Narayana's cows numbers which are concatenations of three repdigits in base $ρ$
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911925327101952 |
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| author | Tiebekabe, Pagdame Adédji, Kouèssi Norbert Pindra, Nadjime Hounkonnou, Mahouton Norbert |
| author_facet | Tiebekabe, Pagdame Adédji, Kouèssi Norbert Pindra, Nadjime Hounkonnou, Mahouton Norbert |
| contents | Narayana's sequence is a ternary recurrent sequence defined by the recurrence relation $\mathcal{N}_n=\mathcal{N}_{n-1}+\mathcal{N}_{n-3}$ with initial terms $\mathcal{N}_0=0$ and $\mathcal{N}_1=\mathcal{N}_2=\mathcal{N}_3=1$. Let $ρ\geqslant2$ be a positive integer. In this study, it is proved that the $n$th Narayana's number $ \mathcal{N}_n$ which is concatenations of three repdigits in base $ρ$ satisfies $n<5.6\cdot 10^{48}\cdot \log^7ρ$. Moreover, it is shown that the largest Narayana's number which is concatenations of three repdigits in base $ρ$ with $1 \leqslant ρ\leqslant 10$ is $58425=\mathcal{N}_{31}=\overline{3332200}_5=\overline{332223}_7.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_00773 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Narayana's cows numbers which are concatenations of three repdigits in base $ρ$ Tiebekabe, Pagdame Adédji, Kouèssi Norbert Pindra, Nadjime Hounkonnou, Mahouton Norbert Number Theory 11B39, 11J86, 11D61 Narayana's sequence is a ternary recurrent sequence defined by the recurrence relation $\mathcal{N}_n=\mathcal{N}_{n-1}+\mathcal{N}_{n-3}$ with initial terms $\mathcal{N}_0=0$ and $\mathcal{N}_1=\mathcal{N}_2=\mathcal{N}_3=1$. Let $ρ\geqslant2$ be a positive integer. In this study, it is proved that the $n$th Narayana's number $ \mathcal{N}_n$ which is concatenations of three repdigits in base $ρ$ satisfies $n<5.6\cdot 10^{48}\cdot \log^7ρ$. Moreover, it is shown that the largest Narayana's number which is concatenations of three repdigits in base $ρ$ with $1 \leqslant ρ\leqslant 10$ is $58425=\mathcal{N}_{31}=\overline{3332200}_5=\overline{332223}_7.$ |
| title | Narayana's cows numbers which are concatenations of three repdigits in base $ρ$ |
| topic | Number Theory 11B39, 11J86, 11D61 |
| url | https://arxiv.org/abs/2304.00773 |