An internal OpenAI reasoning model has done what mathematicians could not for nearly eight decades: autonomously disproved a famous open conjecture in discrete geometry first posed by the prolific mathematician Paul Erdős in 1946.
The announcement came on May 20, 2026, marking what experts are calling a watershed moment for artificial intelligence in frontier scientific research.
The Problem: 80 Years Unsolved
The planar unit distance problem asks a deceptively simple question: among n points placed in a plane, how many pairs can be at exactly distance 1 from each other? Erdős conjectured in 1946 that square grids were the optimal configuration — a belief that held unchallenged across generations of mathematicians.
For decades, the best known constructions produced approximately n^(1+o(1)) unit-distance pairs, where the exponent's bonus term vanished as n grew. The conjecture was not merely a textbook exercise — it sits at the intersection of combinatorics, geometry, and number theory, and remained open despite sustained attention from the world's leading mathematicians.
What the AI Model Found
OpenAI's reasoning model received only the problem statement — no hints, no step-by-step guidance. It returned a 125-page proof demonstrating an infinite family of point configurations yielding n^(1+δ) unit-distance pairs, where δ = 0.014. This fixed positive exponent represents a genuine polynomial improvement over square grids, definitively disproving the long-standing assumption.
The mathematical machinery the model employed was unexpected. It drew on Golod-Shafarevich theory and infinite class field towers — tools from algebraic number theory typically applied to questions about ring theory and class groups, not Euclidean geometry.
As external reviewers noted, these concepts were well known to algebraic number theorists, but their application to geometric questions in the Euclidean plane "came as a great surprise."
Verified by the Field's Best
Four leading mathematicians independently reviewed the work:
- Tim Gowers (Fields Medal, Cambridge): called it "a milestone in AI mathematics"
- Noga Alon (Princeton, combinatorics): described it as "an outstanding achievement"
- Arul Shankar (Toronto): validated the proof's depth
- Jacob Tsimerman (Toronto): confirmed the reasoning's originality
Princeton mathematician Will Sawin subsequently refined the exponent beyond the initial δ = 0.014, building on the AI's framework.
Why This Matters
Previous AI achievements in mathematics — such as AlphaProof's IMO medal-level results — involved competition problems with known solutions. The Erdős unit distance conjecture was an open research frontier, untouched by any solver for 80 years.
The model was general-purpose, not trained specifically for mathematics. That a broadly capable reasoning model could autonomously bridge algebraic number theory and discrete geometry — maintaining coherent logic across a 125-page argument — signals that AI is entering a new phase in scientific discovery.
Researchers note that human expertise remains essential: for identifying which problems matter, interpreting results in broader context, and directing the next steps of inquiry. But the model demonstrated that AI can now contribute original insights to the hardest unsolved problems in mathematics.
What's Next
OpenAI has not disclosed which specific model produced the result, describing it only as an internal general-purpose reasoning model. The company indicated it is working with the mathematics community to explore further applications of reasoning models to open problems.
The result is currently under review for formal publication. Its arrival raises a pointed question for the field: if an 80-year conjecture can fall in a single session, which unsolved problems are next?
Source: OpenAI