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PreprintCC0arXiv

Multiplicative independence in the sequence of k-generalized Pell numbers

  • Cherif B. Deme
  • Kancou D. Fall
  • Khady Faye
  • Bernadette Faye
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Abstract

We study multiplicative dependence between terms of the k-generalized Pell sequence (Pn(k))n≥ 2−k, defined by the linear recurrence Pn(k)=2Pn−1(k)+Pn−2(k)+⋯+Pn−k(k), with initial conditions P0(k)=⋯=P−(k−2)(k)=0 and P1(k)=1. We prove that for every k≥ 2 and all integers n>m≥ 2, the equation (Pn(k))x=(Pm(k))y has no solution in positive integers x,y with x<y<n. The proof combines lower bounds for linear forms in logarithms, the Baker-Davenport reduction method, the LLL algorithm, and a finite computational verification.

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Cite

BibTeXRIS
BibTeX
@misc{deme2026multiplicative,
  title = {{Multiplicative independence in the sequence of $k$-generalized Pell numbers}},
  author = {Cherif B. Deme and Kancou D. Fall and Khady Faye and Bernadette Faye},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.17699},
  eprint = {2605.17699},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.17699},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Multiplicative independence in the sequence of k-generalized Pell numbers
AU  - Cherif B. Deme
AU  - Kancou D. Fall
AU  - Khady Faye
AU  - Bernadette Faye
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2605.17699
UR  - https://arxiv.org/abs/2605.17699
AB  - We study multiplicative dependence between terms of the k-generalized Pell sequence (P_n^(k))_n≥ 2−k, defined by the linear recurrence P_n^(k)=2P_n−1^(k)+P_n−2^(k)+⋯+P_n−k^(k), with initial conditions P_0^(k)=⋯=P_−(k−2)^(k)=0 and P_1^(k)=1. We prove that for every k≥ 2 and all integers n>m≥ 2, the equation (P_n^(k))^x=(P_m^(k))^y has no solution in positive integers x,y with x<y<n. The proof combines lower bounds for linear forms in logarithms, the Baker-Davenport reduction method, the LLL algorithm, and a finite computational verification.
ER  -