Place dots on a flat surface and count the pairs exactly one unit apart. An internal OpenAI model found a construction that overturned a famous prediction about how quickly that count can grow. The result, announced on 20 May 2026, disproved a conjecture Paul Erdős posed in 1946.[1][2]
A small puzzle with a much larger question
Three equally spaced dots on a line produce two unit-distance pairs. Put the same three dots at the corners of an equilateral triangle with unit sides and all three pairs qualify. The arrangement changes the answer before the number of dots changes.[2]
Small arrangements you can check
| Dots | Arrangement | Unit-distance pairs |
|---|---|---|
| 3 | Straight line; adjacent gaps of one unit | 2Source 2 |
| 3 | Equilateral triangle; sides of one unit | 3Source 2 |
| 4 | Square; sides of one unit | 4Source 2 |
| 4 | Two unit-sided equilateral triangles sharing an edge | 5Source 2 |
The conjecture concerned enormous sets: Erdős expected the maximum to grow only slightly faster than the number of dots. The AI established an infinite sequence of arrangements with a fixed positive extra exponent. That persistent growth eventually outruns the conjectured rate.[2]
Why number theory entered a geometry problem
The independent companion paper explains the connection. A square grid can be represented by complex numbers: one component gives the horizontal position, another the vertical position. A difference between two numbers corresponds to the displacement between their dots. Finding many differences of equal length therefore creates many equal-distance pairs.[3]
The new construction uses richer number systems whose degree increases. It builds a lattice in a higher-dimensional space and selects a bounded portion that can be mapped into the ordinary plane. Carefully chosen arithmetic properties supply many displacements that become unit length after that mapping.[3]
The mathematical difficulty is keeping those useful displacements abundant as the construction expands. Existing results about towers of number fields provide the required structure. The companion authors identify earlier number-theoretic ingredients and present a simplified, human-verified argument.[3]
A proof that other mathematicians could develop
Will Sawin then sharpened the construction into an explicit bound. For arbitrarily large numbers n of points, his paper establishes arrangements with more than n raised to the power 1.014 unit-distance pairs. The small-looking extra exponent is fixed; it does not shrink as the set grows.[4]
This settles the old conjecture without determining the exact maximum. A gap remains between the new constructions and the known general upper bound, which grows on the order of n raised to four-thirds. The construction opens further questions about how far the method can be pushed.[4]
The original manuscript attributes the solution to the model; humans subsequently checked and edited its exposition. That leaves a construction other researchers can examine, simplify and extend.[2]
Sources & further reading
- An OpenAI model has disproved a central conjecture in discrete geometry OpenAI · 20 May 2026 · Accessed 12 Oct 2026
- Planar Point Sets with Many Unit Distances OpenAI · 20 May 2026 · Accessed 12 Oct 2026
- Remarks on the disproof of the unit distance conjecture Alon and colleagues, arXiv · 20 May 2026 · Accessed 12 Oct 2026
- An explicit lower bound for the unit distance problem Will Sawin, arXiv · 20 May 2026 · Accessed 12 Oct 2026