Classifying linear division sequences

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.