{
  "id": "2407.19230",
  "version": "v1",
  "published": "2024-07-27T10:57:26.000Z",
  "updated": "2024-07-27T10:57:26.000Z",
  "title": "Distribution and congruences of $(u,v)$-regular bipartitions",
  "authors": [
    "Nabin Kumar Meher"
  ],
  "comment": "First draft of the paper. Comments are welcome. arXiv admin note: substantial text overlap with arXiv:2406.06224, arXiv:2406.07905",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $B_{u,v}(n)$ denote the number of $(u,v)$-regular bipartitions of $n$. In this article, we prove that $B_{p,m}(n)$ is always almost divisible by $p,$ where $p\\geq 5$ is a prime number and $m=p_1^{\\alpha_1} p_2^{\\alpha_2}\\cdots p_r^{\\alpha_r}, $ where $\\alpha_i \\geq 0$ and $p_i \\geq 5$ be distinct primes with $\\gcd(p,m)=1$ . Further, we obtain an infinities families of congruences modulo $3$ for $B_{3,7}(n),$ $B_{3,5}(n)$ and $B_{3,2}(n)$ by using Hecke eigenform theory and a result of Newman \\cite{Newmann1959}. Furthermore, we get many infinite families of congruences modulo $7$, $11$ and $13$ respectively for $B_{2,7}(n)$, $B_{2,11}(n)$ and $B_{2,13}(n),$ by employing an identity of Newman \\cite{Newmann1959}. In addition, we prove infinite families of congruences modulo $2$ for $B_{4,3}(n)$, $B_{8,3}(n)$ and $B_{4,5}(n)$ by applying another result of Newman \\cite{Newmann1962}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-27T10:57:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P83"
    ],
    "keywords": [
      "regular bipartitions",
      "congruences modulo",
      "distribution",
      "infinite families",
      "hecke eigenform theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}