The G.C.D. of $n$ and the $n$th Fibonacci number

Abstract: Let $(F_n)_{n \geq 1}$ be the sequence of Fibonacci numbers, defined as usual by $F_1 = F_2 = 1$ and $F_{n + 2} = F_{n + 1} + F_n$ for all positive integers $n$; and let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$, for some positive integer $n$. In this talk we shall illustrate the following result on $\mathcal{A}$.

\noindent \textbf{Theorem.} \textit{For all $x \geq 2$, we have \begin{equation*} \#\mathcal{A}(x) \gg \frac{x}{\log x} . \end{equation*} On the other hand, $\mathcal{A}$ has zero asymptotic density.} The proofs rely on a result of Cubre and Rouse (PAMS, 2014) which gives, for each positive integer $n$, an explicit formula for the density of primes $p$ such that $n$ divides the rank of appearance of $p$, that is, the smallest positive integer $k$ such that $p$ divides $F_k$.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)