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arXiv · 2605.17699

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 $(P_n^{(k)})_{n\ge 2-k}$, defined by the linear recurrence \[ P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \dots + P_{n-k}^{(k)}, \] with initial conditions $P_0^{(k)} = \dots = P_{-(k-2)}^{(k)} = 0$ and $P_1^{(k)} = 1$. For $k\ge 2$ we determine all pairs $(m,n)$ with $n>m\ge 0$ such that $P_n^{(k)}$ and $P_m^{(k)}$ are multiplicatively dependent. The main result states that the only solutions occur for very small $k,m,n$ (which are listed explicitly). The proof uses lower bounds for linear forms in logarithms (Matveev), the Baker-Davenport reduction algorithm, and a computational search.

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Cherif B. Deme, Kancou D. Fall, Khady Faye, Bernadette Faye. 2026-05-17. Multiplicative independence in the sequence of $k$-generalized Pell numbers. https://arxiv.org/abs/2605.17699

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