Fixed points of K-Fibonacci sequences
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910545245896704 |
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| author | Benfield, Brennan Lippard, Oliver |
| author_facet | Benfield, Brennan Lippard, Oliver |
| contents | A $K$-Fibonacci sequence is a binary recurrence sequence where $F_0=0$, $F_1=1$, and $F_n=K\cdot F_{n-1}+F_{n-2}$. These sequences are known to be periodic modulo every positive integer greater than $1$. If the length of one shortest period of a $K$-Fibonacci sequence modulo a positive integer is equal to the modulus, then that positive integer is called a $\textit{fixed point}$. This paper determines the fixed points of $K$-Fibonacci sequences according to the factorization of $K^2+4$ and concludes that if this process is iterated, then every modulus greater than $3$ eventually terminates at a fixed point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08194 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fixed points of K-Fibonacci sequences Benfield, Brennan Lippard, Oliver Number Theory 11B39, 11B50 A $K$-Fibonacci sequence is a binary recurrence sequence where $F_0=0$, $F_1=1$, and $F_n=K\cdot F_{n-1}+F_{n-2}$. These sequences are known to be periodic modulo every positive integer greater than $1$. If the length of one shortest period of a $K$-Fibonacci sequence modulo a positive integer is equal to the modulus, then that positive integer is called a $\textit{fixed point}$. This paper determines the fixed points of $K$-Fibonacci sequences according to the factorization of $K^2+4$ and concludes that if this process is iterated, then every modulus greater than $3$ eventually terminates at a fixed point. |
| title | Fixed points of K-Fibonacci sequences |
| topic | Number Theory 11B39, 11B50 |
| url | https://arxiv.org/abs/2404.08194 |