{
  "id": "1309.3673",
  "version": "v2",
  "published": "2013-09-14T14:27:24.000Z",
  "updated": "2014-03-22T14:37:41.000Z",
  "title": "Is there an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the number of integer solutions, if the solution set is finite?",
  "authors": [
    "Apoloniusz Tyszka"
  ],
  "comment": "6 pages, two open questions added. arXiv admin note: text overlap with arXiv:1309.2605, arXiv:1309.2682",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let E_n={x_i=1, x_i+x_j=x_k, x_i \\cdot x_j=x_k: i,j,k \\in {1,...,n}}. For a positive integer n, let f(n) denote the greatest finite total number of solutions of a subsystem of E_n in integers x_1,...,x_n. We prove: (1) the function f is strictly increasing, (2) if a non-decreasing function g from positive integers to positive integers satisfies f(n) \\geq g(n) for any n, then a finite-fold Diophantine representation of g does not exist, (3) if the question of the title has a positive answer, then there is a computable strictly increasing function g from positive integers to positive integers such that f(n) \\leq g(n) for any n and a finite-fold Diophantine representation of g does not exist.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-22T14:37:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U05",
      "03D25"
    ],
    "keywords": [
      "positive integer",
      "solution set",
      "diophantine equation",
      "integer solutions",
      "finite-fold diophantine representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.3673T"
    }
  }
}