{
  "abstract": "A Wagstaff prime is a prime number of the form $(2^{\\mathfrak{p}}+1)/3$, where $\\mathfrak{p}$ is an odd prime. For an integer $k\\ge 2$, the sequence of $k$-generalized Pell numbers $\\bigl(P_n^{(k)}\\bigr)_{n\\ge 2-k}$ is the linear recurrence of order $k$ given by $P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\\cdots+P_{n-k}^{(k)}$ for $n\\ge 2$, with initial values $P_{2-k}^{(k)}=\\cdots=P_{0}^{(k)}=0$ and $P_1^{(k)}=1$. For $k=2$ this is the classical Pell sequence. Motivated by the recent determination of the Wagstaff primes in the $k$-Fibonacci and $k$-Lucas sequences, we study the exponential Diophantine equation $P_n^{(k)}=(2^{\\mathfrak{p}}+1)/3$ in positive integers $n,k$ and an odd prime $\\mathfrak{p}\\geq 3$, with $k\\ge 2$. We show that this equation has no solutions, that is no $k$-Pell number is a Wagstaff prime. The proof combines linear forms in logarithms of algebraic numbers with reduction based on the theory of continued fractions, together with a computational verification in the remaining finite range.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01170",
  "author_total": 1,
  "authors": [
    {
      "name": "Prosper Kaggwa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7pjzv9eszqe4b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01170",
  "fulltext_url": null,
  "id": "7pjzv9eszqe4b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01170",
  "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.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01170",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7pjzv9eszqe4b.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01170",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01170"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Wagstaff primes in the $k$-Pell number sequence",
  "title_text": "On Wagstaff primes in the k-Pell number sequence",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/7pjzv9eszqe4b",
  "venue": "arXiv",
  "versions": []
}