  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.
