{
  "abstract": "We solve a fifty-year-old conjecture of Erdős and Graham concerning whether the binomial coefficient $\\binom{n}{k}$ with $1 \\leq k \\leq \\frac{n}{2}$ must always have a divisor $\\leq n$ that is ``close'' to $n$: that is, bigger than a constant times $n$. We show this is the case when $k$ is sufficiently large as a function of $n$. However, we show it is possible to find binomial coefficients $\\binom{n}{k}$, where $k$ is small compared to $n$, such that $\\binom{n}{k}$ does not have divisors $\\leq n$ close to $n$. This latter, more substantial argument involves a restricted covering problem with residue classes, sieve methods, and various exponential sum estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.21221",
  "author_total": 4,
  "authors": [
    {
      "name": "Hung M. Bui"
    },
    {
      "name": "Slava Naprienko"
    },
    {
      "name": "Kyle Pratt"
    },
    {
      "name": "Alexandru Zaharescu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4376pfznaxeb4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.21221",
  "fulltext_url": null,
  "id": "4376pfznaxeb4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.21221",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7161939453",
  "pdf_url": "https://arxiv.org/pdf/2605.21221",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4376pfznaxeb4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2605.21221",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.21221"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Binomial coefficients with divisors avoiding an interval",
  "title_text": "Binomial coefficients with divisors avoiding an interval",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/4376pfznaxeb4",
  "venue": "arXiv",
  "versions": []
}