{
  "annotation_id": "fe359e5e-f800-4aa1-a712-3425ca03d158",
  "date_created": "2026-02-17T05:53:20.395000Z",
  "date_modified": "2026-02-17T05:53:20.395000Z",
  "file_hash": "d7310e38fb8c521b03fe69bebebff65dfdc646525a047fd196efa56b1c64c09d",
  "private": false,
  "record": {
    "abstract": "It is an open problem whether $ \\binom{2n}{n} $ is divisible by 4 or 9 for all $n\u003e256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$\u0027s such that $ m \\nmid \\binom{2^{k+1}}{2^{k}} $ is 0. To do so we examine numbers of the form $\\alpha^{k}$ in base $p$, where $p$ is a prime and $(\\alpha, p)=1$. For every $n$ and $a$ we find an upper bound on the number of $k$\u0027s less than $a$ such that $(\\alpha^{k})_p$ contains less than $n$ digits greater than $\\frac{p}{2}$. This is done by showing that every sequence of the form $\\langle \\sigma_t, \\dots, \\sigma_1,\\sigma_0 \\rangle$, where $0\\leq \\sigma_i\u003cp$ for $i\\geq 1$ and $\\sigma_0$ is in the residue class generated by $\\alpha$ modulo $p$, occurs at specific places in the representation $(\\alpha^k)_p$ as $k$ varies.",
    "arxiv_id": "2601.09510",
    "authors": [
      "Sebastian Tim Holdum",
      "Frederik Ravn Klausen",
      "Peter Michael Reichstein Rasmussen"
    ],
    "categories": [
      "math.NT"
    ],
    "doi": "10.5281/zenodo.10456626",
    "journal_ref": "Integers 15,(2015), Paper No. A43,",
    "license": "http://creativecommons.org/licenses/by-sa/4.0/",
    "title": "Powers in prime bases and a problem on central binomial coefficients",
    "url": "https://arxiv.org/abs/2601.09510",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "b07e1d1e-ec2b-4022-ae00-493f49062d2a",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}