Sums of Powers in Large Finite Fields: A Mix of Methods
Number Theory
2020-12-17 v2
Abstract
Can any element in a sufficiently large finite field be represented as a sum of two 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.
Keywords
Cite
@article{arxiv.2009.09518,
title = {Sums of Powers in Large Finite Fields: A Mix of Methods},
author = {Vitaly Bergelson and Andrew Best and Alex Iosevich},
journal= {arXiv preprint arXiv:2009.09518},
year = {2020}
}
Comments
15 pages, to appear in The American Mathematical Monthly. Change from previous version: fixed a misspelled name in a footnote