On the greatest common divisor of n and the nth Fibonacci number, II

Abstract

AbstractLet $\mathcal {A}$ be the set of all integers of the form $\gcd (n, F_n)$ , where n is a positive integer and $F_n$ denotes the nth Fibonacci number. Leonetti and Sanna proved that $\mathcal {A}$ has natural density equal to zero, and asked for a more precise upper bound. We prove that $$ \begin{align*} \#\big(\mathcal{A} \cap [1, x]\big) \ll \frac{x \log \log \log x}{\log \log x} \end{align*} $$ for all sufficiently large x. In fact, we prove that a similar bound also holds when the sequence of Fibonacci numbers is replaced by a general nondegenerate Lucas sequence.