<s>[INST] Solutions of $x^2 -xy + y^2 \equiv 0 \pmod{43}$ 
$x^2 -xy + y^2 \equiv 0 \pmod{43}$

We could transform this into

$\frac{x^3 + y^3}{x+y} \equiv 0 \pmod{43}$

and thus

$x^3 + y^3 \equiv 0 \pmod{43}$

From what I could manage so far, I know that if $x = 3k$ , $y= 3k+1$, as $43 = 3k+1$.
Also there are no solutions for $43$ itself, $84$ and $127$.
How should I proceed?
 [/INST] We have $x \equiv -y$ iff $x \equiv 0$ iff $y \equiv 0$.
Otherwise, as you have argued, $x^2 - xy + y^2 \equiv 0$ iff $x^3 + y^3 \equiv 0$ and this happens iff $(xy^{-1})^3 \equiv -1$.
So the solutions are $x=yu$, where $u$ is an element of order $6$ mod $43$, that is, $u=7$ or $u=37$.
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