Math isn't ready to solve this problem

Manjul Bhargava and Arul Shankar proved in 2015 that the average rank of elliptic curves is at most 1.15, earning Bhargava the Fields Medal in 2014 for this and related work. This breakthrough bypassed decades of reliance on unproven conjectures.

Mathematician Poincaré discovered in the late 1800s that rational points on elliptic curves can be added using geometric constructions. This breakthrough transformed the study of rational points from hopeless enumeration to structured generation.

Number theorists study elliptic curves as the natural generalization of circle equations, replacing quadratic terms with cubic ones. These curves appear throughout modern mathematics, from cryptography to the proof of Fermat’s Last Theorem.

The rank conjecture predicts the distribution of ranks across all elliptic curves. It remains wide open as of 2025, representing a problem so difficult that mathematics lacks the tools to solve it despite decades of effort.

The rank measures the dimension of the space of rational points on an elliptic curve. It quantifies how many independent generators are needed to produce all rational points through the cord-tangent process.

Ancient Greeks and Babylonians discovered the connection between rational points on the unit circle x² + y² = 1 and Pythagorean triples. Number theorists use this classical problem as motivation for studying more complex Diophantine equations.