On the iterates of shifted Euler's function

Paolo Leonetti; Florian Luca

paper

On the iterates of shifted Euler's function

arXiv:2302.01783

Abstract

Let $φ$ be the Euler's function and fix an integer $k \geq 0$ . We show that, for every initial value $x_{1} \geq 1$ , the sequence of positive integers $(x_{n})_{n \geq 1}$ defined by $x_{n + 1} = φ (x_{n}) + k$ for all $n \geq 1$ is eventually periodic. Similarly, for every initial value $x_{1}, x_{2} \geq 1$ , the sequence of positive integers $(x_{n})_{n \geq 1}$ defined by $x_{n + 2} = φ (x_{n + 1}) + φ (x_{n}) + k$ for all $n \geq 1$ is eventually periodic, provided that $k$ is even.