{
  "id": "1309.7537",
  "version": "v1",
  "published": "2013-09-29T05:12:57.000Z",
  "updated": "2013-09-29T05:12:57.000Z",
  "title": "A variety of Euler's conjecture",
  "authors": [
    "Tianxin Cai",
    "Yong Zhang"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider a variety of Euler's conjecture, i.e., whether the Diophantine system \\[\\begin{cases} n=a_{1}+a_{2}+\\cdots+a_{s-1}, a_{1}a_{2}\\cdots a_{s-1}(a_{1}+a_{2}+\\cdots+a_{s-1})=b^{s} \\end{cases}\\] has solutions $n,b,a_i\\in\\mathbb{Z}^+,i=1,2,\\ldots,s-1,s\\geq 3.$ By using the theory of elliptic curves, we prove that it has no solutions $n,b,a_i\\in\\mathbb{Z}^+$ for $s=3$, but for $s=4$ it has infinitely many solutions $n,b,a_i\\in\\mathbb{Z}^+$ and for $s\\geq 5$ there are infinitely many polynomial solutions $n,b,a_i\\in\\mathbb{Z}[t_1,t_2,\\ldots,t_{s-3}]$ with positive value satisfying this Diophantine system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-29T05:12:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D72",
      "11D41",
      "11G05"
    ],
    "keywords": [
      "eulers conjecture",
      "diophantine system",
      "elliptic curves",
      "polynomial solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.7537C"
    }
  }
}