AI Disproves 80-Year-Old Erdős Conjecture

OpenAI announced on May 20 that one of its AI models disproved a conjecture posed by Hungarian mathematician Paul Erdős in 1946, solving what's known as the planar unit distance problem.

Image: nature.com - AI cracks 80-year-old mathematics challenge

The problem asks: given a set of points on a plane, how many pairs can be exactly the same distance apart? Erdős showed that larger grids could contain same-distance pairs growing slightly faster than the number of points, and he conjectured no arrangement could do better. For nearly 80 years, mathematicians believed him.

OpenAI's model proved otherwise. It used techniques from algebraic number theory to discover a new family of point arrangements that breaks the limit Erdős proposed, according to Nature. The system chose points with coordinates that were solutions to particular equations, finding constructions that outperform square grids.

Image: OpenAI - An OpenAI model has disproved a central conjecture in discrete geometry

"If Erdős were alive, I am sure that he would just be raving about this advance," said Tom Trotter, a mathematician at Georgia Tech who co-authored papers with Erdős, per Nature.

Sebastien Bubeck, a mathematician at OpenAI, said he believes this is the first time AI has autonomously produced a significant result in any research field. The proof came from a single prompt, a machine-rewritten statement of Erdős's question. "It's kind of remarkable to see the model really reasoning through the problem like a human," said OpenAI mathematician Mehtaab Swahney.

Daniel Litt, a mathematician at the University of Toronto who independently verified the proof, called it "the first result produced autonomously by an AI that I find interesting in itself."

What the AI Did and Didn't Do

The broader problem remains unsolved. As the Guardian reported, the AI showed Erdős's proposed limit was too low but did not establish a new answer for how fast the pairs actually grow.

OpenAI has not released the full 125-page chain-of-thought reasoning, nor named the specific model. Bubeck described it as an experimental, general-purpose reasoning model rather than one trained specifically for mathematics.

Image: the Guardian - OpenAI makes breakthrough on 80-year-old maths problem

Independent Verification

The result has been validated by outside mathematicians. Thomas Bloom, who maintains the Erdős Problems website and had previously criticized OpenAI's earlier Erdős claims, co-authored a companion paper. He wrote that the AI achieved its results by "persevering down paths that a human may have dismissed as not worth their time to explore," the Guardian reported.

Bloom added a caveat: "While the original proof produced by AI was completely valid, it was significantly improved by the human researchers at OpenAI and the many other mathematicians involved in the present paper. The human still plays a vital role."

Mathematician Tim Gowers, also writing in the companion paper, described the result as "a milestone in AI mathematics."

OpenAI had been embarrassed last year when it claimed an earlier Erdős breakthrough that turned out to be based on existing literature the model had absorbed. This time, independent verification appears solid.

For the most complete account of the mathematics involved, Nature's coverage by Davide Castelvecchi is the best single read.

Sources: Nature, The Guardian See also https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-remarks.pdf