AI solves 79-year-old math mystery of six-dimensional spheres

Can a six-dimensional sphere be described using imaginary numbers? For decades, experts puzzled over this question. Now an AI has apparently answered it

He’s done it again: mathematician Levent Alpöge, at Anthropic, has solved yet another open math problem using artificial intelligence. He had already tackled the Jacobian conjecture and found two new elliptic curves . Now add to these the 1947 Hopf problem. It asks whether a six-dimensional sphere can be completely described using complex numbers, which include the (seemingly impossible) square root of –1. According to Alpöge, the answer is yes, though the result remains unverified.

Some geometric objects, such as a regular sphere (a two-dimensional surface that lives in three-dimensional space), have a “complex structure.” This means that there’s a simple way to assign a different complex number to every point on the sphere. The advantage of this is that many more mathematical tools are available for complex numbers than the usual “real” numbers, which makes some calculations easier. The complex structure on the sphere has had far-reaching uses, such as connecting complicated, ugly equations that live on its surface to simple, polynomial equations.

In a 1947 paper, German mathematician Heinz Hopf posed the question: What about higher-dimensional spheres? He showed that, no, high-dimensional spheres don’t have a complex structure. But he couldn’t prove it for one case: the 6D sphere.

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Ever since, numerous experts have grappled with this scenario. They’ve proposed evidence both for and against the hypothesis that a complex structure exists for the 6D surface of a sphere. But no one has managed to prove their case.

Now AI may have solved the problem. Alpöge used an internal version of Anthropic’s large language model, Claude, to do what humans could not. On August 23 Alpöge posted a 100-plus-page document with a purported proof that the 6D sphere does, in fact, have a complex structure. For the proof, the AI apparently generated a complex 3D structure and demonstrated that it exactly represents the 6D surface of a sphere.

Unfortunately, the full document was nigh-indecipherable—LLMs tend to explain their proofs in unhelpful ways, spending inordinate time on minor details while glancing past important steps.