Ten advances in mathematics and theoretical computer science

OpenAI

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, each of which resolves or makes substantial progress on a long-standing open problem. 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(opens in a new window). We are also releasing for each solution a model’s narration of its thinking process.

Responsibility to the mathematical community

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.

• 2026