We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Let 
$\varphi $ be 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 all initial values 
$x_1,x_2\ge 1$, 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.
