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arXiv · 2009.09518

Sums of Powers in Large Finite Fields: A Mix of Methods

Vitaly Bergelson·Andrew Best·Alex Iosevich
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Abstract

Can any element in a sufficiently large finite field be represented as a sum of two $d$th powers in the field? In this article, we recount some of the history of this problem, touching on cyclotomy, Fermat's last theorem, and diagonal equations. Then, we offer two proofs, one new and elementary, and the other more classical, based on Fourier analysis and an application of a nontrivial estimate from the theory of finite fields. In context and juxtaposition, each will have its merits.

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Vitaly Bergelson, Andrew Best, Alex Iosevich. 2020-12-16. Sums of Powers in Large Finite Fields: A Mix of Methods. https://arxiv.org/abs/2009.09518

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