Three power summands over a large prime field
= Three power summands over a large prime field
{title2=$p>k^6\Longrightarrow\mathbb F_p=\{u^k+v^k+w^k:u,v,w\in\mathbb F_p\}$}
For $S(a)=\sum_u e(au^k/p)$, <character orthogonality> counts representations of $x$ by $p^{-1}\sum_a S(a)^3e(-ax/p)$. Its zero-frequency term is $p^2$. The <power Gauss sum over a prime field> bounds the other terms in total by $k^3p^{3/2}$, strictly less than $p^2$ when $p>k^6$. Every <residue class> therefore has a representation.