{
  "id": "1503.07281",
  "version": "v1",
  "published": "2015-03-25T05:11:12.000Z",
  "updated": "2015-03-25T05:11:12.000Z",
  "title": "Note on vanshing power sums of roots of unity",
  "authors": [
    "Neeraj Kumar",
    "K. Senthil Kumar"
  ],
  "comment": "4 pages, 1 table, comments are welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a given positive integers $m$ and $\\ell$, we give a complete list of integers $n$ for which their exist $m$th roots of unity $x_1,\\dots,x_n \\in \\mathbb{C}$ such that $x_1^{\\ell} + \\cdots + x_n^{\\ell}=0$. This extends the earlier result of Lam and Leung on vanishing sums of roots of unity. Furthermore, we prove that for which integers $n$ with $2 \\leq n \\leq m$, there are distinct $m$th roots of unity $x_1,\\dots,x_n \\in \\mathbb{C}$ such that $x_1^{\\ell} + \\cdots + x_n^{\\ell}=0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-25T05:11:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11L03",
      "11R18"
    ],
    "keywords": [
      "vanshing power sums",
      "th roots",
      "complete list",
      "earlier result",
      "positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150307281K"
    }
  }
}