{
  "id": "1704.03504",
  "version": "v1",
  "published": "2017-04-12T15:54:06.000Z",
  "updated": "2017-04-12T15:54:06.000Z",
  "title": "Further Results on Hilbert's Tenth Problem",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "35 pages",
  "categories": [
    "math.NT",
    "math.LO"
  ],
  "abstract": "Hilbert's Tenth Problem (HTP) asked for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring $\\mathbb Z$ of the integers. This was finally solved by Matijasevich negatively in 1970. In this paper we obtain some further results on HTP over $\\mathbb Z$. We show that there is no algorithm to determine for any $P(z_1,\\ldots,z_9)\\in\\mathbb Z[z_1,\\ldots,z_9]$ whether the equation $P(z_1,\\ldots,z_9)=0$ has integral solutions with $z_9\\ge0$. Consequently, there is no algorithm to test whether an arbitrary polynomial Diophantine equation $P(z_1,\\ldots,z_{11})=0$ (with integer coefficients) in 11 unknowns has integral solutions, which provides the best record on the original HTP over $\\mathbb Z$. We also show that there is no algorithm to test for any $P(z_1,\\ldots,z_{17})\\in\\mathbb Z[z_1,\\ldots,z_{17}]$ whether $P(z_1^2,\\ldots,z_{17}^2)=0$ has integral solutions, and that there is a polynomial $Q(z_1,\\ldots,z_{20})\\in\\mathbb Z[z_1,\\ldots,z_{20}]$ such that $$\\{Q(z_1^2,\\ldots,z_{20}^2):\\ z_1,\\ldots,z_{20}\\in\\mathbb Z\\}\\cap\\{0,1,2,\\ldots\\}$$ coincides with the set of all primes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-12T15:54:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U05",
      "03D35",
      "03D25",
      "11D99",
      "11A41",
      "11B39"
    ],
    "keywords": [
      "arbitrary polynomial diophantine equation",
      "integral solutions",
      "integer coefficients",
      "original htp",
      "best record"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}