Elliptic Curves and $\mod P$

Question

I'm trying to figure out why the number of points ($Np$) equals any Prime ($P$) when:

$P \equiv 2 \pmod 3$

To the Elliptic Curve $y^2=x^3+17$

Does anyone know why this is?

Answer 1

If $p \neq 1 \pmod 3$ then $x \mapsto x^3$ is a permutation of $\Bbb F_p$, and so for each $y \in \Bbb F_p$ there is a unique $x \in \Bbb F_p$ such that $x^3 = y^2-17$