The quotient problem for linear recurrence sequences

Abstract

Let {U(m)}_{m∈ N} and {V(n)}_{n∈ 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)∈ ℕ² 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.

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BibTeX RIS

BibTeX

@misc{nair2026quotient,
  title = {{The quotient problem for linear recurrence sequences}},
  author = {Parvathi S Nair and S. S. Rout},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.05784},
  eprint = {2605.05784},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.05784},
}

RIS

TY  - UNPB
TI  - The quotient problem for linear recurrence sequences
AU  - Parvathi S Nair
AU  - S. S. Rout
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2605.05784
UR  - https://arxiv.org/abs/2605.05784
AB  - Let {U(m)}_m∈ N and {V(n)}_n∈ 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)∈ ℕ^2 such that there exist non-zero positive integers d_m, n satisfying log |d_m, n|=o(n), and d_m, nU(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, nP(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, nP(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.
ER  -

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