An internal reasoning model developed at OpenAI produced a construction that contradicts a conjecture by Paul Erdős from 1946 about how many pairs of points at exactly unit distance can appear among n points in the plane. The construction reported by the model yields approximately n^{1+0.014} unit-distance pairs, a polynomial improvement beyond the traditional bounds associated with Erdős's conjecture.
What the result says
- Paul Erdős posed the unit-distance question in 1946; for decades, square grid arrangements were treated as a benchmark for upper bounds.
- The OpenAI model's construction produces on the order of n^{1+0.014} unit-distance pairs, exceeding previous constructions that gave roughly linear or near-linear counts in n.
Verification and prior error
Last October, OpenAI publicized a GPT-5 claim related to Erdős's problem, but Thomas Bloom — who maintains a database of Erdős-related literature — demonstrated that the GPT-5 result had merely rephrased existing work, and that claim fell apart. Thomas Bloom is also a co-signer on the verification paper for the current construction.
The announcement prompted strong reactions in the mathematical community: Fields Medalist Timothy Gowers advised mathematicians on X (formerly Twitter) to "sit down before reading," signaling the surprise and potential impact of the claim.
Why this matters
- From a mathematical perspective, the episode illustrates that generative and reasoning models can produce concrete, checkable constructions, not only literature summaries.
- The development also raises broader institutional and economic questions: if century-old open problems can be resolved by model runs, the roles of human researchers, doctoral training, and AI-for-science ventures may shift substantially.
Summary
An OpenAI internal reasoning model has produced a construction claimed to disprove a 1946 conjecture of Paul Erdős by producing about n^{1+0.014} unit-distance pairs among n planar points. The result follows an earlier mistaken GPT-5 claim from October; the current finding has been verified with involvement from Thomas Bloom and prompted notable reactions from mathematicians, highlighting possible changes in how mathematical research is conducted and valued.
