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LSpice
LSpice
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Yes, this is true, and is proved e.g. as Corollary 3 of Small's "Diagonal equations over large finite fields""Diagonal equations over large finite fields" (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\backslash\{0\}$$a,b\in\mathbb{F}_q\setminus\{0\}$ and $q>(\delta-1)^4$, where $\delta=(n,q-1)$.

Yes, this is true, and is proved e.g. as Corollary 3 of Small's "Diagonal equations over large finite fields" (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 $a,b\in\mathbb{F}_q\backslash\{0\}$ and $q>(\delta-1)^4$, where $\delta=(n,q-1)$.

Yes, this is true, and is proved e.g. as Corollary 3 of Small's "Diagonal equations over large finite fields" (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 $\delta=(n,q-1)$.

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Thomas Bloom
Thomas Bloom
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Yes, this is true, and is proved e.g. as Corollary 3 of Small's "Diagonal equations over large finite fields" (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 $a,b\in\mathbb{F}_q\backslash\{0\}$ and $q>(\delta-1)^4$, where $\delta=(n,q-1)$.