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.

Let $$(F_n)$$ ( F n ) be the sequence of Fibonacci numbers and, for each positive integer k, let $${\mathcal {P}}_k$$ P k be the set of primes p such that $$\gcd (p - 1, F_{p - 1}) = k$$ gcd ( p - 1 , F p - 1 ) = k . We prove that the relative density $$\varvec{r}({\mathcal {P}}_k)$$ r ( P k ) of $${\mathcal {P}}_k$$ P k exists, and we give a formula for $$\varvec{r}({\mathcal {P}}_k)$$ r ( P k ) in terms of an absolutely convergent series. Furthermore, we give an effective criterion to establish if a given k satisfies $$\varvec{r}({\mathcal {P}}_k) > 0$$ 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 $$\gcd (n, F_n)$$ 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.

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.

Let A be the set of all integers of the form gcd(𝑛, 𝐹 𝑛 ), where 𝑛 is a positive integer and 𝐹 𝑛 denotes the 𝑛th Fibonacci number. Leonetti and Sanna proved that A has natural density equal to zero, and asked for a more precise upper bound. We prove that𝑥 log log log 𝑥 log log 𝑥 for all sufficiently large 𝑥. In fact, we prove that a similar bound holds also when the sequence of Fibonacci numbers is replaced by a general nondegenerate Lucas sequence.

Cilleruelo conjectured that if [Formula: see text] is an irreducible polynomial of degree [Formula: see text] then, [Formula: see text] In this paper, we investigate the analog of prime arguments, namely, [Formula: see text] where [Formula: see text] denotes a prime and obtain nontrivial lower bounds on it. Further, we also show some results regarding the greatest prime divisor of [Formula: see text]