{
  "id": "2406.06224",
  "version": "v1",
  "published": "2024-06-10T12:40:23.000Z",
  "updated": "2024-06-10T12:40:23.000Z",
  "title": "Arithmetic density and congruences of $\\ell$-regular bipartitions",
  "authors": [
    "Nabin Kumar Meher",
    "Ankita Jindal"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $ B_{\\ell}(n)$ denote the number of $\\ell$-regular bipartitions of $n.$ In this article, we prove that $ B_{\\ell}(n)$ is always almost divisible by $p_i^j$ if $p_i^{2a_i}\\geq \\ell,$ where $j$ is a fixed positive integer and $\\ell=p_1^{a_1}p_2^{a_2}\\ldots p_m^{a_m},$ where $p_i$ are prime numbers $\\geq 5.$ Further, we obtain an infinities families of congruences for $B_3(n)$ and $B_5(n)$ by using Hecke eigen form theory and a result of Newman \\cite{Newmann1959}. Furthermore, by applying Radu and Seller's approach, we obtain an algorithm from which we get several congruences for $B_{p}(n)$, where $p$ is a prime number.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-10T12:40:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P83"
    ],
    "keywords": [
      "regular bipartitions",
      "arithmetic density",
      "congruences",
      "hecke eigen form theory",
      "prime number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}