Greatest common divisors of shifted primes and Fibonacci numbers

Abstract

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.