{
  "abstract": "Let $s(m)$ be the Conway subprime function define the binary operation on the natural numbers $x \\circ y= s(x + y)$, and denote by $C_n$, $n \\ge 0$, the sequence of subsets of natural numbers defined by $C_0 = \\{1\\}$, and $C_{n+1} = C_n \\cup (C_n \\circ C_n)$. We prove the conjecture by Caragiu, Vicol and Zaki that $$\\lim_{n\\to \\infty} \\frac{ | C_{n+1}| }{ | C_n |}= \\frac{1+\\sqrt{5}}{2}.$$ The underlying mathematical proof in this paper was constructed with some algorithmic assistance from GPT-6 Astra and its correctness has been formally verified using the Lean 4 proof assistant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14188",
  "author_total": 1,
  "authors": [
    {
      "name": "Romain Popescu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/42txanqf9xj57.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14188",
  "fulltext_url": null,
  "id": "42txanqf9xj57",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14188",
  "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": "W7213271572",
  "pdf_url": "https://arxiv.org/pdf/2609.14188",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/42txanqf9xj57.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14188",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14188"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Golden-ratio growth of Conway's subprime closure",
  "title_text": "Golden-ratio growth of Conway's subprime closure",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/42txanqf9xj57",
  "venue": "arXiv",
  "versions": []
}