{
  "id": "math/0412036",
  "version": "v1",
  "published": "2004-12-01T21:01:14.000Z",
  "updated": "2004-12-01T21:01:14.000Z",
  "title": "The Diophantine equations $ x^{n}_{1} +x^{n}_{2} +...+x^{n}_{r_{1}}= y ^{n}_{1} +y^{n}_{2} +...+y^{n}_{r_{2}} $",
  "authors": [
    "Michael A. Ivanov"
  ],
  "comment": "11 pages, no figure, Latex. Eprint of published paper of 1996",
  "journal": "Rend. Sem. Mat. Univ. Pol. Torino, Vol. 54, 1 (1996) pp 25-33",
  "categories": [
    "math.NT",
    "math.AC"
  ],
  "abstract": "The aim of this paper is to prove the possibility of linearization of such equations by means of introduction of new variables. For $n=2$ such a procedure is well known, when new variables are components of spinors and they are widely used in mathematical physics. For example, parametrization of Pythagoras threes $a^{2} +b^{2}$, $a^{2} -b^{2}$, $2ab$ may be cited as an example in number theory where two independent variables form a spinor which can be obtained by solution of a system of two linear equations. We also investigate the combinatorial estimate for the smallest sum $r(n)=r _{1}+r_{2} -1 $ for solvable equations of such a type as $r(n) \\leq 2n+1$ (recently the better one with $r(n) \\leq2n-1$ was received by L. Habsieger (J. of Number Theory 45 (1993) 92)). Apart from that we consider two conjectures about $r(n)$ and particular solutions for $n \\leq11$ which were found with the help of the algorithm that is not connected with linearization.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-12-01T21:01:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "diophantine equations",
      "number theory",
      "independent variables form",
      "linearization",
      "smallest sum"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....12036I"
    }
  }
}