Developments in number theory in the last two weeks illustrate how artificial intelligence is helping advance the frontiers of mathematics, but also raising questions about what mathematics means vis-à-vis humans.
The first development is about the Riemann zeta function, a mathematical idea connected to the distribution of prime numbers.
The prime numbers, 2, 3, 5, 7, 11, and so on, are divisible only by 1 and themselves. Scientists have used this property to secure communications and financial transactions, putting prime numbers at the heart of contemporary digital security.
Stepping stones
The Riemann zeta function has many solutions called non-trivial zeroes. The Riemann hypothesis states that all of these solutions lie on a vertical line in the complex plane (on a graph) called the critical line. This remains unproved.
Proving it would enhance experts’ ability to predict the distribution of prime numbers, potentially exposing weaknesses in some cryptographic systems.
Mathematicians have been able to show that 41.6% of the solutions lie on this line. On August 10, an internal version of Anthropic’s AI model Claude showed that at least 67.2% of the non-trivial zeroes lie on the critical line.
Composite numbers can be arranged in rectangles but prime numbers cannot. | Photo Credit: David Eppstein
A plot of the Riemann zeta function shows solutions called trivial zeroes and the twin critical lines of nontrivial zeroes, and the density of their absolute values. | Photo Credit: Conscious (CC BY-SA)
The model had been set on the problem by an Anthropic employee named Jarred Sumner, who was reportedly not a mathematician.
While (human) mathematicians Levent Alpöge and Ralph Furman, both at Anthropic, were able to validate the result, they also found it to be difficult to understand. On September 2, University of Lorraine mathematician Youness Lamzouri published a new proof of the same result, without AI help, that used simpler concepts to arrive at it.
Proving and checking
The second development was also about prime numbers.
Prime numbers become less frequent as numbers get larger but they do not move arbitrarily far apart. Mathematicians had considered for more than a century whether there are infinitely many pairs of prime numbers whose difference is smaller than some fixed number.
In a major breakthrough in 2013, Chinese-American mathematician Yitang Zhang proved that there are infinitely many pairs of prime numbers separated by less than the number 70,000,000. Subsequent work by James Maynard, Terence Tao, and the Polymath collaboration reduced the bound to 4,680, 600, and finally 246.
That is, mathematicians know that there are infinitely many pairs of prime numbers whose gap is at most 246. The twin prime conjecture, a major open problem in mathematics, would replace 246 with 2.
On August 31, U.S. mathematician Julia Stadlmann published a paper reducing the bound from 246 to 240. Days later, San Francisco-based Axiom Math — a company uniting “AI, programming languages, and mathematics into a single system for discovery” — reported a further reduction to 212. Its team said it built on Stadlmann’s work plus extensive computational experiments, and that its AxiomProver tool could formally verify the result against the existing mathematical literature.
Yitang Zhang in 2014. | Photo Credit: Public domain
The role of AI here was different from Claude’s role in the Riemann zeta function case. The engineers and mathematicians behind Axiom Math developed the argument and used large-scale computation to manipulate the relevant numerical parameters. Finally they used an “autonomous theorem prover” named AxiomProver to check the resulting proof.
Such formal verification has been useful because complicated mathematical arguments can contain mistakes that are hard for humans to spot. Tools called theorem-provers check whether a given result follows from an existing set of assumptions and definitions.
Hardest parts
The zeta-function result began with an AI-generated argument, was followed by human verification, and concluded (for now) with a new and simpler human proof. The prime gap result involved mathematicians, engineers, computation, and an AI system designed to check theorems working together on an existing line of research.
While AI models did not replace a mathematician in either development, mathematicians have remarked on their ability to find new arguments in the pursuit of cracking old problems. They have also acknowledged that many parts of the process by which mathematicians ‘discover’ mathematics and develop and check it are becoming increasingly automated.
Improving the size of the gap in the prime-gap problem from 246 to 240 took more than a decade but the next, from 240 to 212, followed within days of Stadlmann’s work. The zeta-function result similarly went from an AI-generated argument to human verification and finally a different, and human, proof in 23 days. Each of these advances has also invoked new methods.
The hardest parts remain human, however, including deciding which problems are worth attacking, understanding what a machine-generated argument actually means, finding simpler formulations, and determining which assumptions and methods are mathematically sound.
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