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On the Minimal Number of Solutions of the Equation φ(n+k)=Mφ(n), M=1,2

TitleOn the Minimal Number of Solutions of the Equation φ(n+k)=Mφ(n), M=1,2
Publication TypeJournal Article
Year of Publication2023
AuthorsFerrari, M, Sillari, L
JournalJournal of Integer Sequences
Volume26
Date Published01/2023
Type of ArticleArticle
ISSN1530-7638
Other NumbersArtcile number: 23.1.6
KeywordsEuler’s phi function
Abstract

We fix a positive integer $k$ and look for solutions $n \in \mathbb{N}$ of the equations $\phi(n + k) = \phi(n)$ and $φ(n + k) = 2 φ(n)$. For $k \le 12 \cdot 10^{100}$, we prove that Fermat primes can be used to build five solutions for the first equation when $k$ is even, and five for the second one when $k$ is odd. Furthermore, for $k \le 4 \cdot 10^{58}$, we show that for the second equation there are at least three solutions when $k$ is even. Our work increases the previously known minimal number of solutions for both equations.

URLhttps://cs.uwaterloo.ca/journals/JIS/VOL26/Sillari/sillari3.html

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