On the iterates of shifted Euler's function

TL;DR

This paper proves that sequences generated by iterating the Euler's totient function with a fixed shift are eventually periodic, extending to second-order recursions under certain conditions.

Contribution

It establishes the eventual periodicity of shifted Euler's function iterates for both first and second-order recursions, a novel result in the dynamics of arithmetic functions.

Findings

Sequences with $x_{n+1}=(x_n)+k$ are eventually periodic for all initial values.

Second-order sequences with $x_{n+2}=(x_{n+1})+(x_n)+k$ are eventually periodic if $k$ is even.

The results hold for all initial positive integers and fixed shifts.

Abstract

Let $\varphi$ be the Euler's function and fix an integer $k\ge 0$. We show that, for every initial value $x_1\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+1}=\varphi(x_n)+k$ for all $n\ge 1$ is eventually periodic. Similarly, for every initial value $x_1,x_2\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+2}=\varphi(x_{n+1})+\varphi(x_n)+k$ for all $n\ge 1$ is eventually periodic, provided that $k$ is even.