Let $q$ be a prime power, let $n$ be a positive integer and let $\mathbb{F}\_q$ be the finite field of cardinality $q$. I have some computational evidence that the set $$\{x^n+(-1)^nay^n:x,y\in\mathbb{F}\_q\}$$ is the whole of $\mathbb{F}\_q$, unless $q$ is small (with respect to $n$).


This does make sense because if, for instance, $q-1=n$, then the set
$\{ x^n+(-1)^nay^n:x,y\in\mathbb{F}\_q \}$ only consists of $\{0,1,a,1+a\}$ because each element of $\mathbb{F}\_q$ raised to the $n$ equals either $0$ or $1$.


Can anyone justify the computational evidence?



Sum of two $n$th powers in finite fields
Yes, this is true, and is proved e.g. as Corollary 3 of Small's "[Diagonal equations over large finite fields](https://doi.org/10.4153/CJM-1984-016-6)" (Can. J. Math. 1984).


Small actually gives explicit bounds on how large $q$ needs to be in terms of $n$ — in particular the equation $ax^n+by^n$ generates all of $\mathbb{F}\_q$ whenever $a,b\in\mathbb{F}\_q\setminus\{0\}$ and $q>(\delta-1)^4$, where $\delta=(n,q-1)$.


