{
  "abstract": "In 1948, Erdös and Straus formulated a conjecture on Egyptian fractions. One only needs to prove the conjecture for any prime number $n$ such that $n\\equiv 1\\;(\\mbox{mod}\\;24)$. In this article, we give a necessary and sufficient polynomial condition for a number of the form $24m+1$ to be a prime number. Then we present 56 solvability polynomials consisting of our earlier tame solutions, wild solutions and the polynomials for composite numbers of the form $24m+1$. Numeric test shows that they imply the conjecture. Moreover, we give a list of 36 exponential-polynomial functions consisting of our earlier 28 tame solutions and the polynomials for composite numbers, whose numeric test shows that Howerton's conjecture on fourteen wild primes holds. Furthermore, we find a version of Goldbach conjecture modulo 24.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32302",
  "author_total": 2,
  "authors": [
    {
      "name": "Yichuan Cao"
    },
    {
      "name": "Xiaoping Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2ynvwvadycx6h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32302",
  "fulltext_url": null,
  "id": "2ynvwvadycx6h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32302",
  "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": "W7223707696",
  "pdf_url": "https://arxiv.org/pdf/2610.32302",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2ynvwvadycx6h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32302",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32302"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Prime Numbers of the Form $24m+1$ and Erdös-Straus Conjecture",
  "title_text": "Prime Numbers of the Form 24m+1 and Erdös-Straus Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/2ynvwvadycx6h",
  "venue": "arXiv",
  "versions": []
}