AI Solves 140-Year-Old Math Problem in a Single Tweet

Sometimes the biggest discoveries come in the smallest packages.

Levent Alpöge, a mathematician at AI company Anthropic, just solved a problem that has puzzled mathematicians for 140 years. Using the company's AI model Claude Fable 5, he found a counterexample to the Jacobian conjecture, a famous puzzle in algebraic geometry first proposed in 1884.

The kicker? His solution was so short it fit into a single post on X (formerly Twitter).

The Jacobian conjecture deals with mathematical functions that move points around in space. Think of it like rearranging furniture in a room. The conjecture predicted that if a function moves things around "nicely" without crushing or folding space, there should always be another function that can put everything back where it started.

For over a century, brilliant mathematicians tried to prove this idea true. Famous mathematicians even published proofs, only to have subtle errors discovered later. The problem was so important that it made Stephen Smale's list of Mathematical Problems for the Next Century in 1998.

Researchers had verified the conjecture worked for two-dimensional cases with extremely complex polynomials (up to the 100th power). But nobody could prove it worked in all situations or find an example where it failed.

Alpöge's AI-generated counterexample showed that in three dimensions, the conjecture is actually false. The function he found has all the "nice" properties but still can't be reversed because it merges multiple starting points into single ending points.

Why This Inspires

What makes this discovery particularly exciting is its simplicity. Unlike other recent AI breakthroughs in mathematics that involved combining complex ideas from different fields, this solution was straightforward enough for other mathematicians to verify immediately.

The challenge wasn't constructing something complicated. It was searching through an enormous universe of possible mathematical functions to find the right one. This is exactly where AI shines: exploring vast spaces of possibilities that would take humans lifetimes to check manually.

This breakthrough joins other recent AI-assisted mathematical discoveries, including OpenAI's disproof of the unit distance conjecture and solutions found by amateur mathematicians working with AI tools. Each success reveals new ways that artificial intelligence can partner with human creativity.

The two-dimensional version of the conjecture remains unsolved, giving mathematicians another puzzle to tackle. But Alpöge's discovery proves that sometimes, as one mathematician predicted in 2017, a smart person with the right tools really can write a simple formula that changes everything.

The tools are just getting smarter at helping us find those formulas.