On common values of $F_n$ and Nathanson's totient function $\Phi (m)$

Abstract

In a recent paper, Chatterjee, the author and Mohan posed the problem of determining all solutions of the Diophantine equation $F_n=\Phi (m)$, where $F_n$ is the $n$-th Fibonacci number and $\Phi (m)$ counts the number of nonempty sets $A\subseteq \{1,2,\dots ,m\}$ for which $\gcd (A)$ is relatively prime to $m$. In this paper, we prove that the Diophantine equation has the only solutions $(n,m)=(1,1),(2,1),(3,2)$. The main tools used in this paper are lower bounds for linear forms in logarithms due to Matveev and Dujella-Peth{ő} version of the Baker-Davenport reduction method in diophantine approximation.

8 pages