Iteration Sums of The Euler Totient Function Regarding Powers of Fermat Primes
Xiang Li, Allison Pacelli
TL;DR
The paper investigates the iteration of Euler's Totient function $\phi$ and the associated sum of iterates $\sum_i \phi^i(n)$, with a focus on Fermat primes and their powers. It establishes that $\phi^k(n)$ converges to $2$ and then to $1$ for all $n$, and proves a clean closed-form for the sum when $n=3^k$, namely $\phi(3^k)+\phi(\phi(3^k))+\cdots+\phi(2)=3^k$, highlighting the special role of $3$ in the pattern. For Fermat primes $p$ (and their powers), the work derives and discusses closed-form expressions for the iterated totient sums, showing $\phi(p)=p-1$ leads to $\phi(p)+\phi(\phi(p))+...+\phi(2)=2p-3$, and extends to higher powers with more intricate, induction-based arguments. The results connect classical multiplicativity and CRT with the arithmetic structure of Fermat primes, offering new avenues for understanding iterated arithmetic functions and their sums, and indicating several open questions requiring rigorous proofs for full generality.
Abstract
Euler Totient function, a cornerstone of number theory, has attracted extensive study and applications across many disciplines. In this paper, we explore the patterns that the iterations of the Totient function exhibit. This paper first covers the foundational definitions and well-established theorems. Then, we build upon those results to investigate applying the Totient function multiple times, such as $φ(φ(φ(n)))$. Theorems regarding the end behavior of such iterations are presented. Next, we apply an innovative summation approach to the iterations of the Totient function, which is in the form of $φ(n)+φ(φ(n))+φ(φ(φ(n)))+\cdots$ that could also be expressed as $\sum φ^i(n)$. We prove novel theorems regarding this sum for all powers of Fermat Primes, and we derive an elegant result for powers of three. This paper initiates investigations into the sums of iterated Totient function values.