OpenAI said Tuesday, October 6, that an unnamed internal model produced a result on the Riemann hypothesis, a major unsolved problem in number theory, and posted 722 manuscripts generated by the model online. The company has not named the model or made it available. The hypothesis, formulated in 1859, concerns the zeros of the zeta function, which govern how prime numbers are distributed.

The model’s manuscript claims a fixed-width margin with no zeros beyond 7/8 of the way across the relevant band, regardless of height along the imaginary axis. Earlier work had ruled out zeros only in a margin near the band’s edge that narrowed at greater heights. A second manuscript uses another method to prove a weaker result, with a boundary at 11/12. The full hypothesis remains unsolved: its boundary must reach 1/2. Rutgers professor and number theorist Alex Kontorovich said a human making the result would receive an immediate Fields Medal, and added that it would also rule out Siegel zeros. Anthropic mathematician Levent Alpöge called it “the most significant moment in the history of mathematics,” despite previously disputing OpenAI’s credit for its September Navier-Stokes breakthrough.

The 722 manuscripts are organized into 372 “families,” which group a main result with its consequences and sometimes an alternative proof. OpenAI received about 4,000 open problems, yielding fewer than one published family per 10 questions. Each selected result used an average of the equivalent of three hours of ChatGPT Pro reasoning. None of the five unsolved Millennium Prize Problems was solved in this batch. Some manuscripts lack a Lean formalization, a computer-checkable proof written in a formal language; OpenAI warned that non-formalized results “could pose problems.”

The publication has prompted debate over how AI-generated mathematics should be released. OpenAI pointed to discussions with an independent Institute for Advanced Study advisory group, whose September 29 recommendations urged labs to disclose model names and prompts, avoid using mathematical results as marketing, and stop testing advanced math problems on proprietary models. OpenAI followed some recommendations by publishing quickly, including Lean formalizations, computing costs and the number of problems posed. The group said publication marks “the beginning, not the completion, of the process of human understanding.” Nature described mathematicians as unsettled both by the reported advances and by mass publication.

Separately, Bloomberg reported on the eve of publication that OpenAI was negotiating to raise at least $30 billion from a consortium of Emirati sovereign wealth funds led by MGX and BlackRock, at a pre-money valuation of about $1.4 trillion. OpenAI confidentially filed for an IPO in June, but it is not expected before 2027; Sam Altman said on September 12 that it was “not a good” time to go public. The GitHub release documents a capability, rather than revenue, for investors.

OpenAI said Tuesday, October 6, that an unnamed internal model produced a result on the Riemann hypothesis, a major unsolved problem in number theory, and posted 722 manuscripts generated by the model online. The company has not named the model or made it available. The hypothesis, formulated in 1859, concerns the zeros of the zeta function, which govern how prime numbers are distributed.

The model’s manuscript claims a fixed-width margin with no zeros beyond 7/8 of the way across the relevant band, regardless of height along the imaginary axis. Earlier work had ruled out zeros only in a margin near the band’s edge that narrowed at greater heights. A second manuscript uses another method to prove a weaker result, with a boundary at 11/12. The full hypothesis remains unsolved: its boundary must reach 1/2. Rutgers professor and number theorist Alex Kontorovich said a human making the result would receive an immediate Fields Medal, and added that it would also rule out Siegel zeros. Anthropic mathematician Levent Alpöge called it “the most significant moment in the history of mathematics,” despite previously disputing OpenAI’s credit for its September Navier-Stokes breakthrough.

The 722 manuscripts are organized into 372 “families,” which group a main result with its consequences and sometimes an alternative proof. OpenAI received about 4,000 open problems, yielding fewer than one published family per 10 questions. Each selected result used an average of the equivalent of three hours of ChatGPT Pro reasoning. None of the five unsolved Millennium Prize Problems was solved in this batch. Some manuscripts lack a Lean formalization, a computer-checkable proof written in a formal language; OpenAI warned that non-formalized results “could pose problems.”

The publication has prompted debate over how AI-generated mathematics should be released. OpenAI pointed to discussions with an independent Institute for Advanced Study advisory group, whose September 29 recommendations urged labs to disclose model names and prompts, avoid using mathematical results as marketing, and stop testing advanced math problems on proprietary models. OpenAI followed some recommendations by publishing quickly, including Lean formalizations, computing costs and the number of problems posed. The group said publication marks “the beginning, not the completion, of the process of human understanding.” Nature described mathematicians as unsettled both by the reported advances and by mass publication.

Separately, Bloomberg reported on the eve of publication that OpenAI was negotiating to raise at least $30 billion from a consortium of Emirati sovereign wealth funds led by MGX and BlackRock, at a pre-money valuation of about $1.4 trillion. OpenAI confidentially filed for an IPO in June, but it is not expected before 2027; Sam Altman said on September 12 that it was “not a good” time to go public. The GitHub release documents a capability, rather than revenue, for investors.