{
  "id": "0904.2530",
  "version": "v1",
  "published": "2009-04-16T16:00:24.000Z",
  "updated": "2009-04-16T16:00:24.000Z",
  "title": "Congruences of the partition function",
  "authors": [
    "Yifan Yang"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p(n)$ denote the partition function. In this article, we will show that congruences of the form $$ p(m^j\\ell^kn+B)\\equiv 0\\mod m \\text{for all} n\\ge 0 $$ exist for all primes $m$ and $\\ell$ satisfying $m\\ge 13$ and $\\ell\\neq 2,3,m$. Here the integer $k$ depends on the Hecke eigenvalues of a certain invariant subspace of $S_{m/2-1}(\\Gamma_0(576),\\chi_{12})$ and can be explicitly computed. More generally, we will show that for each integer $i>0$ there exists an integer $k$ such that for every non-negative integers $j\\ge i$ with a properly chosen $B$ the congruence $$ p(m^j\\ell^kn+B)\\equiv 0\\mod m^i $$ holds for all integers $n$ not divisible by $\\ell$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-04-16T16:00:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P83",
      "11F25",
      "11F37",
      "11P82"
    ],
    "keywords": [
      "partition function",
      "congruence",
      "hecke eigenvalues",
      "invariant subspace"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.2530Y"
    }
  }
}