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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 $k\geq 4$ and $L\geq 3$ there are only finitely many arithmetic progressions of the form $(x_0^{l_0},x_1^{l_1},...,x_{k-1}^{l_{k-1}})$ with $x_i\in{\Bbb Z},$ gcd$(x_0,x_1)=1$ and $2\leq l_i\leq L$ 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.

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N. Bruin, K. Gyory, L. Hajdu, Sz. Tengely. 2005-12-17. Arithmetic progressions consisting of unlike powers. https://arxiv.org/abs/math/0512419

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