Arithmetic progressions consisting of unlike powers |
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| Authors: | N Bruin K Gy&#x;ry L Hajdu Sz Tengely |
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| Institution: | aDepartment of Mathematlcs, Simon Fraser University, Burnaby, BC Canada V5A 1S6;bNumber Theory Research Group of the Hungarian Academy of Sdences, and University of Debrecen, Institute of Mathematics, RO. Box 12, 4010 Debrecen, Hungary |
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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  with xi   , 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. |
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