{
  "id": "2503.11676",
  "version": "v2",
  "published": "2025-03-01T12:30:12.000Z",
  "updated": "2025-05-04T14:45:15.000Z",
  "title": "Note on a theorem of Birch and Erdős",
  "authors": [
    "Yuchen Ding",
    "Honghu Liu"
  ],
  "comment": "In this new version, we obtain a few more results",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p,q>1$ be two relatively prime integers and $\\mathbb{N}$ the set of nonnegative integers. Let $f_{p,q}(n)$ be the number of different expressions of $n$ written as a sum of distinct terms taken from $\\{p^{\\alpha}q^{\\beta}:\\alpha,\\beta\\in \\mathbb{N}\\}$. Erd\\H os conjectured and then Birch proved that $f_{p,q}(n)\\ge 1$ provided that $n$ is sufficiently large. In this note, for all sufficiently large number $n$ we prove $$ f_{p,q}(n)=2^{\\frac{(\\log n)^2}{2\\log p\\log q}\\big(1+O(\\log\\log n/\\log n)\\big)}. $$ We also prove that $f_{2,q}(n+1)\\ge f_{2,q}(n)$ for any $n\\ge 1$ and furthermore classify all the situations of $n$ such that $f_{2,q}(n+1)=f_{2,q}(n)$. Additionally, we will show that $$ \\lim_{n\\rightarrow\\infty}f_{2,q}(n+1)/f_{2,q}(n)=1. $$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-05-04T14:45:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distinct terms taken",
      "relatively prime integers",
      "sufficiently large number",
      "situations",
      "nonnegative integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}