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arXiv · 2606.24908

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

Sagar Mandal
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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{\H{o}} version of the Baker-Davenport reduction method in diophantine approximation.

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Sagar Mandal. 2026-06-18. On common values of $F_n$ and Nathanson's totient function $\Phi(m)$. https://arxiv.org/abs/2606.24908

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