A human set a new mathematical record. Then AI came for it

On the last day of August, a human mathematician kicked off a flurry of advances on one of the most famous problems in number theory — and was almost instantly superseded by AI. Julia Stadlmann of the University of Illinois Urbana–Champaign reported a new record in the quest to solve the twin prime conjecture, which posits that there are infinitely many pairs of prime numbers that are just two numbers apart (3 and 5 are twin primes, for example, as are 17 and 19). It was the first such advance in more than a decade.

It’s a “fiendishly difficult problem,” says mathematician Kannan Soundararajan of Stanford University, who says he admires Stadlmann’s persistence and courage. Within two hours of Axiom’s announcement, both records were surpassed by OpenAI, which just five days later claimed to have solved another of math’s biggest puzzles. For three days, at least.

The rapid-fire series of events has sent mathematicians’ angst over AI skyrocketing. In the view of many researchers, some AI companies are not only subverting professional norms but also undermining researchers’ pursuit of knowledge. Prime numbers, those that can be divided only by one and themselves, are of seemingly endless fascination to mathematicians, who have long puzzled over where along the number line they appear and why.

The twin primes conjecture, which has been wrestled with since at least the 19th century, is widely believed to be true but still unproven by number theorists. An easier version of the problem asks not whether there are infinite pairs of primes with a gap of two but whether there are infinite pairs of primes that all have the same gap of any size. In 2013, mathematician Yitang Zhang, now at Sun Yat-sen University in Guangzhou, China, proved that answer is yes: Some gaps must repeat infinitely many times.

Mathematicians don’t know that a gap of two does, but Zhang showed that a gap of some number less than 70 million does, thus setting the first record for the “small prime-gap bound.” Mathematicians quickly took turns lowering that number. The lower they can push it, the closer they are to solving the twin prime conjecture. By the middle of 2014, massive collaborative efforts had brought the bound down to 246, meaning pairs with some gap size equal to or less than 246 must repeat infinitely.

That is where the record stood for a dozen years. Stadlmann began working intensively on the problem about two years ago, after finishing her doctorate under James Maynard at the University of Oxford in England. (Maynard had helped set the previous record, and won the Fields Medal in 2022, in part for his work on prime gaps.) Stadlmann took on the formidable task of trying to better integrate the more recent techniques with Zhang’s earlier efforts. Both approaches use a “sieve” method, which partially filters out composite numbers using various weights, for example filtering out multiples of 2 more strongly than multiples of 3.