Multiplicative independence in the sequence of $k$-generalized Pell numbers
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913137498783744 |
|---|---|
| author | Deme, Cherif B. Fall, Kancou D. Faye, Khady Faye, Bernadette |
| author_facet | Deme, Cherif B. Fall, Kancou D. Faye, Khady Faye, Bernadette |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17699 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiplicative independence in the sequence of $k$-generalized Pell numbers Deme, Cherif B. Fall, Kancou D. Faye, Khady Faye, Bernadette Number Theory 11B39, 11D61, 11Y50 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. |
| title | Multiplicative independence in the sequence of $k$-generalized Pell numbers |
| topic | Number Theory 11B39, 11D61, 11Y50 |
| url | https://arxiv.org/abs/2605.17699 |