Limiting density of the Fibonacci sequence modulo powers of a prime
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866915457858011136 |
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| author | Bragman, Nicholas Rowland, Eric |
| author_facet | Bragman, Nicholas Rowland, Eric |
| contents | For a given prime $p$, we determine the limit, as $λ\to \infty$, of the density of residues modulo $p^λ$ attained by the Fibonacci sequence. In particular, we show that this limiting density is related to zeros in the sequence of Lucas numbers modulo $p$. The proof uses a piecewise interpolation of the Fibonacci sequence to the $p$-adic numbers and a characterization of Wall-Sun-Sun primes $p$ in terms of the $p$-adic absolute value of a number related to the $p$-adic golden ratio. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_00704 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Limiting density of the Fibonacci sequence modulo powers of a prime Bragman, Nicholas Rowland, Eric Number Theory For a given prime $p$, we determine the limit, as $λ\to \infty$, of the density of residues modulo $p^λ$ attained by the Fibonacci sequence. In particular, we show that this limiting density is related to zeros in the sequence of Lucas numbers modulo $p$. The proof uses a piecewise interpolation of the Fibonacci sequence to the $p$-adic numbers and a characterization of Wall-Sun-Sun primes $p$ in terms of the $p$-adic absolute value of a number related to the $p$-adic golden ratio. |
| title | Limiting density of the Fibonacci sequence modulo powers of a prime |
| topic | Number Theory |
| url | https://arxiv.org/abs/2202.00704 |