{
  "abstract": "In 1948, Erdös and Straus formulated a conjecture : for any positive integer $n>2$, there exist positive integers $n_1,n_2$ and $n_3$ such that \\begin{equation}\\frac{4}{n}=\\frac{1}{n_1}+\\frac{1}{n_2}+\\frac{1}{n_3},\\nonumber\\end{equation} which is still open. It is known that one only needs to prove the conjecture for any prime number $n$ such that $n\\equiv 1\\;(\\mbox{mod}\\;24)$. If $n=24m+1$ and $n_1\\leq n_2,n_3$, then $n_1=6m+k$ with $1\\leq k\\leq 12m$. A solution $(n_1,n_2,n_3)$ of the above equation is called a {\\it tame solution} if $n_2$ and $n_3$ are factors of $(6m+k)(24m+1)$. We call $n=24m+1$ {\\it wild} if it does not have any tame solution. In this paper, we derive twenty-eight families of tame solutions of the above equation. Numeric test shows that they cover all the tame primes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.23601",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiaoping Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0a9aqcksbqgjx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.23601",
  "fulltext_url": null,
  "id": "0a9aqcksbqgjx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.23601",
  "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": "W7162303464",
  "pdf_url": "https://arxiv.org/pdf/2605.23601",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0a9aqcksbqgjx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2605.23601",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.23601"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Congruence Classes of Supporting the Erdös-Straus Conjecture I: Tame Solutions",
  "title_text": "Congruence Classes of Supporting the Erdös-Straus Conjecture I: Tame Solutions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/0a9aqcksbqgjx",
  "venue": "arXiv",
  "versions": []
}