{
  "id": "1401.1717",
  "version": "v1",
  "published": "2014-01-08T15:02:51.000Z",
  "updated": "2014-01-08T15:02:51.000Z",
  "title": "A note on $p$-adic valuations of the Schenker sums",
  "authors": [
    "Piotr Miska"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "A prime number $p$ is called a Schenker prime if there exists such $n\\in\\mathbb{N}_+$ that $p\\nmid n$ and $p\\mid a_n$, where $a_n = \\sum_{j=0}^{n}\\frac{n!}{j!}n^j$ is so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning $p$-adic valuations of $a_n$ in case when $p$ is a Schenker prime. In particular, they asked whether for each $k\\in\\mathbb{N}_+$ there exists the unique positive integer $n_k<p^k$ such that $v_p(a_{m\\cdot 5^k + n_k})\\geq k$ for each nonnegative integer $m$. We prove that for every $k\\in\\mathbb{N}_+$ the inequality $v_5(a_n)\\geq k$ has exactly one solution modulo $5^k$. This confirms the first conjecture stated by the mentioned authors. Moreover, we show that if $37\\nmid n$ then $v_{37}(a_n)\\leq 1$, what means that the second conjecture stated by the mentioned authors is not true.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-08T15:02:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B50",
      "11B83"
    ],
    "keywords": [
      "adic valuations",
      "schenker sum",
      "schenker prime",
      "mentioned authors",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.1717M"
    }
  }
}