On the l.c.m. of shifted Lucas numbers

TL;DR

This paper investigates the asymptotic growth of the least common multiple of shifted Lucas numbers, both for periodic and random shifts, extending previous Fibonacci number results to Lucas sequences.

Contribution

It provides the first asymptotic analysis of the LCM of shifted Lucas numbers, including both periodic and random shift sequences, generalizing earlier Fibonacci number findings.

Findings

Asymptotic behavior of log LCM for periodic shifts determined

Asymptotic analysis extended to random shift sequences

Results generalize previous Fibonacci number studies

Abstract

Let $(L_n)_{n \geq 1}$ be the sequence of Lucas numbers, defined recursively by $L_1 := 1$, $L_2 := 3$, and $L_{n + 2} := L_{n + 1} + L_n$, for every integer $n \geq 1$. We determine the asymptotic behavior of $\log \operatorname{lcm} (L_1 + s_1, L_2 + s_2, \dots, L_n + s_n)$ as $n \to +\infty$, for $(s_n)_{n \geq 1}$ a periodic sequence in $\{-1, +1\}$. We also carry out the same analysis for $(s_n)_{n \geq 1}$ a sequence of independent and uniformly distributed random variables in $\{-1, +1\}$. These results are Lucas numbers-analogs of previous results obtained by the author for the sequence of Fibonacci numbers.