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

Technical Summary: Common Values of $F_n$ and Nathanson's Totient Function $\Phi(m)$

Problem Statement
This paper addresses a specific Diophantine equation posed recently by Chatterjee, the author, and Mohan. The objective is to determine all integer solutions $(n, m)$ to the equation:
$$F_n = \Phi(m)$$
where $F_n$ denotes the $n$-th Fibonacci number (defined by $F_0=0, F_1=1, F_{n+2}=F_{n+1}+F_n$) and $\Phi(m)$ is Nathanson's totient function. The function $\Phi(m)$ counts the number of nonempty subsets $A \subseteq \{1, 2, \dots, m\}$ such that $\gcd(A)$ is relatively prime to $m$. While previous literature has extensively studied Diophantine equations involving Fibonacci numbers and powers, or sums of Fibonacci numbers, this work focuses specifically on the intersection of Fibonacci numbers with the arithmetic properties of $\Phi(m)$.

Methodology
The proof of the main theorem relies on a combination of analytic number theory and computational verification, utilizing the following tools:

1. Lower Bounds for Linear Forms in Logarithms: The primary theoretical tool is Matveev's theorem, which provides explicit lower bounds for linear forms in the logarithms of algebraic numbers. This is used to establish an initial upper bound for the variables in the equation.
2. Reduction Methods: To reduce the initially large upper bound to a computationally manageable range, the author employs the Dujella–Pethő version of the Baker–Davenport reduction method. This involves analyzing continued fraction expansions and using Lemma 2.2 to tighten the bounds on the variables.
3. Properties of $\Phi(m)$ and $F_n$: The proof utilizes specific properties of the functions involved, including:
   • The divisibility property that $6 \mid \Phi(n)$ for $n \ge 3$ (Lemma 2.4), which implies $6 \mid F_n$ and consequently $12 \mid n$.
   • Lower bounds for $\Phi(m)$ established in Theorem 2.3.
   • An auxiliary lemma (Lemma 3.1) bounding the sum of Möbius-weighted powers of 2 for large $m$.
4. Computational Verification: Once the theoretical bounds are reduced to a finite range, the author performs a brute-force search using Python code to verify the absence of solutions within the remaining intervals.

Key Contributions and Results
The paper provides a complete solution to the Diophantine equation $F_n = \Phi(m)$. The main result is stated in Theorem 1.2:

Theorem 1.2: The Diophantine equation $F_n = \Phi(m)$ has only the following solutions:
$$(n, m) = (1, 1), (2, 1), (3, 2)$$

The proof proceeds by contradiction, assuming $m \ge 79$. By establishing that $n$ must be a multiple of 12, the author derives an inequality involving linear forms in logarithms. Through iterative application of the reduction method, the upper bound for the parameter $k$ (where $n=12k$) is reduced from approximately $1.91 \times 10^{14}$ to $k \le 163$. A final computational search confirms that no solutions exist for $79 \le m < 12k$. The remaining cases where $1 \le m \le 78$ are verified directly, yielding the three solutions listed above.

Significance
The paper claims significance by completely resolving the specific problem of finding common values between the Fibonacci sequence and Nathanson's totient function. It extends the body of work on exponential Diophantine equations involving Fibonacci numbers, which has previously focused on sums, differences, and powers of Fibonacci numbers. The work demonstrates the efficacy of combining Matveev's bounds with the Dujella–Pethő reduction method in solving equations involving non-standard arithmetic functions like $\Phi(m)$. The author notes that the manuscript contains no associated data and declares no competing interests.