# On arithmetic progressions in Lucas sequences

## TL;DR

This paper investigates the occurrence of arithmetic progressions within Lucas sequences, establishing finiteness results for most sequences and explicitly characterizing those with infinitely many such progressions.

## Contribution

It provides the first effective bounds on the number of arithmetic progressions in Lucas sequences and explicitly identifies sequences with infinitely many such progressions.

## Key findings

- Most Lucas sequences contain finitely many arithmetic progressions.
- Sequences with dominant root can be explicitly classified.
- Effective bounds are established for the number of progressions.

## Abstract

In this paper, we consider arithmetic progressions contained in Lucas sequences of first and second kind. We prove that for almost all sequences, there are only finitely many and their number can be effectively bounded. We also show that there are only a few sequences which contain infinitely many and one can explicitly list both the sequences and the progressions in them. A more precise statement is given for sequences with dominant root.