OpenAI Chatbot Disproves Erdős Conjecture on Unit-Distance Problem

OpenAI announced on 20 May that one of its chatbot systems had found a configuration of points on a plane that exceeds the maximum number of equal-distance pairs previously established by mathematician Paul Erdős in 1946. The system used algebraic number theory to select coordinates that satisfy specific equations, generating a 125-page chain of reasoning in response to a single open-ended prompt.

Mathematicians unaffiliated with OpenAI reviewed the document and confirmed its validity. Daniel Litt at the University of Toronto stated that the result is the first autonomous AI output he finds interesting in itself. Tom Trotter at the Georgia Institute of Technology, who co-authored papers with Erdős, said Erdős would have been enthusiastic about the advance.

Sebastien Bubeck at OpenAI described the model as a general-purpose reasoning system, not one built specifically for mathematics. ” Mehtaab Swahney at OpenAI noted that the model appeared to reason through the problem in a human-like manner.

Erdős had shown that the number of equal-distance pairs grows slightly faster than the number of points when points are arranged on successively larger square grids. He conjectured that no arrangement could improve on this growth rate. The OpenAI result supplies a different arrangement that produces more equal-distance pairs, thereby disproving the conjecture.

The company has not released the model’s name or the full document.