Iteration Sums of The Euler Totient Function Regarding Powers of Fermat Primes

Abstract

Euler totient function $\varphi (n)$ plays a central role in number theory and is applied in areas such as cryptography. In this paper, we study iterations of the totient function. We first prove that for any integer $n>2$, iteratively applying $\varphi$ eventually yields the value $2$. Motivated by this terminal behavior, we examine sums of iterated totient values of the form $\varphi (n)+\varphi (\varphi (n))+\varphi (\varphi (\varphi (n)))+\cdots +\varphi (2)$, where the summation terminates at $\varphi (2)$. We show that for all integers of the form $n=3^k$, this sum is equal to $n$. We then extend this result to all powers of Fermat primes, deriving a closed-form expression for the corresponding summations.

23 pages, 0 figures