Integers represented by Lucas sequences

TL;DR

This paper investigates the sets of integers that appear as the n-th terms of Lucas sequences, establishing bounds on their size and growth, with results that are sharp for large n and independent of sequence parameters.

Contribution

It introduces bounds on the size and growth of Lucas sequence terms that are independent of sequence parameters, a novel aspect in the study of these sequences.

Findings

Bounds on the size of sets of n-th Lucas sequence terms

Sharp bounds for large n

Growth order bounds independent of sequence parameters

Abstract

In this paper we study the sets of integers which are $n$-th terms of Lucas sequences. We establish lower- and upper bounds for the size of these sets. These bounds are sharp for $n$ sufficiently large. We also develop bounds on the growth order of the terms of Lucas sequences that are independent of the parameters of the sequence, which is a new feature.