AI cracked an 80-year-old conjecture.

based on Sébastien Bubeck on X

An internal OpenAI model resolved the most famous open question in discrete geometry — the unit distance problem. The problem is 80 years old. It is also, on its surface, embarrassingly simple.

This page walks through what the problem is, why everyone thought they knew the answer, and what the model actually did.

How many pairs can sit at the same distance?

Drop n dots on a sheet of paper. Now count: how many pairs of dots are exactly the same distance apart?

By rescaling, we can call that shared distance 1 — hence "unit distance." So the question is just:

The first easy construction (left): pick one point as a hub and put all the others on a circle of radius 1 around it. That gives n − 1 pairs at distance 1. The ceiling is trivially the total number of pairs, about n²/2. The real question is which end of that gap reality lives on.

Erdős's first idea: just use a grid.

In 1946 Paul Erdős asked the question — and immediately tried the most natural configuration he could think of: a square grid.

Every interior grid point has exactly 4 neighbors at distance 1 (north, south, east, west). Counting all such pairs, and dividing by 2 to avoid double counting, gives about 2n unit-distance pairs.

So far the grid only gives a constant times n — same order as the hub construction. To get a real improvement, Erdős had to be cleverer about which distance to fix.

Pick a smarter distance, get more partners.

Instead of distance 1, look at distance $\sqrt{5} = \sqrt{1^2 + 2^2}$. How many grid neighbors does a point have at that distance?

Draw a knight's-move "L" — over 1, up 2 — and rotate / flip it. There are eight different L's, giving 8 grid points at distance √5 from any interior point.

Why does √5 multiply by 2? Because 5 = 1 + 4, and there are essentially two ways to write 5 as a sum of two squares: (±1)² + (±2)² and (±2)² + (±1)². Each sign and swap is a different L. That's not a coincidence — it's the first whisper that prime numbers are doing the work.

Gaussian integers: factoring the distance.

Think of a grid point (a, b) as the complex number $z = a + bi$. Its distance from the origin is $|z| = \sqrt{a^2 + b^2}$. So we're asking: how many complex integers $z$ have $|z|^2 = N$?

The key fact: a prime p that is 1 mod 4 (like 5, 13, 17, 29, …) splits in the Gaussian integers as $p = z \cdot \bar{z}$. For example:

How big can k get?

Take the first k primes that are 1 mod 4: p₁, p₂, …, p_k. Their product R has 2^k distinct factorizations as z₁$\bar{z}$₁ · z₂$\bar{z}$₂ · …, each picking z or $\bar{z}$. So 2^k grid points sit at distance √R from the origin.

How large can k be? We need $\sqrt{R} < \sqrt{n}$ so all 2^k points fit in our n-point grid. By the prime number theorem, the first k primes have $\log R \approx k \log k$. Solving $k \log k < \log n$ gives

Repeating this argument over every interior grid point gives Erdős's bound:

k = 1 2 3 4

Everyone believed the grid was optimal.

Erdős conjectured that you can never do better than the grid: any configuration of n points should have at most n^1+o(1) unit-distance pairs.

For 80 years, zero progress was made on this conjecture. Worse: in the closely related distinct distances problem (introduced in the same 1946 paper), the grid was proven optimal by Guth and Katz in 2015. Everything pointed the same way.

This was the orthodoxy walking into 2025: the grid is the right shape; the only question left was how to prove a matching upper bound. Everyone was looking up. The breakthrough came from looking down — at the grid itself.

"What if there were more ways to split each prime?"

From the chain-of-thought: the model almost immediately decided to attack the lower bound — to beat the grid. That's the opposite of what mathematicians had been trying.

Its strategy comes from staring at Erdős's argument. The 2 in 2^k comes from one fact: each prime splits into two conjugate Gaussian integers, z and $\bar{z}$. What if we worked in a bigger number system where each prime split into many more factors?

Three obstacles, all of which the model navigated:

obstacle 1 · the wrong fields

First guess: cyclotomic fields. The model tried it in the chain of thought, then quickly saw why it doesn't work and abandoned the idea.

obstacle 2 · unique factorization fails

In higher fields, you can't always factor numbers uniquely — but you can uniquely factor ideals. The model switched to the language of ideals, with the failures of unique factorization tracked by the class group.

obstacle 3 · controlling the projection

A high-dimensional lattice has to be projected back to the plane without too many points colliding. To get a number field with a controlled class number and the right projection properties, the model reached for a sledgehammer from class field theory: the infinite class field towers of Golod–Shafarevich.

Knowing everything is itself a kind of creativity.

Step back. The model did not invent class field theory. It did not invent Golod–Shafarevich towers. It did not invent a new branch of mathematics.

What it did do: see a connection between discrete geometry (a combinatorics problem about dots in the plane) and algebraic number theory (rings of integers, class groups, prime splitting). Then it executed a long chain of expert-level calculations end to end, with the right choice of parameters at every step.

That is itself the lesson. A lot of scientific progress — not just in mathematics — looks like this: knowing deeply, chaining expertly, choosing parameters right. If a model can do that, a great many open problems are suddenly within reach.

A coda from the OpenAI team: the proof is not being submitted to arXiv. No human author can claim authorship in the traditional sense. Instead, a companion paper by human mathematicians digests, contextualizes, and slightly simplifies the proof. Separating the AI's proof from the human understanding of it may be a useful template going forward.