{
  "id": "math/0202102",
  "version": "v3",
  "published": "2002-02-12T09:39:54.000Z",
  "updated": "2002-12-17T10:13:05.000Z",
  "title": "Torsion points on curves and common divisors of a^k-1 and b^k-1",
  "authors": [
    "Nir Ailon",
    "Zeev Rudnick"
  ],
  "comment": "Conjecture B and Theorem 3 extended to the case of matrices with arbitrary determinant. Replaced the proof of Theorem 2 with a simpler one",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We study the behavior of the greatest common divisor of a^k-1 and b^k-1, where a,b are fixed integers or polynomials, and k varies. In the integer case, we conjecture that when a and b are multiplicatively independent and in addition a-1 and b-1 are coprime, then a^k-1 and b^k-1 are coprime infinitely often. In the polynomial case, we prove a strong version of this conjecture. To do this we use a result of Lang's on the finiteness of torsion points on algebraic curves. We also give a matrix analogue of these results, where for a unimodular matrix A, we look at the greatest common divisor of the elements of the matrix A^k-I.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2002-12-17T10:13:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D61"
    ],
    "keywords": [
      "torsion points",
      "greatest common divisor",
      "integer case",
      "unimodular matrix",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......2102A"
    }
  }
}