Iteration Sums of The Euler Totient Function Regarding Powers of Fermat Primes
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908742663012352 |
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| author | Li, Xiang Pacelli, Allison |
| author_facet | Li, Xiang Pacelli, Allison |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05698 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Iteration Sums of The Euler Totient Function Regarding Powers of Fermat Primes Li, Xiang Pacelli, Allison General Mathematics 11B37 (Primary) 33B99 (Secondary) 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. |
| title | Iteration Sums of The Euler Totient Function Regarding Powers of Fermat Primes |
| topic | General Mathematics 11B37 (Primary) 33B99 (Secondary) |
| url | https://arxiv.org/abs/2508.05698 |