On Diophantine equations involving intersection of Thabit and Williams numbers base $b$ and some ternary recurrent sequences
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866918160182018048 |
|---|---|
| author | Tripathy, Bibhu Prasad Satapathy, Asutosh Dutta, Utkal Keshari Patel, Bijan Kumar |
| author_facet | Tripathy, Bibhu Prasad Satapathy, Asutosh Dutta, Utkal Keshari Patel, Bijan Kumar |
| contents | Let $\mathcal{P}_{n}$ be the $n$-th Padovan number, $E_{n}$ be the $n$-th Perrin number and $N_{n}$ be the $n$-th Narayana's cows number. Let $b$ be a positive integer such that $b \geq 2$. In this paper, we study the Diophantine equations
\[
\mathcal{P}_{n} = (b \pm 1)\cdot b^{l} \pm 1,
\]
\[
E_{n} = (b \pm 1)\cdot b^{l} \pm 1,
\] and \[
N_{n} = (b \pm 1)\cdot b^{l} \pm 1,
\] in non-negative integers $n, b$ and positive integer $l$. As a result, we determine the Padovan, Perrin and Narayana's cows numbers that are Thabit and Williams numbers base $b$. Moreover, we determine all solutions of the above equations within the range $2 \leq b \leq 10$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12145 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Diophantine equations involving intersection of Thabit and Williams numbers base $b$ and some ternary recurrent sequences Tripathy, Bibhu Prasad Satapathy, Asutosh Dutta, Utkal Keshari Patel, Bijan Kumar Number Theory 11B39, 11D61, 11J86 Let $\mathcal{P}_{n}$ be the $n$-th Padovan number, $E_{n}$ be the $n$-th Perrin number and $N_{n}$ be the $n$-th Narayana's cows number. Let $b$ be a positive integer such that $b \geq 2$. In this paper, we study the Diophantine equations \[ \mathcal{P}_{n} = (b \pm 1)\cdot b^{l} \pm 1, \] \[ E_{n} = (b \pm 1)\cdot b^{l} \pm 1, \] and \[ N_{n} = (b \pm 1)\cdot b^{l} \pm 1, \] in non-negative integers $n, b$ and positive integer $l$. As a result, we determine the Padovan, Perrin and Narayana's cows numbers that are Thabit and Williams numbers base $b$. Moreover, we determine all solutions of the above equations within the range $2 \leq b \leq 10$. |
| title | On Diophantine equations involving intersection of Thabit and Williams numbers base $b$ and some ternary recurrent sequences |
| topic | Number Theory 11B39, 11D61, 11J86 |
| url | https://arxiv.org/abs/2510.12145 |