World News Today: International News Headlines - The Hindu | The Hindu
· collected 2026-09-04 · by Vasudevan Mukunth
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
