{"paper":{"title":"Arithmetic Progressions on Conic Sections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alejandra Alvarado, Edray Herber Goins","submitted_at":"2012-10-24T17:35:59Z","abstract_excerpt":"The set ${1, 25, 49}$ is a 3-term collection of integers which forms an arithmetic progression of perfect squares. We view the set ${(1,1), (5,25), (7,49)}$ as a 3-term collection of rational points on the parabola $y=x^2$ whose $y$-coordinates form an arithmetic progression. In this exposition, we provide a generalization to 3-term arithmetic progressions on arbitrary conic sections $\\mathcal C$ with respect to a linear rational map $\\ell: \\mathcal C \\to \\mathbb P^1$. We explain how this construction is related to rational points on the universal elliptic curve $Y^2 + 4XY + 4kY = X^3 + kX^2$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.6612","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"}