OpenAI Model’s Proof of Erdős Unit Distance Problem
This proof is an important milestone for the math and AI communities. It marks the first time that a prominent open problem, central to a subfield of mathematics, has been solved autonomously by AI. It also demonstrates the depth of reasoning these systems now support.
[…]
The proof is available here. The companion paper by leading external mathematicians is available here. You can find an abridged version of the model’s chain of thought here .
Update (2026-08-17): Timothy Gowers:
I would like to find a coherent explanation of the conjunction of the following facts.
- The most notable mathematical results proved by LLMs have tended to be ones that we would classify as examples or counterexamples, where counterexamples are, broadly speaking, existence statements that disprove statements that we expected to be true.
- Many statements can be formulated as existence statements when we would usually think of them as universal statements, and vice versa, so what we consider to be an example depends on the mathematical context of a statement as well as its logical form.
- LLMs are pretty good at proving universal statements as well: it’s just that the strongest statements they have proved that we would think of as theorems have mainly not been at the level of the strongest statements that we would think of as counterexamples.
3 Comments RSS · Twitter · Mastodon
Countdown until this amazing capability is primarily used to scam and surveil people
Countdown until this amazing achievement is revealed to be not as amazing, irreproducible or mostly done by a human.
"Countdown until this amazing capability is primarily used to scam and surveil people"
No need for a countdown, that's already the case.
"Countdown until this amazing achievement is revealed to be not as amazing, irreproducible or mostly done by a human."
Doubtful, unless they're blatantly lying:
"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 internal 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). The present manuscript is a human-edited exposition of the autonomously produced solution, with references, reorganized proofs, and additional explanatory material added afterward."
I don't think there's any dissembling here. LLMs are just getting good at these types of mathematical reasoning chains.
