{
  "abstract": "Let $0<a_1<\\cdots<a_n$ be integers with $n\\ge 2$ and $\\gcd(a_1,\\ldots,a_n)=1$. Their Frobenius number $F(a_1,\\ldots,a_n)$ is the largest integer that cannot be expressed as a nonnegative integer combination of $a_1,\\ldots,a_n$. Erdős and Graham proved the classical bound $F(a_1,\\ldots,a_n)\\le 2a_{n-1}\\lfloor a_n/n\\rfloor-a_n$, which depends on the two largest generators. Dixmier subsequently proved their conjecture $F(a_1,\\ldots,a_n)\\le a_n^2/(n-1)$, a bound depending only on $a_n$ and $n$ that removes the factor $2$ from the corresponding quadratic estimate. Following the Erdős--Graham approach, we remove the factor $2$ while retaining the dependence on $a_{n-1}$, and prove $F(a_1,\\ldots,a_n)\\le a_{n-1}\\lceil a_n/(n-1)\\rceil-a_n-1$. This bound can be substantially smaller than Dixmier's when $a_{n-1}$ is small relative to $a_n$. For every $n\\ge3$, we also construct infinitely many instances satisfying the reverse inequality, which shows that the bound is sharp.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32219",
  "author_total": 4,
  "authors": [
    {
      "name": "Lin Chen"
    },
    {
      "name": "Tingwei Hu"
    },
    {
      "name": "Yuchen Mao"
    },
    {
      "name": "Guochuan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/619dhada252ek.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32219",
  "fulltext_url": null,
  "id": "619dhada252ek",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32219",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223666947",
  "pdf_url": "https://arxiv.org/pdf/2610.32219",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/619dhada252ek.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32219",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32219"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Sharp Upper Bound for the Frobenius Number",
  "title_text": "A Sharp Upper Bound for the Frobenius Number",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/619dhada252ek",
  "venue": "arXiv",
  "versions": []
}