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

Explore connections, maps & timelines

BibTeXRIS

Xiang Li, Allison Pacelli. 2026-01-01. 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

A Combinatorial Problem in Cinema Seating

We address a problem concerning cinema audiences: ``A cinema has $n$ seats numbered from $1$ to $n$, and there are $n$ people with tickets numbered from $1$ to $n$. People enter the cinema in order. If someone has the ticket number $i$, they can choose seats whose numbers are multiples of $i$. They should exit the cinema if the permitted seats are occupied by previous audience members. In how many ways can they be seated under these conditions?" We give an algorithm to create the list of situations that meet these conditions. We also focus on finding the number of situations in two special cases: when exactly one seat is unoccupied whose total number is denoted by $ω(n)$, and when all audiences $1, \ldots, n-1$ are seated, whose total number is denoted by $ψ(n)$. Giving the recursive formula $ψ(n)=1+\sum_{d|n, d\neq n}ψ(d)$ with the initial value $ψ(1)=1$, we provide an explicit formula for $ψ(p^αq^β)$, where $p$ and $q$ are distinct prime numbers. Furthermore, we show that $ω(n)=-n+\sum_{i=1}^nψ(i)$.

math.GM↗
arXiv2026

Quelques remarques sur les vari{é}t{é}s, fonctions de Green et formule de Stokes

We give some remarks on some manifolds K3 surfaces, Complex projective spaces, real projective space and Torus and the classification of two dimensional Riemannian surfaces, Green functions and the Stokes formula. We also, talk about traces of Sobolev spaces, the distance function, the notion of degree and a duality theorem, the variational formulation and conformal map in dimension 2, the metric on the boundary of a Lipschitz domain and polar geodesic coordinates and the Gauss-Bonnet formula and the positive mass theorem in dimension $ \geq 3 $ and in the flat and non flat case. And the Ricci flow. And fields and their relation to the equations.And obstructions in astronomy. And on strings, superstrings and D-branes. And topological solutions in the negative case, critical, supercritical and superstrings and symmetry. And geometrization. And Decision problem, SAT problem and p=np problem.

math.GM↗
arXiv2026

Counting Truchet Tile Balls

A formula is established that counts the number of different balls that can be made by decorating the pentagons and hexagons of a classic football with Truchet-like patterns.

math.GM↗