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

TL;DR

This paper investigates whether elements in large finite fields can be expressed as sums of two dth powers, presenting two different proofs—one elementary and one classical—highlighting their respective advantages.

Contribution

It introduces a new elementary proof alongside a classical Fourier analysis approach for the problem of sums of powers in finite fields.

Findings

Both proofs establish the representability of elements as sums of two dth powers in large finite fields.

The elementary proof simplifies understanding and broadens accessibility.

The classical proof leverages Fourier analysis and finite field estimates for a rigorous approach.

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.