Unit Distance Conjecture

The Unit Distance Conjecture deals with a math question: if you scatter a bunch of points on a plane, what's the maximum number of point pairs that can be exactly the same distance apart (say, distance 1)? If you place 100 points, the question is how many pairs can end up exactly 1 centimeter apart at most. It sounds simple, but as the number of points grows, proving the answer becomes extremely difficult. Mathematician Paul Erdős posed this problem, and it remained unsolved for nearly 80 years.

What the article covers isn't the conjecture itself, but an event: an unnamed internal math model from OpenAI reportedly disproved this conjecture. Disproving is different from proving — instead of showing the conjecture is true, it means finding a counterexample that shows the conjecture is false. The fact that a counterexample emerged for a problem that had gone untouched for so long drew attention from the academic community.

The article notes that "back in May, an unnamed internal OpenAI model had already disproved the Unit Distance Conjecture, which had gone unsolved for 80 years," and speculates that the model later announced to have solved ten problems belongs to the same lineage. It's worth noting that this event didn't fully prove and resolve the conjecture — it found a counterexample showing it was false. The word "solved" can mean different things depending on the problem.