{
  "id": "1210.6612",
  "version": "v1",
  "published": "2012-10-24T17:35:59.000Z",
  "updated": "2012-10-24T17:35:59.000Z",
  "title": "Arithmetic Progressions on Conic Sections",
  "authors": [
    "Alejandra Alvarado",
    "Edray Herber Goins"
  ],
  "comment": "17 pages, submitted for publication",
  "doi": "10.1142/S1793042113500322",
  "categories": [
    "math.NT"
  ],
  "abstract": "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 $Y^2 + 4XY + 4kY = X^3 + kX^2$ classifying those curves possessing a rational 4-torsion point.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-24T17:35:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B25",
      "11E16",
      "14H52"
    ],
    "keywords": [
      "arithmetic progression",
      "rational points",
      "universal elliptic curve",
      "linear rational map",
      "arbitrary conic sections"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.6612A"
    }
  }
}