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.
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mathematics → illustrate → humans
The first development is about the Riemann zeta function, a mathematical idea connected to the distribution of prime numbers.
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development → connect → numbers
Scientists have used this property to secure communications and financial transactions, putting prime numbers at the heart of contemporary digital security.
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Scientists → use → security
The Riemann zeta function has many solutions called non-trivial zeroes.
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function → have → solutions
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.
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all → state → graph
This remains unproved.
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This → remain → ?
Proving it would enhance experts’ ability to predict the distribution of prime numbers, potentially exposing weaknesses in some cryptographic systems.
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Proving → prove → systems
Mathematicians have been able to show that 41.6% of the solutions lie on this line.
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% → show → 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.
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% → show → line
The model had been set on the problem by an Anthropic employee named Jarred Sumner, who was reportedly not a mathematician.
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who → set → employee
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.
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it → validate → result
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.
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that → publish → it
Proving and checking
The second development was also about prime numbers.
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development → prove → numbers
Prime numbers become less frequent as numbers get larger but they do not move arbitrarily far apart.
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they → become → ?
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.
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difference → consider → 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.
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Zhang → prove → number
Subsequent work by James Maynard, Terence Tao, and the Polymath collaboration reduced the bound to 4,680, 600, and finally 246.
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work → reduce → 4,680
That is, mathematicians know that there are infinitely many pairs of prime numbers whose gap is at most 246.
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gap → know → numbers
The twin prime conjecture, a major open problem in mathematics, would replace 246 with 2.
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conjecture → replace → 2
On August 31, U.S. mathematician Julia Stadlmann published a paper reducing the bound from 246 to 240.
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Stadlmann → publish → 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.
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Math → base → 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.
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tool → say → literature
The engineers and mathematicians behind Axiom Math developed the argument and used large-scale computation to manipulate the relevant numerical parameters.
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engineers → develop → parameters
Finally they used an “autonomous theorem prover” named AxiomProver to check the resulting proof.
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they → use → proof
Such formal verification has been useful because complicated mathematical arguments can contain mistakes that are hard for humans to spot.
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humans → contain → mistakes
Tools called theorem-provers check whether a given result follows from an existing set of assumptions and definitions.
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result → call → assumptions
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.
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result → begin → 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.
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result → involve → 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.
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mathematicians → replace → 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.
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mathematicians → acknowledge → it
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.
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next → improve → 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.
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result → go → days
Each of these advances has also invoked new methods.
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Each → invoke → 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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assumptions → remain → formulations
The Aporia
