The Distribution of G.C.D.s of Shifted Primes and Lucas Sequences

Abstract

Let (u n ) n 0 be a nondegenerate Lucas sequence and g u (n) be the arithmetic function defined by gcd(n, u n ). Recent studies have investigated the distributional characteristics of g u . Numerous results have been proven based on the two extreme values 1 and n of g u (n). Sanna investigated the average behaviour of g u and found asymptotic formulas for the moments of log g u . In a related direction, Jha and Sanna investigated properties of g u at shifted primes.In light of these results, we prove that for each positive integer λ , we havewhere P u,λ is a constant depending on u and λ which is expressible as an infinite series. Additionally, we provide estimates for P u,λ and M u,λ , where M u,λ is the constant for an analogous sum obtained by Sanna [J. Number Theory 191 (2018), 305-315]. As an application of our results, we prove upper bounds on the count #{p x : g u (p − 1) > y} and also establish the existence of infinitely many runs of m consecutive primes p in bounded intervals such that g u (p − 1) > y based on a breakthrough of Zhang, Maynard, Tao, et al. on small gaps between primes. Exploring further in this direction, it turns out that for Lucas sequences with nonunit discriminant, we have max{g u (n) : n x} ≫ x. As an analogue, we obtain that that max{g u (p − 1) : p x} ≫ x 0.4736 unconditionally, while max{g u (p − 1) : p x} ≫ x 1−o(1) under the hypothesis of Montgomery's or Chowla's conjecture.

On the greatest common divisor ofnand thenth Fibonacci number, II

“…Furthermore, Sanna [24] showed that the set of natural numbers n such that gcd(n, u n ) = 1 has a natural density (see [15] for a generalization). Mastrostefano and Sanna [14,23] studied the moments of log(gcd(n, u n )) and gcd(n, u n ) when (u n ) is a Lucas sequence, and Jha and Nath [9] performed a similar study over shifted primes. Part of the interest in studying gcd(n, u n ) resides in the fact that this task can be considered a simpler, albeit nontrivial, case of the general problem of studying the greatest common divisor (GCD) of terms of two linear recurrences, a problem that led to the famous Bugeaud-Corvaja-Zannier bound [5] and the difficult Ailon-Rudnick conjecture [1].…”