXHJDSKQGTSUXIM7H7TTSNR","json":"https://pith.science/pith/LVQJXHJDSKQGTSUXIM7H7TTSNR.json","graph_json":"https://pith.science/api/pith-number/LVQJXHJDSKQGTSUXIM7H7TTSNR/graph.json","events_json":"https://pith.science/api/pith-number/LVQJXHJDSKQGTSUXIM7H7TTSNR/events.json","paper":"https://pith.science/paper/LVQJXHJD"},"agent_actions":{"view_html":"https://pith.science/pith/LVQJXHJDSKQGTSUXIM7H7TTSNR","download_json":"https://pith.science/pith/LVQJXHJDSKQGTSUXIM7H7TTSNR.json","view_paper":"https://pith.science/paper/LVQJXHJD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.0274&json=true","fetch_graph":"https://pith.science/api/pith-number/LVQJXHJDSKQGTSUXIM7H7TTSNR/graph.json","fetch_events":"https://pith.science/api/pith-number/LVQJXHJDSKQGTSUXIM7H7TTSNR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LVQJXHJDSKQGTSUXIM7H7TTSNR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LVQJXHJDSKQGTSUXIM7H7TTSNR/action/storage_attestation","attest_author":"https://pith.science/pith/LVQJXHJDSKQGTSUXIM7H7TTSNR/action/author_attestation","sign_citation":"https://pith.science/pith/LVQJXHJDSKQGTSUXIM7H7TTSNR/action/citation_signature","submit_replication":"https://pith.science/pith/LVQJXHJDSKQGTSUXIM7H7TTSNR/action/replication_record"}},"created_at":"2026-05-18T02:25:10.717070+00:00","updated_at":"2026-05-18T02:25:10.717070+00:00"}