Suppose that p is an odd prime and p≡2 (mod 3). Let E be the elliptic curve defined by $\displaystyle y^2 = x^3 + 17$. Prove that $\displaystyle N_p$, the number of solutions mod p of the elliptic curve E, is exactly equal to p.

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

Does anyone have any idea how to prove this?

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

[also under discussion in math link forum]