{
  "id": "1604.07495",
  "version": "v1",
  "published": "2016-04-26T02:14:10.000Z",
  "updated": "2016-04-26T02:14:10.000Z",
  "title": "Congruences for powers of the partition function",
  "authors": [
    "Madeline Locus",
    "Ian Wagner"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p_{-t}(n)$ denote the number of partitions of $n$ into $t$ colors. In analogy with Ramanujan's work on the partition function, Lin recently proved in \\cite{Lin} that $p_{-3}(11n+7)\\equiv0\\pmod{11}$ for every integer $n$. Such congruences, those of the form $p_{-t}(\\ell n + a) \\equiv 0 \\pmod {\\ell}$, were previously studied by Kiming and Olsson. If $\\ell \\geq 5$ is prime and $-t \\not \\in \\{\\ell - 1, \\ell -3\\}$, then such congruences satisfy $24a \\equiv -t \\pmod {\\ell}$. Inspired by Lin's example, we obtain natural infinite families of such congruences. If $\\ell\\equiv2\\pmod{3}$ (resp. $\\ell\\equiv3\\pmod{4}$ and $\\ell\\equiv11\\pmod{12}$) is prime and $r\\in\\{4,8,14\\}$ (resp. $r\\in\\{6,10\\}$ and $r=26$), then for $t=\\ell s-r$, where $s\\geq0$, we have that \\begin{equation*} p_{-t}\\left(\\ell n+\\frac{r(\\ell^2-1)}{24}-\\ell\\Big\\lfloor\\frac{r(\\ell^2-1)}{24\\ell}\\Big\\rfloor\\right)\\equiv0\\pmod{\\ell}. \\end{equation*} Moreover, we exhibit infinite families where such congruences cannot hold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-26T02:14:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "partition function",
      "natural infinite families",
      "ramanujans work",
      "lins example",
      "congruences satisfy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160407495L"
    }
  }
}