Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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M. Ferrari - L. Sillari

On the minimal number of solutions of the equation $ φ(n+k)= M \, φ(n) $, $ M=1$, $2$

created by sillari on 26 Oct 2023



Inserted: 26 oct 2023
Last Updated: 26 oct 2023

Year: 2021

ArXiv: 2110.05401 PDF


We fix a positive integer $k$ and look for solutions of the equations $\phi(n+k) = \phi(n)$ and $\phi(n + k) = 2\phi(n)$. 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. These results hold for $k \le 2 \cdot 10^{100}$. We also show that for the second equation with even $k$ there are at least three solutions for $k \le 4 \cdot 10^{58}$. Our work increases the previous minimal number of known solutions for both equations.

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