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

Xiang Li; Allison Pacelli

Home/Archive/arXiv

arXiv · 2508.05698

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

Xiang Li·Allison Pacelli

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.GM

Keep this discovery

BibTeX RIS

Xiang Li, Allison Pacelli. 2025-08-06. Iteration Sums of The Euler Totient Function Regarding Powers of Fermat Primes. https://arxiv.org/abs/2508.05698

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗

arXiv2026

The decrease in sectional curvature of the Fisher--Rao manifold of beta distributions

In the paper, the author develops a general convexity lemma and applies it to demonstrate the monotonic decrease of a function involving polygamma functions, arising from the sectional curvature of the Fisher--Rao manifold of beta distributions.

math.GM↗

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗