Non-standard quaternary representations and the Fibonacci numbers
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912364806275072 |
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| author | Anders, Katie Dawsey, Madeline L. Gupta, Rajat Lebowitz-Lockard, Noah Vandehey, Joseph |
| author_facet | Anders, Katie Dawsey, Madeline L. Gupta, Rajat Lebowitz-Lockard, Noah Vandehey, Joseph |
| contents | Let $f_4(n)$ be the number of hyperquaternary representations of $n$ and $b_4(n)$ be the number of balanced quaternary representations of $n$. We show that there is no integer $k$ such that $f_4(n+k)=b_4(n)$ for all $n\ge -k$, in contrast to the binary case. Nevertheless, there do exist integers $k$ such that $f_4(n+k)=b_4(n)$ for arbitrarily large intervals of $n$. We generalize these results to any even base $d$. We also study the rate of growth of $b_4(n)$ and show that maximal values of this function correspond to certain Fibonacci numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04589 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-standard quaternary representations and the Fibonacci numbers Anders, Katie Dawsey, Madeline L. Gupta, Rajat Lebowitz-Lockard, Noah Vandehey, Joseph Number Theory 11B37, 11A63 Let $f_4(n)$ be the number of hyperquaternary representations of $n$ and $b_4(n)$ be the number of balanced quaternary representations of $n$. We show that there is no integer $k$ such that $f_4(n+k)=b_4(n)$ for all $n\ge -k$, in contrast to the binary case. Nevertheless, there do exist integers $k$ such that $f_4(n+k)=b_4(n)$ for arbitrarily large intervals of $n$. We generalize these results to any even base $d$. We also study the rate of growth of $b_4(n)$ and show that maximal values of this function correspond to certain Fibonacci numbers. |
| title | Non-standard quaternary representations and the Fibonacci numbers |
| topic | Number Theory 11B37, 11A63 |
| url | https://arxiv.org/abs/2505.04589 |