OpenAI releases 722 AI-generated maths manuscripts, drawing praise and pushback
In short
OpenAI has published on GitHub 722 mathematics manuscripts generated by an unreleased internal AI model, grouped into 372 results across fields such as number theory and algebraic geometry. A result on the 'quasi-Riemann hypothesis' drew praise from Rutgers professor Alex Kontorovich, but not all manuscripts have been formally verified, and OpenAI said unverified ones may contain errors. Mathematicians' groups and Fields Medal winners have criticised how AI companies are using unsolved maths problems.
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OpenAI has released 722 mathematics manuscripts generated by an unreleased internal AI model, grouped into 372 results spanning number theory, algebraic geometry, differential equations, mathematical logic and other fields. Chosun Ilbo, Asia Economy and Newsis said the release came on October 6 local time; Korea Economic Daily, citing OpenAI and the Wall Street Journal, said the day before October 8. The manuscripts were published on GitHub. OpenAI said it gave the model about 4,000 problems and that each result used, on average, compute equal to about three hours of ChatGPT Pro reasoning. Newsis noted that the 722 manuscripts do not mean 722 separate problems were solved. [ 1 , 2 , 3 , 4 ]
The most-watched result concerns the 'quasi-Riemann hypothesis', a weakened form of the Riemann hypothesis. Korea Economic Daily said the manuscript does not prove the Riemann hypothesis itself but claims to extend the region where zeros of the Riemann zeta function and Dirichlet L-functions do not exist. Alex Kontorovich, a professor at Rutgers University, said on X that if a human had done this, they would have immediately won the Fields Medal. Korea Economic Daily said the release includes progress related to three unsolved Millennium Prize problems, which have not themselves been solved. Chosun Ilbo said OpenAI had earlier announced results on 10 problems in August and claimed to have solved more than 100 long-standing problems in September. [ 1 , 2 ]
Many manuscripts include formal verification in the Lean language, but not all, and OpenAI said unverified manuscripts may contain errors. Oum Sang-il (엄상일) of the Institute for Basic Science told Asia Economy that about 42% had Lean formalisation linked. Asia Economy reported that OpenAI said, as of 2:20 p.m. on October 8, it had withdrawn three results and revised or supplemented about a dozen proofs. Baek Hyung-ryul (백형렬) of KAIST said the release contained several problems he would call Fields-Medal-level in his field and that the impact would be very large if the main results were verified, but that manuscripts he read were hard to follow. [ 1 , 3 , 4 ]
The release has drawn criticism. An independent maths and AI advisory group, launched at OpenAI's suggestion, said on October 6 that the future of mathematical research should not consist only of understanding results produced by AI companies, Chosun Ilbo reported. A mathematicians' group launched last month, the AHM (인간을 위한 수학협회), said on October 7 that releasing more than 700 files at once was a show of power and urged mathematicians to stop working with OpenAI. Last month, 25 Fields Medal winners including Terence Tao, Peter Scholze and June Huh issued a joint statement criticising AI companies for using maths problems as performance benchmarks. Nature reported on October 7 that mathematicians reacted with both surprise and backlash, according to Newsis. [ 2 , 4 ]
OpenAI said it published papers and summaries of reasoning for some problems, will fund workshops and conferences, and is preparing to release the model. It did not accept calls to stop testing advanced maths problems on unreleased models, saying such evaluations are needed to build better research tools. [ 1 , 2 ]
Why it matters
KAIST professor Baek Hyung-ryul said the impact on mathematics would be very large if the main results are verified, while Newsis reported that the burden of checking the results falls on mathematicians. Parts of the mathematics community object to AI companies testing unreleased models on unsolved problems, which OpenAI said it will continue to do.
Key facts
- OpenAI released 722 mathematics manuscripts generated by an unreleased internal AI model, grouped into 372 results. [ 1 , 2 , 3 , 4 ]
- The manuscripts were published on GitHub. [ 1 , 3 , 4 ]
- OpenAI gave the model about 4,000 problems; producing one result took on average compute equal to about three hours of ChatGPT Pro reasoning. [ 1 , 4 ]
- The most-watched result concerns the 'quasi-Riemann hypothesis', a weakened form of the Riemann hypothesis; it does not prove the Riemann hypothesis itself. [ 1 , 2 ]
- Many manuscripts include formal verification in the Lean language, but not all; OpenAI said unverified manuscripts may contain errors. [ 1 , 3 , 4 ]
- Asia Economy reported that OpenAI said, as of 2:20 p.m. on October 8, it had withdrawn three results and revised or supplemented about a dozen proofs. [ 3 ]
- Twenty-five Fields Medal winners, including Terence Tao and June Huh, issued a joint statement last month criticising AI companies' goals in mathematics. [ 2 , 4 ]
- OpenAI did not accept calls to stop testing advanced maths problems on its unreleased models. [ 1 , 2 ]
Confirmed by several sources
- OpenAI released 722 AI-generated mathematics manuscripts grouped into 372 results. [ 1 , 2 , 3 , 4 ]
- The manuscripts are on GitHub. [ 1 , 3 , 4 ]
- About three hours' worth of reasoning compute was used per result on average. [ 1 , 3 , 4 ]
- Alex Kontorovich of Rutgers University said a human who had done the quasi-Riemann work would have immediately won the Fields Medal. [ 1 , 2 ]
- Not all manuscripts have been formally verified with Lean. [ 1 , 3 , 4 ]
- Twenty-five Fields Medal winners issued a joint statement last month criticising AI companies' approach to mathematics. [ 2 , 4 ]
- OpenAI intends to keep testing maths and science problems on unreleased models. [ 1 , 2 ]
Still unclear
- The date of the release. Chosun Ilbo, Asia Economy and Newsis say October 6 local time; Korea Economic Daily, citing OpenAI and the WSJ, says the day before October 8.
- Whether the results are correct and how far they advance existing research. The outlets say independent review is still needed, and Asia Economy reported some results have already been withdrawn or revised.
- Share of manuscripts with Lean formalisation. The figure of about 42% comes from one mathematician quoted by Asia Economy only.
- The time zone of the 2:20 p.m. October 8 update on withdrawals. Asia Economy gives the time without saying which time zone it refers to.
- How many Fields Medal winners joined last month's warning. Chosun Ilbo and Newsis say 25; Korea Economic Daily says mathematicians including about 20 Fields Medal winners.
What local media are saying
Timeline, local time
- October 8: Korea Economic Daily reports OpenAI's release of 722 maths manuscripts. [ 1 ]
- October 8: as of this time, OpenAI said it had withdrawn three results and revised about a dozen proofs, according to Asia Economy. [ 3 ]
- October 8: Chosun Ilbo reports mathematicians' concerns and the AHM call to stop working with OpenAI. [ 2 ]
- October 9: Asia Economy publishes comments from Korean mathematicians on verification. [ 3 ]
- October 9: Newsis reports on Fields Medal winners' warnings and the verification burden on mathematicians. [ 4 ]