{
  "abstract": "For the Pell numbers one has $P_{n+1}-P_{n-1}=2P_n$ and $P_{n+1}^2-P_{n-1}^2=2P_{2n}$. We look for all the solutions of $\\big(P_{n+1}^{(k)}\\big)^x-\\big(P_{n-1}^{(k)}\\big)^x=2P_m^{(k)}$ in positive integers $k\\ge2$, $n$, $m$, $x$, where $P^{(k)}$ is the $k$-generalized Pell sequence. We prove that for $k=2$ the two identities above give all the solutions, and that for $k\\ge3$ the only solutions are $(n,m,x)=(1,1,1),(1,2,2),(2,2,1)$. In contrast with the analogous equation for $k$-generalized Fibonacci numbers, there are no parametric families of solutions. The proof uses an auxiliary equation with Fibonacci numbers which does not depend on $k$, Matveev's lower bound for linear forms in logarithms, the reduction of Dujella and Pethő, and, for large $k$, the fact that the dominant root of $P^{(k)}$ is very close to $φ^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02299",
  "author_total": 2,
  "authors": [
    {
      "name": "Fallou Faye"
    },
    {
      "name": "Bernadette Faye"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/apgc7jhymbzze.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02299",
  "fulltext_url": null,
  "id": "apgc7jhymbzze",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02299",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02299",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/apgc7jhymbzze.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02299",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02299"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the exponential Diophantine equation $\\left(P_{n+1}^{(k)}\\right)^{x}-\\left(P_{n-1}^{(k)}\\right)^{x}=2P_{m}^{(k)}$",
  "title_text": "On the exponential Diophantine equation (P_n+1^(k))^x−(P_n−1^(k))^x=2P_m^(k)",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/apgc7jhymbzze",
  "venue": "arXiv",
  "versions": []
}