Elliptic Curve y^2 = x^3 +17; show N_p = p
Suppose that p is an odd prime and p≡2 (mod 3). Let E be the elliptic curve defined by . Prove that , the number of solutions mod p of the elliptic curve E, is exactly equal to p.
[hint: if p is a prime, then 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]