Arithmetic progressions consisting of unlike powers

In this paper we present some new results about unlike powers in arithmetic progression. We prove among other things that for given k ⩾ 4 and L ⩾ 3 there are only finitely many arithmetic progressions of the form (x0l0,x1l1,...,xk−1lk−1) with xi ∈ ℤ, gcd(x0, xl) = 1 and 2 ⩽ li ⩽ L for i = 0, 1, …, k − 1. Furthermore, we show that, for L = 3, the progression (1, 1,…, 1) is the only such progression up to sign. Our proofs involve some well-known theorems of Faltings [9], Darmon and Granville [6] as well as Chabauty's method applied to superelliptic curves.