{
  "id": "1311.0717",
  "version": "v1",
  "published": "2013-11-04T14:52:05.000Z",
  "updated": "2013-11-04T14:52:05.000Z",
  "title": "On certain diophantine equations of diagonal type",
  "authors": [
    "Andrew Bremner",
    "Maciej Ulas"
  ],
  "comment": "16 pages, revised version will appear in the Journal of Number Theory",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this note we consider Diophantine equations of the form \\begin{equation*} a(x^p-y^q) = b(z^r-w^s), \\quad \\mbox{where}\\quad \\frac{1}{p}+\\frac{1}{q}+\\frac{1}{r}+\\frac{1}{s}=1, \\end{equation*} with even positive integers $p,q,r,s$. We show that in each case the set of rational points on the underlying surface is dense in the Zariski topology. For the surface with $(p,q,r,s)=(2,6,6,6)$ we prove density of rational points in the Euclidean topology. Moreover, in this case we construct infinitely many parametric solutions in coprime polynomials. The same result is true for $(p,q,r,s)\\in\\{(2,4,8,8), (2,8,4,8)\\}$. In the case $(p,q,r,s)=(4,4,4,4)$, we present some new parametric solutions of the equation $x^4-y^4=4(z^4-w^4)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-04T14:52:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D57",
      "11D85"
    ],
    "keywords": [
      "diophantine equations",
      "diagonal type",
      "rational points",
      "parametric solutions",
      "zariski topology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.0717B"
    }
  }
}