How many pairs of n points can be exactly 1 apart? Erdős bet on the square grid for eighty years, and in 2026 an AI model found something better
Put n points in the plane. How many pairs can be exactly one unit apart? For eighty years the answer was believed to be barely more than n. In May 2026 an AI model showed it is not.
Choose a distance and a grid size, and click points to see their circles. A distance that can be written as a sum of two squares in many ways wins once the grid is large.
Click any point to draw its circle. The orange web joins every pair at distance √m; m = 5·13·17 = 1105 has 32 representations, but needs a big grid before it pays off.
Paul Erdős posed the problem in 1946. Write u(n) for the largest number of unit distances n points can have. A row of points gives n − 1. A square grid with nearest neighbours as the unit gives about 2n (a 10 × 10 grid has 180). Neither is anywhere near the trivial ceiling of n²/2, and the real question is how far above linear you can go.
Erdős’s trick is to stop using the nearest neighbours. Take an L × L grid and a number m, and count pairs at distance √m. Each such pair is an integer vector (a, b) with a² + b² = m, so the count is driven by r₂(m), the number of ways of writing m as a sum of two squares. Fermat and Jacobi tell you exactly what that is: r₂(m) is four times the excess of divisors of m that are 1 mod 4 over those that are 3 mod 4. Multiply together many primes that are 1 mod 4 — 5, 13, 17, 29, … — and r₂ doubles with each new prime: 1105 = 5 · 13 · 17 already has 32 representations.
Balancing m against the size of the grid gives Erdős’s lower bound u(n) ≥ n^(1 + c / log log n). The exponent creeps above 1 by an amount that shrinks to zero as n grows, so the grid’s count is n^(1+o(1)): more than linear, but less than n^(1+δ) for any fixed δ. Erdős conjectured that this was essentially the truth.
The best upper bound is O(n^(4/3)), due to Spencer, Szemerédi and Trotter in 1984. The cleanest proof, by Székely, draws a unit circle around every point, treats the circles as a drawing of a graph, and applies the crossing lemma: two unit circles cross at most twice, so too many point-circle incidences would force more crossings than the drawing can have. For forty years nobody has improved either bound in a way that changed the exponent.
Each point is a sum of unit steps in k directions. Switch between unrelated and related directions and watch the unit pairs per point.
With unrelated directions every sum is a different point, and each point gains about one unit neighbour per direction. Related directions make sums collide, packing more unit pairs onto fewer points.
On 20 May 2026 OpenAI announced that an internal model had produced point sets with at least n^(1+δ) unit distances for a fixed δ > 0, disproving the conjecture. Lijie Chen ran the model; Mark Sellke and Mehtaab Sawhney checked the argument. Soon after, nine mathematicians (Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood) posted a short, human-verified digest of the proof (arXiv:2605.20695), and pointed out that its methods trace back to work of Ellenberg and Venkatesh, of Golod and Shafarevich, and of Hajir, Maire and Ramakrishna.
The demo above is the right picture, made far more extreme. A point that is a sum of unit steps has a unit-distance neighbour for every step you can add or remove, so the more independent unit directions you have, the more unit pairs. With k unrelated directions you get at most k unit pairs per point: linear, no better than a row. The gain has to come from arithmetic: directions that are related, so that sums collide and the same point is reached many ways.
The construction works in a CM field: a number field K of large degree that is a totally imaginary quadratic extension of a totally real field. Its ring of integers, embedded in all of its complex coordinates at once, is a lattice in ℂ^f. The key input is a huge supply of elements u with |u| = 1 in every complex embedding — in a CM field, being on the unit circle in one embedding forces it in all of them. Adding such a u moves every coordinate by a unit-length step, so a window of the lattice contains many pairs that differ by one. Project the window to a single complex coordinate and those pairs become unit distances in the plane.
Everything hinges on making the degree f large while keeping the field’s root discriminant bounded, so that the lattice stays well-shaped. Class field towers, whose existence Golod and Shafarevich proved in 1964, supply exactly such fields. The digest computes the exponent this gives as about 1 + 6 × 10⁻³⁸ — tiny, but a fixed constant, which is all a disproof needs.
Drag the number of points up and watch the grid’s exponent drain toward zero while the new constructions stay put.
Constants are ignored here; this is purely the race between exponents. The grid’s δ only drops below 0.014 once ln ln n passes about 50, a number of points with more than 10²¹ digits.
Settled: Erdős’s conjecture is false. There is a fixed δ > 0 with u(n) ≥ n^(1+δ) for infinitely many n, and the argument has been checked by professional number theorists and combinatorialists, not only by its authors. Will Sawin optimised the choice of number fields and obtained more than n^1.014 unit distances (arXiv:2605.20579).
Not settled: the true exponent. It now lies somewhere between 1.014 and 4/3, and nothing in the new argument suggests where. The constructions only win for absurdly large n — the grid’s exponent drops below 0.014 only once ln ln n passes about 50 — so for any configuration you could draw, the grid is still the champion.
It is worth being careful about the source. This was one of several mathematical claims OpenAI made in 2026, and they have had very different levels of scrutiny. This one is the best vetted: a short proof, human-digested, and confirmed by independent experts.
Spot a mistake or have an idea?