This problem was solved in a completely automated fashion. Our internal model was given an AI-written statement of the problem, and its output was sent to an AI grading pipeline, which indicated high confidence that the solution was correct. It was only after this point that inernal human researchers and mathematicians began to examine the solution carefully. After preliminary AI-assisted verification and rewriting, a draft was sent to external mathematicians, including several number theory experts, who confirmed the proof’s correctness (and have already simplified and strengthened the argument).
Noga Alon:
Arul Shankar:
Thomas Bloom:
Tim Gowers:
Gowers then imagines a “hint sequence” as “something like a multi-part question on a question sheet, designed to help a suitably expert mathematician work through the proof by reducing it to a sequence of exercises”, names 3 such hints (look for a counterexample; take the best known construction and generalize it; Will Sawin’s less obvious hint to try a sequence of number fields of increasing degree, but work with prime ideals of bounded norm), and then concludes:
Among all the Fields medalists and comparable-tier (Tsimerman etc) I usually find Gowers’ remarks the most interesting as he’s the most AGI-pilled in this reference class.
Jacob Tsimerman:
“They can play for longer and in more treacherous waters without getting overwhelmed” reminds me of these anecdotes about Terry Tao and John von Neumann. The Tao anecdote is by his frequent collaborator Allen Knutson:
More specifically, one thing I learned from Terry that I was not taught in school is the importance of bad proofs. I would say “I think this is true”, work on it, see that there was no nice proof, and give up. Terry would say “Here’s a criterion that eliminates most of the problem. Then in what’s left, here’s a worse one that handles most of the detritus. One or two more epicycles. At that point it comes down to fourteen cases, and I checked them.” Yuck. But we would know it was true, and we would move on. (Usually these would get cleaned up a fair bit before publication.) …
Since we were working in my field of expertise rather than his, I knew better what the interesting questions were, and could translate them into combinatorics, then sic Terry on them. He would beat them to a bloody death as described above, and then it would be my job to dress the carcass for public viewing back in the original field.
Another notable and enviable trait of von Neumann’s was his mathematical courage. If, in the middle of a search for a counterexample, an infinite series came up, with a lot of exponentials that had quadratic exponents, many mathematicians would start with a clean sheet of paper and look for another counterexample. Not Johnny! When that happened to him, he cheerfully said: “Oh, yes, a theta function...’’, and plowed ahead with the mountainous computations. He wasn’t afraid of anything.
Tsimerman may be more AGI-pilled than Gowers, it’s probably just that the latter has a more approachable communication style. E.g. https://arxiv.org/abs/2507.09369
Some remarks by various leading mathematicians on OpenAI’s internal model solving the planar unit distance problem, one of Erdos’ favorite problems (proof, abridged CoT, companion paper, MathOverflow discussion). The proof itself was ~2.5 pages.
Noga Alon:
Arul Shankar:
Thomas Bloom:
Tim Gowers:
Gowers then imagines a “hint sequence” as “something like a multi-part question on a question sheet, designed to help a suitably expert mathematician work through the proof by reducing it to a sequence of exercises”, names 3 such hints (look for a counterexample; take the best known construction and generalize it; Will Sawin’s less obvious hint to try a sequence of number fields of increasing degree, but work with prime ideals of bounded norm), and then concludes:
Among all the Fields medalists and comparable-tier (Tsimerman etc) I usually find Gowers’ remarks the most interesting as he’s the most AGI-pilled in this reference class.
Jacob Tsimerman:
“They can play for longer and in more treacherous waters without getting overwhelmed” reminds me of these anecdotes about Terry Tao and John von Neumann. The Tao anecdote is by his frequent collaborator Allen Knutson:
The von Neumann anecdote I’ve mentioned before, from P. R. Halmos’ The Legend of John von Neumann:
Tsimerman may be more AGI-pilled than Gowers, it’s probably just that the latter has a more approachable communication style. E.g. https://arxiv.org/abs/2507.09369
Oh wow, I had no idea. Thanks for the pointer.