Unit distances: a simple question, an unexpected construction
An Erdős problem draws an answer from algebraic number theory.
OpenAI · Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang & Matchett Wood
Start with the question
The problem
Place n points in the plane. How many pairs can be exactly one unit apart? The long-standing conjecture predicted an upper bound of n to the power 1 + o(1). Finding an infinite family with a polynomially larger number refutes that prediction.
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The old benchmark was the grid
Erdős posed the planar unit-distance problem in 1946. OpenAI’s announcement reports an infinite family beating the prevailing square-grid-based expectation by a polynomial factor. It attributes the construction to a general-purpose model being evaluated on research problems. [1]
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External mathematicians explain the idea
The companion remarks are authored by mathematicians across combinatorics and number theory. They place the algebraic-number-theory construction in context. Their explanatory work is a separate contribution from the model-generated argument and provides a more useful route into the mathematics than the headline alone. [2]
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A disproof does not give the final extremal answer
Refuting a proposed asymptotic bound need not determine the exact maximum number of unit distances for every n. The result changes what configurations can achieve and opens follow-up questions about the true growth rate. [2]
Inspect the evidence
The Lean formalization
No linked Lean artifact located in the reviewed release
The announcement links a proof and companion remarks, but does not link a Lean development. Read those documents for the argument and review context. This article does not label the result Lean-verified or substitute an illustrative code snippet for a formal proof.
Open the research source ↗Sources & further reading
Reviewed 9 Sept 2026. X permalinks were recovered from indexed posts and linked discussions; direct X pages were not consistently accessible. They document the conversation, not proof correctness.
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