Arithmetic progressions consisting of unlike powers

N. Bruin, K. Gyory, L. Hajdu, Sz Tengely

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Abstract

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.

Original languageEnglish
Pages (from-to)539-555
Number of pages17
JournalIndagationes Mathematicae
Volume17
Issue number4
DOIs
Publication statusPublished - Dec 18 2006

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ASJC Scopus subject areas

  • Mathematics(all)

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