Arithmetic geometry
Millennium Prize Problem
Status: Open
Status last verified
Birch and Swinnerton-Dyer Conjecture
The number of independent rational points on an elliptic curve (its rank) equals the order of vanishing of its L-function at s = 1.
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Suggested experiments
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Count points modulo p
For a curve such as y² = x³ − x, count #E(F_p) for primes p up to a bound and compute a_p = p + 1 − #E(F_p). Compare the product Π N_p / p with curves of different rank — the original Birch–Swinnerton-Dyer experiment.
Establishes: The values you computed for the primes you used.
Does not establish: The conjecture for this or any other curve.