{
  "abstract": "For a positive integer $n$, we write $L(n)$ in place of the maximum number of distinct primes whose sum is $n$, assuming there is a way of writing $n$ as a sum of distinct primes, with $L(n) = 0$ otherwise. Again if $n$ admits a partition into distinct primes, then we let $K(n)$ denote the number of parts in the unique partition that has a largest product, with $K(n) = 0$ otherwise. In the latest editions of Guy's classic text on unsolved problems in number theory, the problem as to whether or not $L(n) - K(n)$ is unbounded was left open, and it appears that this has remained open. We solve this problem in the affirmative. Our proof relies, to a large extent, on a consequence of Vinogradov's theorem together with a 1948 result due to Sierpiński on the existence of primes whose gaps to the immediately preceding and following primes both exceed any prescribed positive bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19534",
  "author_total": 1,
  "authors": [
    {
      "name": "John M. Campbell"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0xzzamqrjtd31.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19534",
  "fulltext_url": null,
  "id": "0xzzamqrjtd31",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19534",
  "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": "W7222346538",
  "pdf_url": "https://arxiv.org/pdf/2610.19534",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0xzzamqrjtd31.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19534",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19534"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On a problem on partitions into distinct primes",
  "title_text": "On a problem on partitions into distinct primes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/0xzzamqrjtd31",
  "venue": "arXiv",
  "versions": []
}