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 , Darmon and Granville  as well as Chabauty's method applied to superelliptic curves.
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