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__Theorem:__Suppose that p is an odd prime and p≡2 (mod 3). Let E be the elliptic curve defined by [tex]y^2 = x^3 + 17[/tex]. Then [tex]N_p[/tex], the number of solutions mod p of the elliptic curve E, is exactly equal to p.

[hint: if p is a prime, then [tex]1^k, 2^k, ..., (p-1)^k[/tex] form a reduced residue system (mod p) if and only if gcd(k, p-1)=1.]

Does anyone have any idea how to prove that [tex]N_p = p[/tex]?

Any help is greatly appreciated! :) (If possible, please explain in simpler terms. In particular, I have no background in abstract algebra.)