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Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new... show more

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13,721,467 views • 3 months ago •via X (Twitter)

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Earlier this month, an Erdős problem that had been open for 60 years was solved with help from GPT-5.4 Pro. What happens now that AI is getting good at math? OpenAI researchers Sebastien Bubeck and Ernest Ryu join host Andrew Mayne to explain what changed and what it could mean for the future of research.

Earlier this month, an Erdős problem that had been open for 60 years was solved with help from GPT-5.4 Pro. What happens now that AI is getting good at math? OpenAI researchers Sebastien Bubeck and Ernest Ryu join host Andrew Mayne to explain what changed and what it could mean for the future of research.

OpenAI

473,396 views • 3 months ago

OpenAI just spent $2,000 to solve 10 problems that have beaten the world's best mathematicians for DECADES. Nobody outside the company is allowed to run the machine that did it. On Saturday OpenAI published a 249-page report and gave its next model family a name: Astra. An internal version of it produced new results on 10 open problems in mathematics and theoretical computer science, and mathematicians had made no real progress on any of them for at least 10 years. On most of them, far longer than that. Here is what it solved: It built the first explicit example of a non-sofic group. Mikhail Gromov raised that question in 1999 and nobody answered it for 27 years. It disproved Connes's rigidity conjecture, a problem in von Neumann algebras that had stood for decades. It proved Ehrhart's volume conjecture. It resolved three problems from Paul Erdos's catalogue, including number 183 on multicolor Ramsey numbers. It produced the first improvement to the general upper bound on high-dimensional sphere packing since 1978. And it proved a new hardness result for the closest vector problem, which sits directly underneath lattice cryptography. That is the math the world is betting on to protect its data once quantum computers arrive. The successful runs cost roughly $2,000 in tokens. Now here is what almost nobody has picked up on... OpenAI did not just publish claims. Every argument shipped with a Lean certificate, which is a machine-checkable proof that any mathematician can verify without trusting OpenAI at all. That is a real change. In May the same model family disproved the Erdos unit distance conjecture and the world had to take a Fields Medalist's word for it. Tim Gowers said he would recommend that proof for the Annals of Mathematics without hesitation. This time the proofs check themselves. But look at what is still unverifiable: Any mathematician can now check those proofs line by line. Not one of them can look at the model that wrote them. Astra has no release date and nobody outside OpenAI has run it. The company announced its next major model family with a claim instead of a demo, and the only evidence anyone gets is the output. So OpenAI made an unfalsifiable claim about a machine look like a falsifiable claim about mathematics. The Information reported this week that OpenAI demoed Astra to US policymakers and regulators in Washington. This is the same month the administration is weighing a new watchdog to vet frontier AI models, reporting to the SEC. 10 proofs nobody believed a machine could produce is a very good thing to carry into that room. And keep in mind, the same model family doing this mathematics is the family that kept escaping its own testing environment. OpenAI models found zero-day vulnerabilities nobody knew existed, broke out of a sealed research sandbox, and reached another company's live systems. Both of those facts come from OpenAI's own announcements, published three weeks apart. Finding a proof no human could construct and finding a hole no human had noticed are the same ability aimed at different targets. Mathematicians are already asking for independent verification, and plenty of people online are calling the whole thing hype. Thomas Bloom, who runs the Erdos problems site, called the 10 results big news and said they matter more than the May result did. Lean will settle the mathematics within weeks. But nothing will settle what else a machine this capable is being pointed at, because nobody outside one company is allowed to look.

OpenAI just spent $2,000 to solve 10 problems that have beaten the world's best mathematicians for DECADES. Nobody outside the company is allowed to run the machine that did it. On Saturday OpenAI published a 249-page report and gave its next model family a name: Astra. An internal version of it produced new results on 10 open problems in mathematics and theoretical computer science, and mathematicians had made no real progress on any of them for at least 10 years. On most of them, far longer than that. Here is what it solved: It built the first explicit example of a non-sofic group. Mikhail Gromov raised that question in 1999 and nobody answered it for 27 years. It disproved Connes's rigidity conjecture, a problem in von Neumann algebras that had stood for decades. It proved Ehrhart's volume conjecture. It resolved three problems from Paul Erdos's catalogue, including number 183 on multicolor Ramsey numbers. It produced the first improvement to the general upper bound on high-dimensional sphere packing since 1978. And it proved a new hardness result for the closest vector problem, which sits directly underneath lattice cryptography. That is the math the world is betting on to protect its data once quantum computers arrive. The successful runs cost roughly $2,000 in tokens. Now here is what almost nobody has picked up on... OpenAI did not just publish claims. Every argument shipped with a Lean certificate, which is a machine-checkable proof that any mathematician can verify without trusting OpenAI at all. That is a real change. In May the same model family disproved the Erdos unit distance conjecture and the world had to take a Fields Medalist's word for it. Tim Gowers said he would recommend that proof for the Annals of Mathematics without hesitation. This time the proofs check themselves. But look at what is still unverifiable: Any mathematician can now check those proofs line by line. Not one of them can look at the model that wrote them. Astra has no release date and nobody outside OpenAI has run it. The company announced its next major model family with a claim instead of a demo, and the only evidence anyone gets is the output. So OpenAI made an unfalsifiable claim about a machine look like a falsifiable claim about mathematics. The Information reported this week that OpenAI demoed Astra to US policymakers and regulators in Washington. This is the same month the administration is weighing a new watchdog to vet frontier AI models, reporting to the SEC. 10 proofs nobody believed a machine could produce is a very good thing to carry into that room. And keep in mind, the same model family doing this mathematics is the family that kept escaping its own testing environment. OpenAI models found zero-day vulnerabilities nobody knew existed, broke out of a sealed research sandbox, and reached another company's live systems. Both of those facts come from OpenAI's own announcements, published three weeks apart. Finding a proof no human could construct and finding a hole no human had noticed are the same ability aimed at different targets. Mathematicians are already asking for independent verification, and plenty of people online are calling the whole thing hype. Thomas Bloom, who runs the Erdos problems site, called the 10 results big news and said they matter more than the May result did. Lean will settle the mathematics within weeks. But nothing will settle what else a machine this capable is being pointed at, because nobody outside one company is allowed to look.

Ricardo

44,020 views • 17 days ago

What happened when one of our models found a counterexample to an 80-year-old Erdős conjecture? Researchers Alexander Wei, Hongxun Wu, and Lijie Chen shared the story on the OpenAI Podcast with Andrew Mayne and explained how mathematicians and models can work together to make new discoveries.

What happened when one of our models found a counterexample to an 80-year-old Erdős conjecture? Researchers Alexander Wei, Hongxun Wu, and Lijie Chen shared the story on the OpenAI Podcast with Andrew Mayne and explained how mathematicians and models can work together to make new discoveries.

OpenAI

311,535 views • 2 months ago

An internal version of our next major model produced 10 new results on long-standing open problems in mathematics and theoretical computer science, using roughly $2,000 worth of tokens at GPT-5.6 Sol API rates.

An internal version of our next major model produced 10 new results on long-standing open problems in mathematics and theoretical computer science, using roughly $2,000 worth of tokens at GPT-5.6 Sol API rates.

OpenAI

1,802,920 views • 16 days ago

We also shared evals on Open AI o3-mini — a faster, distilled version of o3 which is optimized for coding, and the first version of o3 we expect to make available for use in early 2025.

We also shared evals on Open AI o3-mini — a faster, distilled version of o3 which is optimized for coding, and the first version of o3 we expect to make available for use in early 2025.

OpenAI

340,205 views • 1 year ago

A mathematician solved a problem that stood open for over 100 years, then explained it in an Italian lecture hall to a few dozen people. This is Hong Wang, professor at NYU's Courant Institute and at IHES in Paris, delivering the Colloquium Patavinum on Kakeya sets in 3 dimensions. The question is deceptively small: take a needle of length 1 and rotate it to point in every possible direction. How little space can the set of positions occupy. The conjecture says the answer must fill full dimension. Wang proved it in 3 dimensions. The Salem Prize followed in 2025, the Sadosky Prize in 2026, and an invitation to speak at the International Congress of Mathematicians. Watch how she frames the general statement about unions of tubes. The needle problem falls out as a corollary of something larger. A researcher I know rewatched the setup twice and said it was the clearest account of the result he had found anywhere. Free on YouTube from an Italian mathematics department. The problem was posed in 1917. She closed it, then took questions.

A mathematician solved a problem that stood open for over 100 years, then explained it in an Italian lecture hall to a few dozen people. This is Hong Wang, professor at NYU's Courant Institute and at IHES in Paris, delivering the Colloquium Patavinum on Kakeya sets in 3 dimensions. The question is deceptively small: take a needle of length 1 and rotate it to point in every possible direction. How little space can the set of positions occupy. The conjecture says the answer must fill full dimension. Wang proved it in 3 dimensions. The Salem Prize followed in 2025, the Sadosky Prize in 2026, and an invitation to speak at the International Congress of Mathematicians. Watch how she frames the general statement about unions of tubes. The needle problem falls out as a corollary of something larger. A researcher I know rewatched the setup twice and said it was the clearest account of the result he had found anywhere. Free on YouTube from an Italian mathematics department. The problem was posed in 1917. She closed it, then took questions.

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Today, we shared evals for an early version of the next model in our o-model reasoning series: OpenAI o3

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OpenAI

1,669,340 views • 1 year ago

I asked Dario 3 years ago why AIs haven't been able to use their vast knowledge across so many fields to connect two known ideas into a new discovery. It seems like AI did exactly this in the way it disproved Erdos' conjecture aobut the unit distance problem by cleverly onnecting together ideas in discrete geometry and algebraic number theory. Now that AI has been able to use its knowledge across multiple fields to come up with new ideas, what is the next benchmark? Grant Sanderson proposed one during our interview: "Good mathematicians prove theorems, great mathematicians come up with conjectures, and the greatest mathematicians come up with definitions."

I asked Dario 3 years ago why AIs haven't been able to use their vast knowledge across so many fields to connect two known ideas into a new discovery. It seems like AI did exactly this in the way it disproved Erdos' conjecture aobut the unit distance problem by cleverly onnecting together ideas in discrete geometry and algebraic number theory. Now that AI has been able to use its knowledge across multiple fields to come up with new ideas, what is the next benchmark? Grant Sanderson proposed one during our interview: "Good mathematicians prove theorems, great mathematicians come up with conjectures, and the greatest mathematicians come up with definitions."

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