OpenAI's AI Just Cracked 10 Math Problems That Sat Unsolved for Over a Decade!

Hi, it's Shiichan! I found some seriously advanced news today: OpenAI just shared new results on a batch of long-standing open problems in math and theoretical computer science. It sounds intimidating, but let's dig in together!

OpenAI News openai.com

What was announced?

OpenAI's News shared new results on ten open problems in mathematics and theoretical computer science that had seen no progress on the main result for at least a decade, and in most cases much longer. The problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics. Each one matters a lot to its own mathematical community, and several are of broad interest across math as a whole.

This comes alongside OpenAI's ChatGPT for Academic Researchers initiative, which gives 100,000 scientists and mathematicians free access to OpenAI's best ChatGPT models, and reflects OpenAI's ongoing practice of evaluating its models on open research problems during development.

Why it matters

Back in May, OpenAI shared an AI-generated disproof of the Erdős unit-distance conjecture, discovered while evaluating an unreleased model. That result already inspired follow-up research, including a disproof of the sum-product conjecture for real numbers (Bloom, Sawin, Schildkraut, and Zhelezov), a paper on split primes and the Elekes-Rónyai problem (Pohoata), and other papers on communication complexity, all posted to arXiv.

This new batch of ten results builds on that momentum, offering a much broader body of evidence for how far AI can go in contributing to real mathematical research.

What changes

Here's what the ten results actually cover:

• High-dimensional sphere packing: New upper bounds on sphere-packing density, down to the Cohn–Elkies threshold
• Binary and spherical codes: Exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with similar results for high-dimensional spherical codes
• Non-sofic groups: A construction proving non-sofic groups exist, settling a central open question in group theory
• Connes's rigidity conjecture: A disproof of the long-standing conjecture that certain groups are uniquely determined by their von Neumann algebras
• Arithmetic circuit complexity: New lower bounds for computing the permanent with arithmetic circuits and formulas, including an arithmetic-formula lower bound on the order of n^4/log n
• Quantum parallel repetition: An exponential parallel repetition theorem for general two-player quantum games, extending a foundational principle from classical complexity theory
• Closest vector problem: Polynomial-factor hardness of approximation for this foundational lattice problem, closely tied to post-quantum cryptography
• Ehrhart's volume conjecture: The maximum possible volume, in every dimension, of a convex body whose centroid is its only interior lattice point
• Multicolor Ramsey numbers: A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183
• Extremal number conjectures: Results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180

These are all problems that experts had been stuck on for years, so this is a big deal within each of those research communities.

Dive Deep

These results came from an internal version of Astra, OpenAI's next major model. OpenAI says the total token cost to find solutions to all ten problems would come to roughly $2,000 at Sol API rates.

After the model produced the mathematical arguments, humans worked together with the same model to turn them into manuscripts. The model then formalized each proof as a verified Lean certificate, published in the openai/ten-proofs GitHub repository. OpenAI is also releasing the model's own narration of its thinking process for each result.

In a section on responsibility to the mathematical community, OpenAI takes a clear stance: presenting an AI-generated proof as purely human work would misrepresent both the system's contribution and genuine human intellectual effort. While acknowledging concerns raised by signers of the Leiden declaration on AI and Mathematics, OpenAI says it helped prepare the manuscripts and formalize the proofs in Lean, and takes responsibility for their correctness, while the underlying mathematical arguments were generated by the system itself.

Wrap-up

• OpenAI's internal "Astra" model produced new results on 10 math and theoretical CS problems that had sat unsolved for a decade or more
• The problems span high-dimensional geometry, coding theory, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics
• Highlights include new sphere-packing bounds, a construction proving non-sofic groups exist, a disproof of Connes's rigidity conjecture, and solutions to Erdős problems 183, 146, and 180
• The model's arguments were turned into manuscripts and formally verified as Lean certificates, published on GitHub as openai/ten-proofs
• OpenAI is explicit that these are AI-generated proofs and takes responsibility for their correctness

If you're into the frontier of math, theoretical CS, or how far AI can really go as a research collaborator, this one's worth digging into!