{
  "abstract": "For an integer $k\\ge2$, let $(P_n^{(k)})_{n\\ge0}$ be the $k$-generalized Pell sequence and $(T_m)_{m\\ge1}$ the Tribonacci sequence. We determine, for every $k\\ge2$, all integers $c$ having at least two representations $c=P_n^{(k)}-T_m$ with $m\\ge2$. The starting point is an identity of Kılıç: $P_n^{(k)}=F_{2n+1}$ for $0\\le n\\le k$. In the range $n\\le k$ the problem therefore becomes, for all $k$ at once, Pillai's equation between odd-indexed Fibonacci numbers and Tribonacci numbers, which has exactly six solutions. In the range $n>k$ we solve the problem for $k=2$ and $k=3$ using the exact heights of the Binet coefficients; for $k\\ge4$ we prove a bound for $n$ polynomial in $k$. We also use the fact that $P_n^{(k)}$ is extremely close to $F_{2n+1}$ when $k$ is large to show that $k\\le414$, and treat the remaining values one by one. For $k\\ge4$ there are, besides the solutions common to all $k$, exactly four further solutions, for $k=4$, $5$ and $9$. The proofs combine Matveev's theorem with the reduction method of Dujella and Pethő.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24333",
  "author_total": 3,
  "authors": [
    {
      "name": "Cherif B. Deme"
    },
    {
      "name": "Kancou D. Fall"
    },
    {
      "name": "Bernadette Faye"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5qpmr8dh40pg5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24333",
  "fulltext_url": null,
  "id": "5qpmr8dh40pg5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24333",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223000729",
  "pdf_url": "https://arxiv.org/pdf/2610.24333",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5qpmr8dh40pg5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24333",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24333"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Pillai's equation between k-generalized Pell numbers and Tribonacci numbers",
  "title_text": "Pillai's equation between k-generalized Pell numbers and Tribonacci numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/5qpmr8dh40pg5",
  "venue": "arXiv",
  "versions": []
}