On the GCD-s of k consecutive terms of Lucas sequences

Abstract

Let u = (u n ) ∞ n=0 be a Lucas sequence, that is a binary linear recurrence sequence of integers with initial terms u 0 = 0 and u 1 = 1. We show that if k is large enough then one can find k consecutive terms of u such that none of them is relatively prime to all the others. We even give the exact values g u and G u for each u such that the above property first holds with k = g u ; and that it holds for all k G u , respectively. We prove similar results for Lehmer sequences as well, and also a generalization for linear recurrence divisibility sequences of arbitrarily large order. On our way to prove our main results, we provide a positive answer to a question of Beukers from 1980, concerning the sums of the multiplicities of 1 and −1 values in non-degenerate Lucas sequences. Our results yield an extension of a problem of Pillai from integers to recurrence sequences, as well.

On common factors within a series of consecutive terms of an elliptic divisibility sequence

“…Later, Ohtomo and Tamari [21] proved that for any coprime integers a, b the arithmetic progression an+b (n ≥ 1) is also a Pillai sequence. Recently, Hajdu and Szikszai [17] together with other related results proved that Lucas and Lehmer sequences of the first kind are Pillai sequences, as well. However, they also demonstrated that being a Pillai sequence is a special property, at least Lucas and Lehmer sequences of the second kind do not have this property in general.…”

“…The coincidence of these parameters seems to be a common behavior; see [16] and [17] for related situations. However, we are pretty sure that there are elliptic divisibility sequences B with g B < G B , similarly to the cases considered in [16] and [17].…”

“…As before, if G u (T ) exists then so does g u (T ), and we have g u (T ) ≤ G u (T ). A sequence u for which G u (T ) exists is called a T -Pillai sequence (see [17]). Obviously, for T = {1} we get back the notion of Pillai sequences.…”

“…Moreover, we explicitly give the values g B and G B for several particular elliptic divisibility sequences B. Our results rely on divisibility properties of elliptic divisibility sequences shown by Everest, Mclaren and Ward [9], a classical theorem of Baker [2] bounding the solutions of elliptic diophantine equations, a theorem of Ingram and Silverman [19] concerning the existence of primitive prime divisors of B, and results and algorithms of Hajdu and Saradha [16] and Hajdu and Szikszai [17] concerning the T -Pillai property.…”

“…However, if G u exists then so does g u , and we have g u ≤ G u . Sequences u for which G u exists are called Pillai sequences (see [17]) after Pillai, who was the first to investigate this property in N (i.e. in the increasing sequence of positive integers).…”