Uniform Bounds for Digit-Appending Fibonacci Walks
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917131106385920 |
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| author | Kominers, Scott Duke |
| author_facet | Kominers, Scott Duke |
| contents | Building on the work of Miller et al. [Fibonacci Quarterly, 2022], we show that it is impossible to "walk to infinity" along the Fibonacci sequence in any integer base $b\geq 2$ when at most $N$ digits are appended per step. Our proof method is base-independent, yielding the bound \[L \;\leq\; 2N\log_φb \,+\, O(1),\] uniformly in the starting term, without relying on base-specific periodicity computations (here, $φ=\frac{1+\sqrt{5}}{2}$). Our approach extends to certain Lucas sequences. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_06446 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform Bounds for Digit-Appending Fibonacci Walks Kominers, Scott Duke Number Theory 11B39 (Primary) 11B37, 11A07 (Secondary) Building on the work of Miller et al. [Fibonacci Quarterly, 2022], we show that it is impossible to "walk to infinity" along the Fibonacci sequence in any integer base $b\geq 2$ when at most $N$ digits are appended per step. Our proof method is base-independent, yielding the bound \[L \;\leq\; 2N\log_φb \,+\, O(1),\] uniformly in the starting term, without relying on base-specific periodicity computations (here, $φ=\frac{1+\sqrt{5}}{2}$). Our approach extends to certain Lucas sequences. |
| title | Uniform Bounds for Digit-Appending Fibonacci Walks |
| topic | Number Theory 11B39 (Primary) 11B37, 11A07 (Secondary) |
| url | https://arxiv.org/abs/2512.06446 |