OpenAI model claims to disprove Erdős conjecture

OpenAI announced in May 2026 that its general-purpose reasoning model has disproven an 80-year-old conjecture in discrete geometry originally posed by Paul Erdős in 1946, marking the first time AI has autonomously solved a prominent open problem central to a mathematical subfield. The proof constructs configurations with at least n^(1+δ) unit-distance pairs for a fixed exponent δ > 0, later refined by Princeton professor Will Sawin to δ = 0.014, surpassing the previous best bound that had remained essentially unchanged since 1946. The proof employs advanced algebraic number theory including infinite class field towers and Golod–Shafarevich theory, surprising mathematicians with the deep connection between number theory and geometric questions. Fields medalist Tim Gowers called the result a milestone in AI mathematics, while OpenAI states the breakthrough demonstrates AI's potential as a research partner across multiple scientific disciplines.

Erdős unit distance conjecture disproved

• ▪The previously best known construction for the unit distance problem came from a rescaled square grid and gave n^(1 + C / log log(n)) unit-distance pairs for a constant C.
• ▪Paul Erdős posed the planar unit distance problem in 1946, asking how many pairs of points can be exactly distance 1 apart among n points in the plane.
• ▪The best known lower bound for the unit distance problem had been essentially unchanged since Erdős's original 1946 construction.
• ▪Princeton mathematics professor Will Sawin refined the OpenAI proof to show the exponent δ can be taken as 0.014.
• ▪An OpenAI model disproved the longstanding Erdős conjecture that the unit distance problem has an upper bound of n^(1+o(1)) by constructing configurations with at least n^(1+δ) unit-distance pairs for some fixed exponent δ > 0.

Algebraic number theory proof technique

• ▪The OpenAI proof replaces the Gaussian integers with more complicated algebraic number fields that have richer symmetries capable of creating many more unit-length differences.
• ▪The OpenAI proof of the unit distance problem uses algebraic number theory concepts including infinite class field towers and Golod–Shafarevich theory.

AI autonomous mathematical discovery milestone

• ▪The OpenAI proof of the unit distance problem marks the first time a prominent open problem central to a subfield of mathematics has been solved autonomously by AI.
• ▪The unit distance problem proof came from a new general-purpose reasoning model rather than from a system trained specifically for mathematics or targeted at the unit distance problem.
• ▪OpenAI evaluated the model on a collection of Erdős problems as part of a broader effort to test whether advanced models can contribute to frontier research.

Expert mathematician verification process

• ▪Fields medalist Tim Gowers called the unit distance problem result a milestone in AI mathematics.
• ▪A group of external mathematicians checked the OpenAI proof of the unit distance problem and wrote a companion paper explaining the argument and providing background and context.
• ▪Number theorist Arul Shankar stated that the unit distance problem paper demonstrates current AI models are capable of having original ingenious ideas and carrying them out to fruition, going beyond just helpers to human mathematicians.

Perspective of External mathematicians reviewing the proof

• ▪Thomas Bloom states that the AI-generated proof of the unit distance problem reveals number theoretic constructions have more to say about discrete geometry questions than mathematicians suspected.

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