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One great problem, three kinds of immersion

From a story to your own conjecture

Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.

  1. 1 · DiscoverYou are hereMeet the questionLearn when it appeared and why it still holds people’s attention.
  2. 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
  3. 3 · DevelopBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
Partially solved1965

BSD Conjecture — How Integer Solutions on Elliptic Curves Distribute

"Are rational points on an elliptic curve finite or infinite? The L-function knows." Conjectured 1965, open for 60 years.

Challenge passport
First posed
1965
Time it held mathematicians
61 years open · as of 2026
Starting level
Research frontier

Small experiments are possible, but a complete proof sits at the frontier of modern mathematics.

The starting level measures how easily you can understand and test small cases. It is not the difficulty of a complete proof.

Jump to your first five minutes

Clay Mathematics Institute — $1 million (Millennium Problem)

See the puzzle visually

y² = x³ − x (rank 0, 4 points)y² = x³ − 25x (rank 1, ∞)

The BSD conjecture says the abundance of rational solutions (rank) equals the order at which the L-function vanishes at s=1. Purple is rank 0 with finitely many rational points; cyan is rank 1 with an infinite family generated by (−4, ±6).

[y² = x³ - x (rank 0) · y² = x³ - 25x (rank 1)]

Problem statement

For an elliptic curve E: y² = x³ + ax + b, the rank of the group of rational points E(ℚ) equals the order of vanishing of L(E, s) at s = 1.

The story of this puzzle

In the early 1960s at Cambridge, Birch and Swinnerton-Dyer used EDSAC 2 to test many elliptic curves, comparing point counts to L-function behavior. A striking pattern emerged: the L-function appears to encode the count exactly. Important for cryptography (ECC, Bitcoin). Coates-Wiles, Gross-Zagier, and Kolyvagin proved BSD for rank 0 and 1; rank ≥ 2 remains open. Clay $1M. Bridges geometry and analysis through the L-function — a deep unification.

Try it yourself

Mini challenge

Find integer points: y² = x³ - x has only (-1,0), (0,0), (1,0) — finite. y² = x³ - 25x has (-5,0), (-4,6), (0,0), (5,0), (10,±30), … infinite. Why finite vs infinite? That's BSD.

Beyond MathVoyage

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