{
  "abstract": "The Pell numbers satisfy $P_n^2+P_{n+1}^2=P_{2n+1}$, and Bravo, Luca and Pomeo recently found all the cases in which the sum of the $x$-th powers of two consecutive $k$-generalized Pell numbers is again a $k$-generalized Pell number. We ask what happens when the two indices are arbitrary, that is, we solve $\\big(P_n^{(k)}\\big)^x+\\big(P_r^{(k)}\\big)^x=P_m^{(k)}$ in positive integers $k\\ge2$, $n\\ge r$, $m$ and $x$. The answer is that nothing new appears: apart from $1^x+1^x=2$, every solution has consecutive indices. The proof combines linear forms in logarithms with an auxiliary Fibonacci equation which does not depend on $k$, and, for large $k$, with the fact that the dominant root of the sequence is very close to $φ^2$. For the reduction step, the method of Dujella and Pethő fails for large indices and has to be replaced by a criterion of Legendre type.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02295",
  "author_total": 2,
  "authors": [
    {
      "name": "Fallou Faye"
    },
    {
      "name": "Bernadette Faye"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fawjytmnjjhaf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02295",
  "fulltext_url": null,
  "id": "fawjytmnjjhaf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02295",
  "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/2611.02295",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fawjytmnjjhaf.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02295",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02295"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the exponential Diophantine equation $\\left(P_{n}^{(k)}\\right)^{x}+\\left(P_{r}^{(k)}\\right)^{x}=P_{m}^{(k)}$",
  "title_text": "On the exponential Diophantine equation (P_n^(k))^x+(P_r^(k))^x=P_m^(k)",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/fawjytmnjjhaf",
  "venue": "arXiv",
  "versions": []
}