Three power summands over a large prime field ($p>k^6⟹{\mathbb{F}}_p=\{u^k+v^k+w^k:u,v,w\in {\mathbb{F}}_p\}$)

For $S(a)={\sum }_{u}e(a{u}^k/p)$, character orthogonality counts representations of $x$ by $p^{-1}{\sum }_{a}S(a{)}^{3}e(-ax/p)$. Its zero-frequency term is $p^2$. The power Gauss sum over a prime field bounds the other terms in total by $k^3{p}^{3/2}$, strictly less than $p^2$ when $p>k^6$. Every residue class therefore has a representation.