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 P k be the set of primes p such that gcd(p − 1, F p−1 ) = k. We prove that the relative density r(P k ) of P k exists, and we give a formula for r(P k ) in terms of an absolutely convergent series. Furthermore, we give an effective criterion to establish if a given k satisfies r(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.