1/ Today, an internal @OpenAI model has refuted Erdős’s unit distance conjecture — a research result that one could recommend “acceptance without any hesitation” to the Annals of Mathematics, one of the most prestigious journals in mathematics.

We came across it in a side quest to push our model on the hardest problems. 2/ It is an elegant conjecture that remained unsolved for 80 years:

Draw n points on a piece of paper. What is the largest possible number of pairs at distance exactly 1?

Erdős conjectured that the answer is almost linear, i.e. n^(1+o(1)). Our model gives a counterexample, drawing on ideas from algebraic number theory, showing that one can have n^(1+δ) such pairs for some small δ > 0.