{
  "id": "1410.1712",
  "version": "v1",
  "published": "2014-10-07T13:01:48.000Z",
  "updated": "2014-10-07T13:01:48.000Z",
  "title": "A New Super Congruence Involving Multiple Harmonic Sums",
  "authors": [
    "Liuquan Wang"
  ],
  "comment": "10 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let ${\\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p\\ge 5$ and $r\\ge 2$, we prove that \\begin{equation} \\sum\\limits_{\\begin{smallmatrix} {{l}_{1}}+{{l}_{2}}+\\cdots +{{l}_{5}}={{p}^{r}} {{l}_{1}},\\cdots ,{{l}_{5}}\\in {\\mathcal{P}_{p}} \\end{smallmatrix}}{\\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}{{l}_{5}}}}\\equiv -\\frac{5!}{6}{{B}_{p-5}}{{p}^{r-1}} \\pmod{{{p}^{r}}}. \\end{equation} This gives an extension of a family of super congruences found by Wang, Cai and Zhao.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-07T13:01:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11A41"
    ],
    "keywords": [
      "multiple harmonic sums",
      "super congruence",
      "th bernoulli number",
      "positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.1712W"
    }
  }
}