OpenAI Reports Advances on Ten Long-Standing Mathematical Problems

Ten advances in mathematics and theoretical computer science

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We want to empower scientists and mathematicians with tools that accelerate discovery. That is why we recently announced ChatGPT for Academic Researchers, an initiative providing 100,000 scientists and mathematicians with free access to our best ChatGPT models. We also continue to evaluate our models on open research problems during development.

In May, we shared an AI-generated disproof of the Erdős unit-distance conjecture, discovered while evaluating an unreleased model. This work has already inspired further developments in mathematics and theoretical computer science1. Today, we are sharing a selection of ten results to problems that have been open and have seen no progress on the main result for at least a decade, and in most cases much longer. These problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics. All of these problems are of substantial interest to their respective mathematical communities, and several are of broad interest across mathematics as a whole.

The results

We provide new results for the following problems. The results were achieved by an internal version of Astra, our next major model. The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates. These arguments were then prepared into manuscripts by humans with the same model. Afterward, the model formalized each argument in a Lean certificate. We are also releasing for each solution a model’s narration of its thinking process.

1. High-dimensional sphere packing. New upper bounds on sphere-packing density down to the Cohn–Elkies threshold.
2. Binary and spherical codes: Exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous results for high-dimensional spherical codes.
3. Non-sofic groups. A construction establishing the existence of non-sofic groups, addressing a central open question in group theory.
4. Connes’s rigidity conjecture. Disproof of a longstanding conjecture that certain groups are uniquely determined by their von Neumann algebras
5. Arithmetic circuit complexity. New lower bounds for computing the permanent using arithmetic circuits and formulas, including an arithmetic-formula lower bound of order n4/log n.
6. Quantum parallel repetition. An exponential parallel repetition theorem for general two-player quantum games, extending a foundational principle from classical complexity theory.
7. Closest vector problem. Polynomial-factor hardness of approximation for the closest vector problem, a foundational lattice question related to post-quantum cryptography.
8. Ehrhart’s volume conjecture. Determining, in every dimension, the maximum possible volume of a convex body whose centroid is its only interior lattice point
9. Multicolor Ramsey numbers. A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183.
10. Extremal number conjectures. Results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180.

Responsibility to the mathematical community

The emergence of systems capable of contributing to mathematical research raises questions that cannot be answered by a technology company alone. There are many views as to the role of AI in mathematics, and we have deep respect and understanding for those concerned with its impact, including the signers of the Leiden declaration on AI and Mathematics. We believe attribution should honestly reflect how a result was produced: claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system’s contribution and the nature of genuine human intellectual work. We helped prepare the manuscripts and formalize the proofs in Lean, and we take responsibility for their correctness, while the mathematical arguments themselves were generated by our system. We hope the mathematical community will engage deeply with these results, place them in context, and bring the ideas behind them to life through new research and discovery.

As AI systems evolve into more sophisticated research collaborators, ensuring widespread access is fundamental to supporting scientists and mathematicians as they navigate and define the future of their disciplines during this transformative era.

Footnote

1. Subsequent research includes Bloom, Sawin, Schildkraut, and Zhelezov, “The sum-product conjecture is false for real numbers”; Pohoata, “Split primes and the Elekes-Rónyai problem”; Saha, Xu, and Ye, “Furthest Pair Requires Quadratic Time in Superconstant Dimension under SETH”; Goh and Hatami, “Communication complexity of point-line incidences over the reals”; and Lee, Pohoata, and Zhu, “The Minkowski grid has robustly many repeated distances.”