Multiplicative dependence in linear recurrence sequences

TL;DR

This paper establishes an upper bound on the number of s-tuples within certain ranges for which elements of a broad class of integer linear recurrence sequences are multiplicatively dependent.

Contribution

It provides a new upper bound on the frequency of multiplicative dependence among elements of linear recurrence sequences within specified intervals.

Findings

Derived an explicit upper bound for multiplicative dependence in recurrence sequences.

Applicable to a wide class of integer linear recurrence sequences.

Enhances understanding of multiplicative relations in recurrence sequences.

Abstract

For a wide class of integer linear recurrence sequences $\left(u(n)\right)_{n=1}^\infty$, we give an upper bound on the number of $s$-tuples $\left(n_1, \ldots, n_s\right) \in \left(\mathbb Z\cap [M+1,M+ N]\right)^s$ such that the corresponding elements $u(n_1), \ldots, u(n_s)$ in the sequence are multiplicatively dependent.