{
  "id": "1311.5646",
  "version": "v1",
  "published": "2013-11-22T03:32:59.000Z",
  "updated": "2013-11-22T03:32:59.000Z",
  "title": "On the Products $(1^\\ell+1)(2^\\ell+1)\\cdots (n^\\ell +1)$, II",
  "authors": [
    "Yong-Gao Chen",
    "Ming-Liang Gong"
  ],
  "comment": "13pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, the following results are proved: (i) For any odd integer $\\ell$ with at most two distinct prime factors and any positive integer $n$, the product $(1^\\ell+1)(2^\\ell+1)\\cdots (n^\\ell +1)$ is not a powerful number; (ii) For any integer $r\\ge 1$, there exists a positive integer $T_r$ such that, if $\\ell$ is a positive odd integer with at most $r$ distinct prime factors and $n$ is an integer with $n\\ge T_r$, then $(1^\\ell+1)(2^\\ell+1)\\cdots (n^\\ell +1)$ is not a powerful number.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-22T03:32:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25"
    ],
    "keywords": [
      "distinct prime factors",
      "powerful number",
      "positive integer",
      "positive odd integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.5646C"
    }
  }
}