{
  "abstract": "Let $q_n$ denote the $n$th product of two distinct primes. By completing a square in Sono's sieve and solving the resulting one-dimensional variational problem, we prove unconditionally that, for every $\\varepsilon>0$ and all sufficiently large integers $ρ$, \\[ \\liminf_{n\\to\\infty}(q_{n+ρ}-q_n) \\leq \\exp\\left((π+\\varepsilon)\\sqrtρ\\right). \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35548",
  "author_total": 1,
  "authors": [
    {
      "name": "Genheng Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/38652sd8a6v2c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35548",
  "fulltext_url": null,
  "id": "38652sd8a6v2c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35548",
  "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": "W7214767559",
  "pdf_url": "https://arxiv.org/pdf/2609.35548",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/38652sd8a6v2c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35548",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35548"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A note on large clusters of $E_2$-numbers",
  "title_text": "A note on large clusters of E_2-numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/38652sd8a6v2c",
  "venue": "arXiv",
  "versions": []
}