OpenAI has announced that an internal version of OpenAI Astra, its next major model, has solved ten mathematical problems which have remained open with no progress on the main result, for at least a decade and in most cases far longer.
None of the ten results has yet been independently verified or peer-reviewed by the mathematical community.
The problems span areas such as high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography and extremal combinatorics.
According to the company, the AI generated the mathematical arguments, which were then prepared into manuscripts by OpenAI staff working with the same model. The proofs were subsequently formalized by the model itself in the Lean proof assistant, a software tool used to formally check mathematical proofs, with the certificates published on GitHub.
Among the ten results are a disproof of Connes’s rigidity conjecture in operator algebras, a construction establishing the existence of non-sofic groups, new upper bounds on high-dimensional sphere-packing density, and solutions to three problems from Paul Erdős’s list, numbered 146, 180 and 183.
OpenAI said the tokens needed to find the solutions would have cost about $2,000 at its Sol API rates.The company did not say whether that figure accounts for failed attempts or the cost of developing the model. The company is also releasing the model’s narration of its thinking process for each solution.
The announcement follows the launch of ChatGPT for Academic Researchers, an initiative which gives 100,000 scientists and mathematicians free access to OpenAI’s most advanced ChatGPT models to support research.
Earlier this year, OpenAI said one of its models had produced a disproof of the Erdős unit-distance conjecture, a longstanding mathematical problem. The company said that work has already inspired further developments in mathematics and theoretical computer science, citing five arXiv preprints published between May and July.
“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 science,” the company said.
“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,” it mentioned.
Mathematicians across the world have warned that the growing use of artificial intelligence in mathematics is putting the field’s fundamental values under threat, raising concerns over unreliable proofs, missing citation and research bias. Those concerns led to the drafting of the Leiden Declaration on Artificial Intelligence and Mathematics by a working group of sixteen researchers from fifteen universities.
OpenAI said that it recognizes the concerns about the growing role of AI in mathematics, including those raised by researchers who signed 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,” OpenAI said.
Astra has not been released publicly and OpenAI has given no launch date. The company has not published independent assessments of the ten results, and the Lean certificates have not yet been checked by outside mathematicians.
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