Orbits of Second Order Linear Recurrences over Finite Fields
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arXiv
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| Format: | Preprint |
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2024
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| author | Panraksa, Chatchawan Somasunderam, Naveen |
| author_facet | Panraksa, Chatchawan Somasunderam, Naveen |
| contents | Let $Q$ be the matrix $\displaystyle \begin{pmatrix} a & b \\ 1 & 0 \end{pmatrix}$ in $GL_2(\mathbb{F}_q)$ where $\mathbb{F}_q$ is a finite field, and let $G$ be the finite cyclic group generated by $Q$. We consider the action of $G$ on the set $\mathbb{F}_q \times \mathbb{F}_q$. In particular, we study certain relationships between the lengths of the non-trivial orbits of $G$, and their frequency of occurrence. This is done in part by investigating the order of elements of a product in an abelian group when the product has prime power order. For $q$ a prime and $b=1$, the orbits correspond to Fibonacci type linear recurrences modulo $q$ for different initial conditions. We also derive certain conditions under which the roots of the characteristic polynomial of $Q$ are generators of $\mathbb{F}_q^\times$. Examples are included to illustrate the theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_09561 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orbits of Second Order Linear Recurrences over Finite Fields Panraksa, Chatchawan Somasunderam, Naveen Number Theory 11B37, 11B39, 11B50, 11T06, 11T30, 20K01, 37P25 Let $Q$ be the matrix $\displaystyle \begin{pmatrix} a & b \\ 1 & 0 \end{pmatrix}$ in $GL_2(\mathbb{F}_q)$ where $\mathbb{F}_q$ is a finite field, and let $G$ be the finite cyclic group generated by $Q$. We consider the action of $G$ on the set $\mathbb{F}_q \times \mathbb{F}_q$. In particular, we study certain relationships between the lengths of the non-trivial orbits of $G$, and their frequency of occurrence. This is done in part by investigating the order of elements of a product in an abelian group when the product has prime power order. For $q$ a prime and $b=1$, the orbits correspond to Fibonacci type linear recurrences modulo $q$ for different initial conditions. We also derive certain conditions under which the roots of the characteristic polynomial of $Q$ are generators of $\mathbb{F}_q^\times$. Examples are included to illustrate the theory. |
| title | Orbits of Second Order Linear Recurrences over Finite Fields |
| topic | Number Theory 11B37, 11B39, 11B50, 11T06, 11T30, 20K01, 37P25 |
| url | https://arxiv.org/abs/2408.09561 |