{
  "abstract": "Call a set $A$ of positive integers admissible if, whenever $a_1\\cdots a_r=b_1\\cdots b_s$ with $a_1,\\dots,a_r$ distinct elements of $A$ and $b_1,\\dots,b_s$ distinct elements of $A$, necessarily $r=s$. Erdős asked whether admissible sets can have density $1-\\varepsilon$ for every $\\varepsilon>0$, and whether $\\{1,\\dots,N\\}$ always contains an admissible subset of size $(1-o(1))N$. For the variant in which repetitions are allowed both questions were answered negatively by Erdős, Ruzsa and Sárközy and by Granville and Soundararajan; for products of distinct elements, the first question was answered only recently (with density bound $7/8$), and the second has remained open. We show that every admissible $A\\subseteq\\{1,\\dots,N\\}$ satisfies $\\sum_{a\\in A}1/a\\le\\tfrac12\\log N+(\\log\\log N+2)^2$, and that there is an absolute constant $η>0$ such that every admissible $A\\subseteq\\{1,\\dots,N\\}$ has $|A|<(1-η)N$ for all large $N$. Both questions therefore have negative answers. The proofs are elementary; the second rests on a coupling that replaces the largest divisor of an integer composed of small primes, which avoids the divisor-function losses inherent in counting quotients along a multiplication table. Both negative answers are formally verified in Lean 4 against the statements of the Formal Conjectures project.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37471",
  "author_total": 1,
  "authors": [
    {
      "name": "Shisheng Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5pabbdqhcq1xv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37471",
  "fulltext_url": null,
  "id": "5pabbdqhcq1xv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37471",
  "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": "W7214890764",
  "pdf_url": "https://arxiv.org/pdf/2609.37471",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5pabbdqhcq1xv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37471",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37471"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A negative answer to Erdős Problem #786",
  "title_text": "A negative answer to Erdős Problem #786",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/5pabbdqhcq1xv",
  "venue": "arXiv",
  "versions": []
}