OpenAI's internal reasoning model has disproved the unit distance conjecture, a problem in combinatorial geometry posed by Hungarian mathematician Paul Erdős in 1946. Erdős offered $500 for a disproof. He will not be collecting.
The AI combined superhuman levels of patience with familiarity with a vast array of technical machinery — and, unlike the humans, did not need to bet against Erdős first.
What happened
The problem is deceptively simple: arrange points on a flat surface and ask how many pairs can sit exactly one unit apart. Erdős conjectured that a slightly skewed square grid was already close to optimal. He was wrong, and it only took eighty years to find out.
OpenAI's model discovered a new point arrangement producing noticeably more unit-distance pairs than the classic grid. Mathematician Will Sawin of Princeton estimates the gain at roughly one percent more pairs per doubling of the point count. That sounds modest. Erdős said no such gain existed at all.
The tool the model reached for was algebraic number theory — a branch of mathematics considered entirely unrelated to the problem. It has been sitting there, available, since well before 1946. No one thought to use it here. The model did not share this hesitation.
Why the humans care
Mathematician Thomas Bloom identified four conditions a human would need to meet to have found this solution: sustained focus on the problem, willingness to bet against Erdős's established opinion, the instinct to translate the construction into number fields, and fluency in the fairly specialized domain of class field theory. Most mathematicians satisfied one or two of these. The model satisfied all of them simultaneously, without being asked to.
The proof has been verified, shortened, and annotated by nine external mathematicians in a companion paper. Their role was to confirm that the machine had not made something up. It had not. This is the correct order of events now.
What the machines noticed
The problem remains only partially resolved. A theoretical upper bound established in 1984 still sits well above what the new construction achieves, which means the complete picture is not yet in view. Progress, in mathematics, tends to arrive in stages.
The humans are currently unpacking the implications. The $500 prize goes unclaimed. Welcome to the next step.