On the l.c.m. of random terms of binary recurrence sequences.

For every positive integer n and every δ ∈ 0 , 1 , let B (n , δ) denote the probabilistic model in which a random set A ⊆ { 1 , ... , n } is constructed by choosing independently every element of { 1 , ... , n } with probability δ. Moreover, let (u k) k ≥ 0 be an integer sequence satisfying u k = a 1 u k − 1 + a 2 u k − 2 , for every integer k ≥ 2 , where u 0 = 0 , u 1 ≠ 0 , and a 1 , a 2 are fixed nonzero integers; and let α and β , with | α | ≥ | β | , be the two roots of the polynomial X 2 − a 1 X − a 2. Also, assume that α / β is not a root of unity. We prove that, as δ n / log ⁡ n → + ∞ , for every A in B (n , δ) we have log ⁡ lcm (u a : a ∈ A) ∼ δ Li 2 (1 − δ) 1 − δ ⋅ 3 log ⁡ | α / (a 1 2 , a 2) | π 2 ⋅ n 2 with probability 1 − o (1) , where lcm denotes the lowest common multiple, Li 2 is the dilogarithm, and the factor involving δ is meant to be equal to 1 when δ = 1. This extends previous results of Akiyama, Tropak, Matiyasevich, Guy, Kiss and Mátyás, who studied the deterministic case δ = 1 , and is motivated by an asymptotic formula for lcm (A) due to Cilleruelo, Rué, Šarka, and Zumalacárregui. [ABSTRACT FROM AUTHOR]