OpenAI 内部模型推翻 Erdős 单位距离猜想，数学界称里程碑

Last week, OpenAI announced that an internal AI model had disproved the Erdős unit distance conjecture, a famous problem in discrete geometry that had stumped human mathematicians for the last 80 years.

OpenAI gave several mathematicians early access to the result and published their reactions. Tim Gowers — who won the Fields Medal, the most prestigious prize in mathematics — wrote that “there is no doubt that the solution to the unit-distance problem is a milestone in AI mathematics.”

University of Toronto professor Daniel Litt wrote that “this is the first example of a result produced autonomously by an AI that I find exciting in itself, as opposed to as a leading indicator.”

It’s arguably the first time that an AI system has found a proof resolving a major open conjecture. That’s impressive, but I don’t view it as a radical break from the previous trajectory of AI progress in mathematics.

Three years ago, LLMs struggled to solve arithmetic problems. It was only last year that LLMs started acing high school mathematics competitions.

When I attended the Joint Mathematics Meetings — the largest annual mathematics conference in the world — in January, I learned that AI systems were starting to contribute to mathematical research, but only in constrained settings. It took significant human interpretation to turn an AI output into a publishable theorem.

OpenAI’s new result is the next step in this progression. The AI model cleverly applied existing ideas drawn from several subfields of mathematics to create a full proof. But it didn’t pioneer any genuinely new techniques. The result has since been cleaned up and extended by human mathematicians.

This points to a medium-term future where human mathematicians and AI models complement each other: AIs have a broader knowledge of past work than any human alive and much more willingness to grind through tedious proof strategies that aren’t likely to work. But humans can still think more deeply about any one problem and ask more interesting questions.

That might not last. AI systems have been improving at math so rapidly that it’s unclear what role — if any — human mathematicians will play a decade from now.

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The unit distance problem

Paul Erdős was one of the most prolific mathematicians in history. He wrote over 1,500 papers in his lifetime, the most ever. One of his greatest talents was coming up with problems that are simple to state but have deep roots.

In 1946, he introduced the unit distance problem. Imagine you have some points in a 2D plane and you measure the distance between each pair of points:

In this diagram, there are five points and ten pairs of points. Three pairs happen to be exactly 1 unit apart: AD, BE, and CE.

Can we rearrange the points so that more pairs of points are exactly 1 unit apart?

Yes. For instance, we could move points A and D to be closer to the B, C, and E cluster. With a bit more work, we could further rearrange the points so that there are seven pairs exactly one unit apart. But that’s the most we can do.

We could do the same analysis with 6 points, 7 points, and so on. But as the number of points grows, the problem very quickly becomes too complicated to find the exact answer.

The arrangements of 5, 6, 7, 8, and 9 points that have the most pairs of points exactly one unit apart. Figure from the appendix of “The Erdős unit distance problem for small point sets” by Boris Alexeev, Dustin G. Mixon, and Hans Parshall showing the optimal arrangements for 5 through 9 points. Alexeev et al. give the optimal solutions through 21 points; the question is open after that. (CC BY 4.0)

So instead of asking exactly how many unit distances are possible for a given number of points, Erdős tried to calculate upper and lower bounds on the number of length-one lines for n points, assuming that n is a large number.

To help calculate a lower bound, Erdős assumed that the points would be laid out in a grid. This is probably not the optimal layout, but if he could demonstrate that points in a grid have a certain number of pairs with unit distance, then the optimal arrangement must have at least that number.

If we make the grid smaller, we can intersect more grid points with the unit circle. This gives more unit distances. (Diagram by Kai Williams)

The simplest option is to space the grid so that every point is distance 1 from its neighbors directly above, below, left, and right. However, Erdős saw that you could do even better if you took diagonals into account. If you make the grid spacing smaller, you can make each point be distance 1 from a greater number of neighbors. In the diagram above, if the grid spacing is 1, then each individual point is one unit away from four neighbors (the left panel). Instead, if the grid spacing is ⅕ (as shown on the right), then each individual point is one unit away from 12 neighbors:

An animation of the distance-one neighbors of nine central points in a 13×13 grid. You can draw similar circles for other points in the grid to get the remaining distance-one pairs, but some points on the circle won’t land on grid points. (Animation by Kai Williams)

OpenAI’s writeup of its new result included a confusing diagram showing points in a grid with a bunch of lines connecting them. The diagram becomes easier to understand if we superimpose a circle like this:

A diagram from OpenAI’s announcement of the AI’s disproof of the unit distance conjecture, onto which I superimposed a circle showing the distance-one neighbors for one point. The grid spacing here is 1/√65, which produces unit circles that intersect 16 points on the grid (or would if the grid were larger). (Animation by Kai Williams)

This works because of the Pythagorean theorem, which states that if we have a point that is a units to the right and b units above another point, the distance c between those two points satisfies a² + b² = c². The trick is to choose some number c² so that there are a whole bunch of pairs of whole numbers a and b such that a² + b² = c². Then, if we scale the grid down so that each point is 1/c from its neighbors, there will be a bunch of unit distances.

For example, if we choose c² = 25, then the Pythagorean equation can be satisfied by either 0² + 5² = 25 or 3² + 4² = 25. This corresponds to the 12-grid-point circle I showed earlier, with points at (0,5), (3,4), (4,3), (5,0), (-4,3), (-3,4), and so forth. (Technically, these lengths should all be divided by 5 — (⅗, ⅘) for example — but I’m leaving the denominators out for clarity.)

OpenAI’s diagram is based on choosing c² = 65, which can be satisfied by either 1² + 8² = 65 or 4² + 7² = 65. This means that if the grid spacing is 1/√65, each point will be one unit away from 16 other points: (1,8), (4,7), (7,4), (8,1), (-1,8), (-4,7), and so forth. Larger values for c² — if they’re chosen carefully — enable more whole-number diagonals and hence more unit-distance pairs.

However, if c² is too large, compared to the number of points in the grid, then many of the potential one-unit-away neighbors will be outside the grid.1

In short, we want to choose a c² that’s large enough but not too large. Using insights from number theory, including Jacobi’s two-square theorem, Erdős was able to show that an optimally sized circle will enable the number of unit-distance pairs to grow faster than the number of points, but only barely.

The question became: can you do better? To find an upper bound, Erdős used an argument from a quite different area of mathematics called graph theory to show that you could only have so many unit distances. But his upper bound grows much, much faster than the best lower bound he was able to construct.