{
  "abstract": "Lee and O'Clarey recently conjectured that every integer $n>24$ can be written as the sum of a prime and a positive integer that is not square-free. They proved this for odd $n$, and for all $n>24$ assuming the generalised Riemann hypothesis for Dirichlet $L$-functions. We prove their conjecture unconditionally for every integer $n>24$ that is not divisible by $997\\#$, where $997\\#=\\prod_{p\\leq 997}p$. In particular, any counterexample must be divisible by every prime up to $997$, and hence exceeds $10^{415}$. The proof uses an elementary congruence argument together with finite computation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14714",
  "author_total": 1,
  "authors": [
    {
      "name": "Peter J. Campbell"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4cjmmsfah7bwg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14714",
  "fulltext_url": null,
  "id": "4cjmmsfah7bwg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14714",
  "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": "W7213281841",
  "pdf_url": "https://arxiv.org/pdf/2609.14714",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4cjmmsfah7bwg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14714",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14714"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Prime Plus a Non-Square-Free Integer: An Elementary Approach",
  "title_text": "Prime Plus a Non-Square-Free Integer: An Elementary Approach",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/4cjmmsfah7bwg",
  "venue": "arXiv",
  "versions": []
}