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

TL;DR

This paper investigates the behavior of iterated Euler totient functions, proving that repeated application reduces any number greater than two to two, and explores sums of these iterates for powers of Fermat primes, deriving explicit formulas.

Contribution

It establishes the terminal value of iterated totients for all integers greater than two and derives closed-form sums for powers of Fermat primes, extending previous results.

Findings

Iterated totient function eventually reaches 2 for all n > 2.

Sum of iterated totients of n = 3^k equals n.

Closed-form expressions for sums of iterated totients of 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.