{
  "abstract": "In this paper, we discuss two related Diophantine tuple problems involving star numbers $S_n=6n(n-1)+1$. First, we investigate classical $D(k)$-tuples whose elements are star numbers. We construct infinitely many $D(k)$-pairs for several positive and negative admissible values of $k$, and we prove several nonexistence results for $D(k)$-triples and quadruples. Second, we introduce a modified notion of a star number $D(k)$-tuple, requiring $a_ia_j+k$ itself to be a star number. We construct several infinite families of $D(1)$-triples, and study their extensions to quadruples. In particular, we prove that $\\{4,108,165\\}$ has the unique extension $\\{4,108,165,360\\}$ to a star number $D(1)$-quadruple. The main tools used in this paper are generalized Pell equations, Nagell's theorem, lower bounds for linear forms in logarithms due to Matveev and Dujella-Pethő version of the Baker-Davenport reduction method in diophantine approximation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27033",
  "author_total": 2,
  "authors": [
    {
      "name": "Tapas Chatterjee"
    },
    {
      "name": "Sagar Mandal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e5gjs8ntpe17k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27033",
  "fulltext_url": null,
  "id": "e5gjs8ntpe17k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27033",
  "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": "W7223117340",
  "pdf_url": "https://arxiv.org/pdf/2610.27033",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e5gjs8ntpe17k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27033",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27033"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Algebraic Geometry and Number Theory",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On Diophantine tuples involving star numbers",
  "title_text": "On Diophantine tuples involving star numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/e5gjs8ntpe17k",
  "venue": "arXiv",
  "versions": []
}