New Delhi: OpenAI on Tuesday released 722 mathematics research papers produced by an internal Artificial Intelligence (AI) model not available to the public. The company claims the papers settle hundreds of open problems across 17 areas of mathematics, from prime numbers and geometry to computer science and quantum physics.
The release comes a month after OpenAI claimed its model had solved the Navier–Stokes problem, one of the seven Millennium Prize Problems in mathematics. It also comes amid growing concern among mathematicians over how AI companies announce such results.
What has OpenAI released?
The papers have been posted on GitHub, a public platform for sharing code and documents. They are grouped into 372 ‘families’, each built around one main result along with related papers. For 10 of the results, OpenAI has also released shortened summaries of the model’s reasoning.
None of the papers has been peer-reviewed, the process in which independent experts check a proof before a journal publishes it. OpenAI has said in the release that some of the results “could have issues”, and that it will correct them as they are found.
OpenAI Chief Executive Officer Sam Altman has highlighted four results: the quasi-Riemann hypothesis, the Unique Games Conjecture, a case of the Hodge conjecture, and the free group factor problem.
How were the results produced?
According to OpenAI, it began testing the model on unsolved research problems after its scores on the company’s existing math tests reached their ceiling. Over the course of the exercise, about 4,000 problems were put to it. Each result used, on average, about three hours of computing in ChatGPT Pro’s ‘thinking’ mode. OpenAI then selected the results it considered significant. Most of the papers are dated 23 and 24 September.
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What is the quasi-Riemann hypothesis?
Prime numbers, such as 2, 3, 5 and 7, can be divided only by 1 and themselves. They are the building blocks of all whole numbers and are used to secure online banking. Their pattern along the number line, however, is irregular.
In 1859, German mathematician Bernhard Riemann linked this pattern to a mathematical function, now known as the Riemann zeta function. He proposed that the points where the function equals zero all lie on a single straight line. This is the Riemann hypothesis, one of the Millennium Prize Problems, each of which carries a reward of $1 million from the Clay Mathematics Institute.
The model does not claim to prove the Riemann hypothesis. It claims a smaller step: that these zeros stay at least a fixed distance away from one edge of the region in which they can appear. This is known as the quasi-Riemann hypothesis, and it has remained unproven since Riemann’s time. A related paper claims to rule out a type of rare zero that has held back research on prime numbers since the 1930s.
What does the model claim in computer science?
The model claims a proof of the Unique Games Conjecture, proposed in 2002 by Subhash Khot. Born in Ichalkaranji in Maharashtra, Khot studied at the Indian Institute of Technology, Bombay, and is a professor at New York University. He was awarded the Nevanlinna Prize, one of the highest honours in computer science, in 2014, partly for this work.
Many practical problems, such as planning delivery routes or drawing up exam timetables, cannot be solved perfectly by computers in a reasonable amount of time. Computers instead settle for answers that come close. Khot’s conjecture, if true, fixes how close any efficient method can get for a long list of such problems.
The model also claims a faster method, in theory, for multiplying large grids of numbers called matrices. This is the basic operation behind AI systems and much of scientific computing. Researchers have been reducing the time it takes since 1969. The previous record was set in August by a team that included Google DeepMind researchers. OpenAI’s model claims a further reduction, though it would not directly speed up computers in use today.
What are the other claims?
The Hodge conjecture, proposed by British mathematician WVD Hodge in 1950, is another Millennium Prize Problem. It asks whether complex shapes defined by equations can always be built from simpler pieces. The model claims a proof for one special family of shapes, not for the conjecture as a whole.
Another paper extends the line of work that led to the proof of Fermat’s Last Theorem by Andrew Wiles in 1995. Wiles worked with ordinary numbers; the model claims a similar result for number systems that include the square root of minus one.
The model also claims to settle a version of the tenth problem on a list of 23 set by German mathematician David Hilbert in 1900. The problem asked for a method to tell whether a given equation has a solution. In 1970, it was shown that no such method exists when the solutions must be whole numbers. The model claims the same holds for fractions.
Among the other claims is an answer to a 1950 puzzle: what is the fewest number of colours needed to paint a flat surface so that no two points exactly 1 cm apart share a colour? The answer was known to lie between 5 and 7. The model claims that 5 colours are not enough. It also claims that pi cannot be approximated by fractions any better than a typical number can, and that a number called Catalan’s constant cannot be written as a fraction.
In at least 44 cases, the model claims to have found an example that disproves an existing conjecture.
How were the results verified?
OpenAI has used Lean, a computer program that checks each step of a proof. Nearly two-thirds of the result families come with a Lean check. These include the results on prime numbers, the Unique Games Conjecture, matrix multiplication, pi and the colouring puzzle. The results on the Hodge conjecture, Fermat-type equations and Hilbert’s tenth problem do not yet have one.
A Lean check, however, has limits. It confirms that the logic holds, but experts must still confirm that the computer checked the correct statement. The Lean checks in this release were themselves written by AI, and OpenAI lists their review status as ‘unchecked’.
Why have mathematicians raised concerns?
In August, OpenAI announced 10 results from its model. Some mathematicians said parts of that work drew on earlier research without giving credit. OpenAI said it would update the papers.
On 8 September, OpenAI said its model had solved the Navier–Stokes problem, which deals with how liquids and gases flow. The proof has not been independently verified and is not part of the new release. Tristan Buckmaster, a mathematician at New York University, said he and a colleague had reached a similar result, leading to a dispute over credit. OpenAI has said its team acted properly.
On 11 September, 25 winners of the Fields Medal, the highest honour in mathematics, signed a declaration warning that the rush by AI companies to announce results was harming the discipline. On 29 September, an independent group of nine mathematicians hosted at the Institute for Advanced Study in Princeton issued guidelines. These asked AI companies to publish results in journals and to disclose how they were produced.
What now?
OpenAI has said it will record corrections as new versions and is exploring moving the papers to a repository run by the mathematics community. Since the model has not been released, researchers outside OpenAI cannot repeat the process. The results will now have to be checked by mathematicians reading the proofs, a process that can take months for a single paper.
(Edited by Nardeep Singh Dahiya)
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What are we going to do with AI ? No one is discussing about it our country. The opposition is filled with bunch of monkeys so I don’t expect them to talk about it so can anyone really start a serious debate on it ?
It’s truly terrifying how rapidly it has become more intelligent.