{
  "abstract": "Let $C>3$ be an odd square-free integer, $q$ an odd prime not dividing the class number of $\\mathbb Q(\\sqrt{-C})$, and $p_1,\\dots,p_k$ distinct odd primes not dividing $C$. We describe all solutions of $Cx^2+p_1^{2m_1}\\cdots p_k^{2m_k}=y^q$ in positive integers with $\\gcd(x,y)=1$ and $y$ odd. Writing $N=p_1^{m_1}\\cdots p_k^{m_k}$, every solution comes from a unique factorization $x\\sqrt C+N\\sqrt{-1}=\\pm(u\\sqrt C\\pm v\\sqrt{-1})^q$, or from a half-integral analogue when $q=3$, in which each $p_i\\ne q$ divides exactly one of $v$ and $N/v$. Using primitive divisors of Lehmer sequences, we show that for $q\\ge5$ some $p_i$ satisfies $p_i\\equiv\\leg{-C}{p_i}\\pmod q$; in particular, there is no solution when $q\\ge\\max(5,p_1+2,\\dots,p_k+2)$. For $q=3$ the equation reduces to four explicit auxiliary equations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04923",
  "author_total": 2,
  "authors": [
    {
      "name": "Usha K. Sangale"
    },
    {
      "name": "Priyanka R. Satpute"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/exqf8jhcn4f7j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04923",
  "fulltext_url": null,
  "id": "exqf8jhcn4f7j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04923",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04923",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/exqf8jhcn4f7j.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04923",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04923"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the Diophantine equation $Cx^2+p_1^{2m_1}p_2^{2m_2}\\cdots p_k^{2m_k}=y^q$",
  "title_text": "On the Diophantine equation Cx^2+p_1^2m_1p_2^2m_2⋯ p_k^2m_k=y^q",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/exqf8jhcn4f7j",
  "venue": "arXiv",
  "versions": []
}