论文标题
素数和斐波那契数的最大共同分数
Greatest common divisors of shifted primes and Fibonacci numbers
论文作者
论文摘要
令$（f_n）$为fibonacci数字的顺序，对于每个正整数$ k $，令$ \ nathcal {p} _k $是Primes $ p $的集合，使得$ \ gcd（p -1，f_1，f_ {p -1}）= k $。我们证明存在$ \ Mathcal {p} _k）的相对密度$ \ text {r}（\ Mathcal {p} _k）$，并且存在$ \ text {r}（\ nathcal {p} _k）$的公式，以绝对收敛的系列为单位。此外，我们给出一个有效的标准，以确定给定的$ k $是否满足$ \ text {r}（\ Mathcal {p} _K）> 0 $，并且我们为此类$ k $的集合功能提供了上下界限。作为结果的应用，我们为某些正整数$ n $ $ \ gcd（n，f_n）$的整数的计数函数提供了新的证明。我们的证明比Leonetti和Sanna给出的先前的证明更为基础，后者依赖于Cibre and Rouse的结果。
Let $(F_n)$ be the sequence of Fibonacci numbers and, for each positive integer $k$, let $\mathcal{P}_k$ be the set of primes $p$ such that $\gcd(p - 1, F_{p - 1}) = k$. We prove that the relative density $\text{r}(\mathcal{P}_k)$ of $\mathcal{P}_k$ exists, and we give a formula for $\text{r}(\mathcal{P}_k)$ in terms of an absolutely convergent series. Furthermore, we give an effective criterion to establish if a given $k$ satisfies $\text{r}(\mathcal{P}_k) > 0$, and we provide upper and lower bounds for the counting function of the set of such $k$'s. As an application of our results, we give a new proof of a lower bound for the counting function of the set of integers of the form $\gcd(n, F_n)$, for some positive integer $n$. Our proof is more elementary than the previous one given by Leonetti and Sanna, which relies on a result of Cubre and Rouse.
