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Consider the elliptic curve defined by $\ y^2 = x^3 + 1\ $ over $\ \mathbb{Z}_p,\ $ where $\ p \equiv 2 \pmod{3}\ $ is prime. Prove that the number of points on the curve is exactly $\ p + 1.\ $

Hint: for $\ y \in \mathbb{Z}_p,\ $ prove that there is exactly one $\ x \in \mathbb{Z}_p\ $ satisfying the equation.

Prove without showing that $\ x \mapsto x^3\ $ is a bijection

Somos
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Apple
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  • trying to prove without showing that x maps to x^3 is a bijection – Apple Apr 23 '19 at 18:10
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    You must use the fact that $x\mapsto x^3$ is a bijection in that of another form, this is the essence of the problem. – W-t-P Apr 23 '19 at 18:26
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    Why not use the bijection? Will some evil being open his eyes, catch you and kill you? (From a video game where he did that if you did not time your jumps properly.) – Oscar Lanzi Apr 23 '19 at 19:25

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