An internal artificial intelligence model developed by OpenAI has successfully generated solutions to ten long-standing mathematical and theoretical computer science problems, each open for at least a decade, according to an announcement from the company. The results—which span high-dimensional geometry, lattice cryptography, and quantum complexity—were achieved using an internal version of an AI model called Astra at a computational token cost of roughly $2,000 at Sol API rates, before being formatted into human-prepared manuscripts and Lean certificates.
AI Solves Ten Decade-Old Mathematical Problems
The ten mathematical problems tackled by the Astra model had seen no main-result progress for at least ten years. According to OpenAI, the breakthrough problems cover high-dimensional sphere packing, binary and spherical codes, non-sofic groups, and Connes’s rigidity conjecture. Additional solved problems include arithmetic circuit complexity, quantum parallel repetition, the closest vector problem, Ehrhart’s volume conjecture, multicolor Ramsey numbers, and extremal number conjectures resolving Erdős problems 146, 180, and 183.
The total cost to find these solutions using Sol API rates reached approximately $2,000. Following the generation of these arguments, human researchers prepared manuscripts using the model. Afterward, the system formalized each argument into a Lean certificate. OpenAI is also releasing the model’s narration of its thinking process for each solution.
Broader Research Initiatives and Mathematical Impact
This computational advance follows an AI-generated disproof of the Erdős unit-distance conjecture shared in May, which was discovered while evaluating an unreleased model. That previous work inspired further developments in mathematics and theoretical computer science. To foster continued research, OpenAI recently launched ChatGPT for Academic Researchers, an initiative designed to provide 100,000 scientists and mathematicians with free access to the company’s best ChatGPT models.
“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 stated regarding attribution standards. The organization noted that while the mathematical arguments were generated by the system, researchers helped prepare manuscripts and formalize proofs in Lean, taking responsibility for their correctness.
Addressing Community Concerns and Attribution
The emergence of AI systems contributing to mathematical research raises questions that cannot be answered by a technology company alone. OpenAI acknowledged diverse views on the role of AI in mathematics, noting respect for concerns expressed by signers of the Leiden declaration on AI and Mathematics. Ensuring widespread access to sophisticated research collaborators remains fundamental as scientists navigate the future of their disciplines.
Did you know?
The total token cost to find solutions to ten major mathematical problems using OpenAI’s Astra model was roughly $2,000 at Sol API rates. Each argument was subsequently formalized in a Lean certificate.
Frequently Asked Questions
What models and tools were used to solve these mathematical problems?
The results were achieved by an internal version of Astra, OpenAI’s next major model, and verified through Lean certificates.
How many problems were solved, and how long were they open?
Ten problems spanning geometry, group theory, and quantum complexity were solved, each of which had seen no progress on the main result for at least a decade.
Who is eligible for free access under the new academic initiative?
OpenAI’s initiative provides 100,000 scientists and mathematicians with free access to advanced ChatGPT models.
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