{"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"}