论文标题
在$ n $和$ n $ th fibonacci号码的最大共同除数上，ii
On the greatest common divisor of $n$ and the $n$th Fibonacci number, II
论文作者
论文摘要
令$ \ Mathcal {a} $为$ \ gcd（n，f_n）$的所有整数的集合，其中$ n $是一个正整数，$ f_n $表示$ n $ th fibonacci编号。 Leonetti和Sanna证明了$ \ Mathcal {a} $的自然密度等于零，并要求更精确的上限。我们证明\ begin {equation*} \＃\ big（\ mathcal {a} \ cap [1，x] \ big）\ ll \ frac {x \ log \ log \ log x} {\ log x} {\ log x} {\ log x}
Let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th 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{equation*} \#\big(\mathcal{A} \cap [1, x]\big) \ll \frac{x \log \log \log x}{\log \log x} \end{equation*} for all sufficiently large $x$.
