{
  "abstract": "Let $\\{U(m)\\}_{m\\in \\N}$ and $\\{V(n)\\}_{n\\in \\N}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set of natural numbers $n$ such that the ratio $U(n)/V(n)$ is an integer. We study the finiteness problem for the set $(m, n)\\in \\mathbb{N}^2$ such that there exist non-zero positive integers $d_{m, n}$ satisfying $\\log |d_{m, n}|=o(n)$, and $d_{m, n}U(m)/V(n)$ is an element from a finitely generated subring of $\\C$. In particular, we prove that for $m\\neq n $, there exists a polynomial $P$ such that $d_{m, n}P(n)U(m)/V(n)$ is a multi-recurrence and $V(n)/P(n)$ is a linear recurrence and for $m=n$ both $d_{m, n}P(n)U(m)/V(n)$ and $V(n)/P(n)$ are linear recurrences. To prove our results, we employ Schmidt's subspace theorem, and the concept of moving hyperplanes, moving polynomials, and moving points.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.05784",
  "author_total": 2,
  "authors": [
    {
      "name": "Parvathi S Nair"
    },
    {
      "name": "S. S. Rout"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/btk4wd88dke80.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.05784",
  "fulltext_url": null,
  "id": "btk4wd88dke80",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.05784",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7160647805",
  "pdf_url": "https://arxiv.org/pdf/2605.05784",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/btk4wd88dke80.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2605.05784",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.05784"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The quotient problem for linear recurrence sequences",
  "title_text": "The quotient problem for linear recurrence sequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/btk4wd88dke80",
  "venue": "arXiv",
  "versions": []
}