{
  "abstract": "We study multiplicative dependence between terms of the \\(k\\)-generalized Pell sequence \\((P_n^{(k)})_{n\\ge 2-k}\\), defined by the linear recurrence \\[ P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\\cdots+P_{n-k}^{(k)}, \\] with initial conditions \\(P_0^{(k)}=\\cdots=P_{-(k-2)}^{(k)}=0\\) and \\(P_1^{(k)}=1\\). We prove that for every \\(k\\ge 2\\) and all integers \\(n>m\\ge 2\\), the equation \\[ \\bigl(P_n^{(k)}\\bigr)^x=\\bigl(P_m^{(k)}\\bigr)^y \\] has no solution in positive integers \\(x,y\\) with \\(x<y<n\\). The proof combines lower bounds for linear forms in logarithms, the Baker-Davenport reduction method, the LLL algorithm, and a finite computational verification.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.17699",
  "author_total": 4,
  "authors": [
    {
      "name": "Cherif B. Deme"
    },
    {
      "name": "Kancou D. Fall"
    },
    {
      "name": "Khady Faye"
    },
    {
      "name": "Bernadette Faye"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45jqkjhxv8ddm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.17699",
  "fulltext_url": null,
  "id": "45jqkjhxv8ddm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.17699",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7161711450",
  "pdf_url": "https://arxiv.org/pdf/2605.17699",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45jqkjhxv8ddm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2605.17699",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.17699"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Multiplicative independence in the sequence of $k$-generalized Pell numbers",
  "title_text": "Multiplicative independence in the sequence of k-generalized Pell numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/45jqkjhxv8ddm",
  "venue": "arXiv",
  "versions": []
}