Cullen and Woodall numbers in Padovan and Perrin sequences
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| Format: | Preprint |
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2026
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| _version_ | 1866913154511929344 |
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| author | Batte, Herbert Bravo, Eric F. Luca, Florian |
| author_facet | Batte, Herbert Bravo, Eric F. Luca, Florian |
| contents | Let $\{P_n\}_{n\ge 0}$ and $\{R_n\}_{n\ge 0}$ denote the Padovan and Perrin sequences, both satisfying the recurrence $U_{n+3} = U_{n+1} + U_n$, but with initial values $P_0 = P_1 = P_2 = 1$ and $R_0 = 3$, $R_1 = 0$, $R_2 = 2$, respectively. A \textit{Cullen number} is a positive integer of the form $m\cdot 2^m + 1$ for some integer $m \ge 1$, while a \textit{Woodall number} is a positive integer of the form $m\cdot 2^m - 1$ for some integer $m \ge 1$. In this paper, we determine all Woodall numbers in the Padovan sequence and all Cullen numbers in the Perrin sequence. Specifically, we prove that $1$ and $7$ are the only Woodall numbers in the Padovan sequence, and that $3$ is the only Cullen number in the Perrin sequence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_23084 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cullen and Woodall numbers in Padovan and Perrin sequences Batte, Herbert Bravo, Eric F. Luca, Florian Number Theory 11B39, 11D61, 11D45, 11J86 Let $\{P_n\}_{n\ge 0}$ and $\{R_n\}_{n\ge 0}$ denote the Padovan and Perrin sequences, both satisfying the recurrence $U_{n+3} = U_{n+1} + U_n$, but with initial values $P_0 = P_1 = P_2 = 1$ and $R_0 = 3$, $R_1 = 0$, $R_2 = 2$, respectively. A \textit{Cullen number} is a positive integer of the form $m\cdot 2^m + 1$ for some integer $m \ge 1$, while a \textit{Woodall number} is a positive integer of the form $m\cdot 2^m - 1$ for some integer $m \ge 1$. In this paper, we determine all Woodall numbers in the Padovan sequence and all Cullen numbers in the Perrin sequence. Specifically, we prove that $1$ and $7$ are the only Woodall numbers in the Padovan sequence, and that $3$ is the only Cullen number in the Perrin sequence. |
| title | Cullen and Woodall numbers in Padovan and Perrin sequences |
| topic | Number Theory 11B39, 11D61, 11D45, 11J86 |
| url | https://arxiv.org/abs/2605.23084 |