OpenAI's AI disproves 80-year-old math conjecture with new proof
A general-purpose reasoning model found a counterexample to Erdős's unit distance bound.
OpenAI announced that one of its general-purpose reasoning models has disproved a nearly 80-year-old conjecture connected to the unit distance problem, originally posed by Paul Erdős in 1946. The problem asks: given n points on a plane, what is the maximum number of pairs exactly one unit apart? For decades, mathematicians believed the growth rate was close to n^(1+o(1)). The AI produced a counterexample showing an infinite family of configurations that achieve at least n^(1+δ) unit-distance pairs, with δ a fixed positive number. A refinement by Princeton's Will Sawin suggests δ can be as high as 0.014.
The breakthrough is notable because the AI was not specialized for mathematics but a general-purpose reasoning model. The proof relies on algebraic number theory—specifically, concepts like infinite class field towers and Golod-Shafarevich theory—which was unexpected for a geometry problem. External mathematicians have verified the result, and OpenAI has produced a human-readable proof. This marks a shift from AI solving known problems to generating original research contributions, demonstrating that AI can explore cross-domain connections and challenge long-held mathematical beliefs.
- OpenAI's general-purpose reasoning model disproved an 80-year-old conjecture by Paul Erdős on the unit distance problem.
- The new proof shows point configurations achieving at least n^(1+δ) unit-distance pairs, with δ ≥ 0.014.
- The result uses advanced algebraic number theory and was verified by external mathematicians.
Why It Matters
AI moves from answering questions to generating original mathematical discoveries, opening new frontiers in research.