{
  "abstract": "A negative Pell equation produces positive integers with $a_n^3+b_n^3+c_n^3=561N_n^3+(-1)^{n+1}$. We classify all positive solutions satisfying the natural linear constraints of the construction and prove an exact pairwise coprimality law, with exceptional factor $137$ in one class modulo $68$. The associated sextic $K=(1-13u+26u^2)^3+(6+182u^2)^3$ defines an elliptic K3 surface with six type-IV fibres. Its genus-four cubic cover has a geometrically simple Prym threefold; exact good-reduction arithmetic establishes simplicity over every finite extension of the residue field. We determine the full geometric Mordell-Weil group, the integral Neron-Severi and transcendental lattices, and the arithmetic Picard lattice over $\\mathbb{Q}$, which is $U\\oplus\\langle-4\\rangle^{\\oplus4}$. Every two-cube representation by rational functions over $\\mathbb{Q}$ is obtained from a multiple of one section, with numerator degrees $2m^2$ and denominator degree $2m^2-2$. Polynomial rigidity follows without a degree assumption. Along the specified Pell orbit all specialized points are nontorsion, their heights are $\\frac{4}{3}|n|\\log(8+\\sqrt{65})+O(1)$, and infinitely many rational cubic-twist classes occur. We also identify the local zeta function at $19$ and the generic rank and rank-jump criterion for the corresponding two-parameter base-change family.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.11925",
  "author_total": 1,
  "authors": [
    {
      "name": "K. Srinivasa Raghava"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/967d0ajwetewr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.11925",
  "fulltext_url": null,
  "id": "967d0ajwetewr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.11925",
  "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.GM"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7168408786",
  "pdf_url": "https://arxiv.org/pdf/2607.11925",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/967d0ajwetewr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2607.11925",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.11925"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Algebraic Geometry and Number Theory",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Ramanujan Pell K3 Surface and Cube Representations",
  "title_text": "A Ramanujan Pell K3 Surface and Cube Representations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/967d0ajwetewr",
  "venue": "arXiv",
  "versions": []
}