{
  "id": "1902.10672",
  "version": "v1",
  "published": "2019-02-27T18:14:03.000Z",
  "updated": "2019-02-27T18:14:03.000Z",
  "title": "On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-$p$ digits",
  "authors": [
    "Bernd C. Kellner",
    "Jonathan Sondow"
  ],
  "comment": "21 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We give a new characterization of the set $\\mathcal{C}$ of Carmichael numbers in the context of $p$-adic theory, independently of the classical results of Korselt and Carmichael. The characterization originates from a surprising link to the denominators of the Bernoulli polynomials via the sum-of-base-$p$-digits function. More precisely, we show that such a denominator obeys a triple-product identity, where one factor is connected with a $p$-adically defined subset $\\mathcal{S}$ of the squarefree integers that contains $\\mathcal{C}$. This leads to the definition of a new subset $\\mathcal{C}'$ of $\\mathcal{C}$, called the \"primary Carmichael numbers\". Subsequently, we establish that every Carmichael number equals an explicitly determined polygonal number. Finally, the set $\\mathcal{S}$ is covered by modular subsets $\\mathcal{S}_d$ ($d \\geq 1$) that are related to the Kn\\\"odel numbers, where $\\mathcal{C} = \\mathcal{S}_1$ is a special case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-27T18:14:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B68",
      "11B83"
    ],
    "keywords": [
      "bernoulli polynomials",
      "primary carmichael numbers",
      "carmichael number equals",
      "triple-product identity",
      "adic theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}