Greatest common divisors of shifted primes and Fibonacci numbers

TL;DR

This paper investigates the distribution of primes related to Fibonacci numbers and shifted primes, establishing density formulas, criteria for positivity, and bounds, with applications to gcd-related integer sets.

Contribution

It provides a new, elementary proof for the lower bound on the count of integers of the form gcd(n, F_n), extending understanding of prime-Fibonacci gcd properties.

Findings

Existence of relative density for sets of primes with gcd conditions

Explicit formula for the density in terms of an absolutely convergent series

Elementary proof of lower bounds for gcd(n, F_n) counts

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.