Uma GENERALIZAÇÃO DO PEQUENO TEOREMA DE FERMAT VIA SISTEMAS DINÂMICOS E A SOLUÇÃO DE UM PROBLEMA DE L. LEVINE

Abstract

Given an integer $k\geq 1$, Levine \cite{Levine} considers the dynamical system defined by the function $f(z)=z^k$ on the unit circle $\mathbb{S}^1$ and proves that $\sum_{m|n}\mu(n/m)\mathcal{N}_m$ is divisible by $n$, thus generalizing Fermat's little theorem. The notation $\mathcal{N}_m$ indicates the number of fixed points of $f^m$ in $\mathbb{S}^1$ and $\mu$ is the Möbius function. At the same time, the author leaves an open question: given a sequence of non-negative integers $(p_m)_m$, is there any function $f$ that performs this sequence, that is, $p_m=\mathcal{N}_m$ and does it satisfy the divisibility criterion? In this article we revisit Euler's well-known theorem using Chebyshev's polynomials, following Carrillo and Guzmán \cite{Carrillo} and Frame \emph{et al} \cite{Frame}, and answer Levine's question in the negative with an argument based on Sharkovsky's theorem .