{
  "abstract": "For $n\\ge2$ let $F(n)$ be the least $k\\ge2$ such that $n!$ is the largest factor in a product of $k$ distinct factorials that is a perfect square, and let $D_k(X)$ be the number of $n\\le X$ with $F(n)=k$. Erdos and Graham asked for the order of growth of $D_k(X)$ for $3\\le k\\le6$, and conjectured that $D_6(X)\\gg X$. We prove that $D_3(X)=κ_3\\sqrt X+O_\\varepsilon(X^{2/5+\\varepsilon})$ with an explicit constant $κ_3=2.7097\\ldots$, and that $D_5(X)\\asymp D_6(X)\\asymp X$. Together with classical facts, this determines the order of growth of $D_k(X)$ for every $k$. The exponent $2/5$ comes from balancing a uniform bound for Pell equations against Gallagher's larger sieve, with residue restrictions supplied by the Weil bound. For five and six factors we restrict to integers with a prime factor exceeding $X^{1-α}$, where $α>0$ is small and fixed. We exclude shorter representations by combining an equidistribution estimate for primes of Matomaki, Radziwill, Shao, Tao and Teravainen with the large sieve. In an appendix we use a zero-sum theorem for finite abelian groups to construct, for every $m\\ge2$, perfect $m$-th powers that are products of a bounded number of distinct factorials with arguments given by fixed affine functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01899",
  "author_total": 1,
  "authors": [
    {
      "name": "Fedir Yudin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/75vg4mqz7c99t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01899",
  "fulltext_url": null,
  "id": "75vg4mqz7c99t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01899",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7216431441",
  "pdf_url": "https://arxiv.org/pdf/2610.01899",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/75vg4mqz7c99t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01899",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01899"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Square products of factorials and a conjecture of Erdős and Graham",
  "title_text": "Square products of factorials and a conjecture of Erdős and Graham",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/75vg4mqz7c99t",
  "venue": "arXiv",
  "versions": []
}