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

Abstract

Euler totient function $\phi(n)$ 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 $\phi$ eventually yields the value $2$. Motivated by this terminal behavior, we examine sums of iterated totient values of the form $\phi(n)+\phi(\phi(n))+\phi(\phi(\phi(n)))+\cdots+\phi(2)$, where the summation terminates at $\phi(2)$. 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.