{
  "id": "0901.2093",
  "version": "v152",
  "published": "2009-01-14T19:16:02.000Z",
  "updated": "2014-10-20T12:23:40.000Z",
  "title": "A hypothetical upper bound for the solutions of a Diophantine equation with a finite number of solutions",
  "authors": [
    "Apoloniusz Tyszka"
  ],
  "comment": "Unchanged text, the conjecture with the bound 2^(2^(n-1)) is false, see http://dx.doi.org/10.13140/2.1.1707.2640 arXiv admin note: substantial text overlap with arXiv:1105.5747, arXiv:1102.4122, arXiv:1011.4103, arXiv:1109.3826",
  "journal": "Fund. Inform. 125 (2013), no. 1, pp. 95-99 (an altered version with a new title)",
  "doi": "10.3233/FI-2013-854",
  "categories": [
    "math.NT",
    "math.LO"
  ],
  "abstract": "We conjecture that if a system S \\subseteq {x_i=1, x_i+x_j=x_k, x_i \\cdot x_j=x_k: i,j,k \\in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n| \\leq 2^{2^{n-1}}. By the conjecture, if a Diophantine equation has only finitely many solutions in integers (non-negative integers, rationals), then their heights are bounded from above by a computable function of the degree and the coefficients of the equation. The conjecture implies that the set of Diophantine equations which have infinitely many solutions in integers (non-negative integers) is recursively enumerable. The conjecture stated for an arbitrary computable bound instead of 2^{2^{n-1}} remains in contradiction to Matiyasevich's conjecture that each recursively enumerable set M \\subseteq {\\mathbb N}^n has a finite-fold Diophantine representation.",
  "revisions": [
    {
      "version": "v151",
      "updated": "2014-03-03T23:28:11.000Z",
      "comment": "LaTeX2e, 15 pages, the conjecture with the bound 2^(2^(n-1)) is false, a counterexample was communicated to the author. arXiv admin note: substantial text overlap with arXiv:1105.5747, arXiv:1102.4122, arXiv:1011.4103, arXiv:1109.3826"
    },
    {
      "version": "v152",
      "updated": "2014-10-20T12:23:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03D20",
      "11U05"
    ],
    "keywords": [
      "diophantine equation",
      "hypothetical upper bound",
      "finite number",
      "non-negative integers",
      "arbitrary computable bound"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0901.2093T"
    }
  }
}