A twelve-year prime-gap record changed hands three times inside four days

A bound of 240 says that some gap of 240 or less between consecutive primes recurs infinitely often ^[2]. It does not say that a gap of two does. That is what the twin prime conjecture asserts, and it is still unproved ^[3][9]. Between the record and the conjecture sit 238 integers ^[23].

Each further cut in the bound costs more work. The year after Yitang Zhang's result cut the bound by a factor of roughly 280,000 ^[24]. The twelve years after that cut it by 6, a reduction of about 2.4 percent ^[22].

Those 6 came out of the weights. Sieve methods filter composite numbers partially rather than completely, weighting the filter so that multiples of 2 are removed more strongly than multiples of 3, and the partial filter describes the distribution of primes better than a complete one would ^[10]. Stadlmann found the optimal weights by piecing together regions of high-dimensional space where the competing approaches each prevail, then calculating the volumes of those regions with only a rough idea of what the regions look like ^[11]. Andrew Granville, a number theorist at the University of Montreal, said of it: "It goes beyond finding a needle in a haystack." ^[12] Kannan Soundararajan of Stanford called the problem "fiendishly difficult" and said, "It was really quite impressive." ^[13]

Her methods could have done better with more time, and she did not have it ^[15]. Kevin Ford, her postdoctoral mentor at Illinois, gave her the instruction in plain words: "Drop everything you're doing and put this result on the math preprint archive. Get it out there." ^[17] Counting from that posting to OpenAI's result, the whole sequence fits inside about 74 hours ^[25].

Ford also said, "It's a David-versus-Goliath story where the human gets the gold." ^[18] Terence Tao of UCLA, a Fields Medalist, wrote on mathstodon.xyz that "what we are seeing here is a consequence of some very deliberate choices to abandon any pretense of gaining human understanding ... and using AI tools for the sole purpose of achieving a benchmark" ^[19]. Science News reports that many researchers now see some AI companies as subverting professional norms and undermining the pursuit of knowledge ^[26].

Neither Axiom Math nor OpenAI has given the bound it reached, and OpenAI's later result on another famous problem stands as a claim ^[21][6]. Stadlmann's 240 arrived with a method other people can read and extend, and Axiom Math extended it in three days ^[11][4].