{
  "abstract": "Let $\\mathbb{K}=\\mathbb{Q}(\\sqrt{-d})\\text{ or }\\mathbb{Q}(\\sqrt{d})$, where $d$ is a positive square-free integer, and denote by $\\mathcal{O}_\\mathbb{K}$ the ring of integers of $\\mathbb{K}$. I investigate the solution of the equation $x_1^{3}-x_2^{2}x_1+1=0$ where $x_1\\in \\mathbb{K}$ and $x_2\\in\\mathcal{O}_\\mathbb{K}$. The case $\\mathbb{K}=\\mathbb{Q}(\\sqrt{d})$ for $d\\equiv 2,3\\pmod{4}$ faces infinite units that require a separate treatment. Using the arithmetic of the quadratic integer rings $\\mathbb{Z}[\\sqrt{d}]]$, together with norm arguments, divisibility properties, and the explicit structure of its unit group, I prove that the equation has exactly two solutions, namely $(x_1,x_2)=(-1,0)~\\text{ and }~(1,\\sqrt{2})$ for $\\mathbb{K}=\\mathbb{Q}(\\sqrt{2})$ and one solution $(-1,0)$ for $\\mathbb{K}=\\mathbb{Q}(\\sqrt{d})$ As an application, I consider the family of elliptic curves $C_m:Y^{2}=X^{3}-m^{2}X+1,~ m\\in\\mathcal{O}_\\mathbb{K},$ and deduce that, for every $m\\neq0,\\sqrt{2}$ the Mordell--Weil group $C_m(\\mathbb{K})$ contains no rational point of order two.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.26079",
  "author_total": 1,
  "authors": [
    {
      "name": "Pinki Khatun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1bwk5zkfwwt0w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.26079",
  "fulltext_url": null,
  "id": "1bwk5zkfwwt0w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.26079",
  "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.GM"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7171802852",
  "pdf_url": "https://arxiv.org/pdf/2607.26079",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1bwk5zkfwwt0w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2607.26079",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.26079"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Algebraic Geometry and Number Theory",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Diophantine Equation $x_1^{3}-x_2^{2}x_1+1=0$ over number fields",
  "title_text": "Diophantine Equation x_1^3−x_2^2x_1+1=0 over number fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/1bwk5zkfwwt0w",
  "venue": "arXiv",
  "versions": []
}