{
  "id": "0707.2505",
  "version": "v2",
  "published": "2007-07-17T12:47:12.000Z",
  "updated": "2007-07-26T15:10:24.000Z",
  "title": "Primitive Divisors in Arithmetic Dynamics",
  "authors": [
    "Patrick Ingram",
    "Joseph H. Silverman"
  ],
  "comment": "Version 2 is substantial revision. The proof of the main theorem has been simplified and strengthened. (16 pages)",
  "journal": "Proc. Camb. Philos. Soc. 146 #2 (2009), 289--302",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in lowest terms. We prove that for all but finitely many n > 0, the numerator A_n has a primitive divisor, i.e., there is a prime p such that p divides A_n and p does not divide A_i for all i < n. More generally, we prove an analogous result when F is defined over a number field and 0 is a periodic point for F.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-07-26T15:10:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B37",
      "11G99",
      "14G99",
      "37F10"
    ],
    "keywords": [
      "primitive divisor",
      "arithmetic dynamics",
      "periodic point",
      "lowest terms",
      "rational function"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1017/S0305004108001795",
      "journal": "Mathematical Proceedings of the Cambridge Philosophical Society",
      "year": 2008,
      "month": "Sep",
      "volume": 146,
      "number": 2,
      "pages": 289
    },
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008MPCPS.146..289I"
    }
  }
}