Classifying linear division sequences

TL;DR

This paper classifies all linear division sequences in integers and explores related properties of linear recurrence sequences, using advanced number theory tools to establish comprehensive results.

Contribution

It provides a complete classification of linear division sequences in integers and extends results to polynomial sequences, employing Ritt's theorem and the subspace theorem.

Findings

Classified all linear division sequences in integers.

Characterized linear recurrence sequences with specific gcd properties.

Established divisibility relations for sequences with large common factors.

Abstract

We classify all linear division sequences in the integers, a problem going back to at least the 1930s. As a corollary we also classify those linear recurrence sequences in the integers for which $(x_m,x_n)=\pm x_{(m,n)}$. We also show that if two linear division sequences have a large common factor infinitely often then they are each divisible by a common linear division sequence on some arithmetic progression. Moreover our proofs also work for polynomials. The key to our proofs are Ritt's irreducibility theorem and the subspace theorem (of Schmidt and Schlickewei), in a direction developed by Bugeaud, Corvaja and Zannier.