AI disproves Erdős unit distance conjecture, guardrails demanded

AI disproves – An OpenAI-trained reasoning model produced a proof posted May 20 on OpenAI.com that discredits Paul Erdős’ 1946 conjecture about the unit distance problem. The result—described by Harvard mathematician Melanie Matchett Wood as “a beautiful piece of mathematics

It starts with a question that feels almost playful: place dots on a flat surface and try to maximize the number of pairs separated by exactly the same distance. For decades, mathematicians have treated that unit distance problem as deceptively simple—easy to state, stubborn to resolve.

Back in 1946, Paul Erdős proposed a conjectured answer. No one could prove it. No one could disprove it.

Then, with a single change in how the question was approached, the ground shifted.

Researchers at OpenAI gave an AI model Erdős’ conjecture and walked away. When they returned, they discovered the model had disproved the conjecture in a mathematical proof posted May 20 on OpenAI.com.

“It’s a beautiful piece of mathematics that has been discovered,” said Melanie Matchett Wood of Harvard University. Wood contributed remarks to an accompanying paper where outside experts reviewed the AI’s result. For Wood, the outcome is not just a win for a long-standing problem. It also bolsters hopes that AI can contribute to scientific understanding.

But the proof has come with friction. A group of experts published a declaration calling for tight guardrails around AI in mathematical research on June 2. By June 5, the declaration had 1,590 signatures.

The uneasy mix—math breakthrough and technology unease—was already baked into the way the proof happened.

The AI model that produced the result isn’t publicly available yet. OpenAI said it is a general-purpose large language model trained for reasoning. OpenAI also said the model did not use any math-specific tools or software. “We didn’t guide the model in any particular way,” said OpenAI researcher Sébastien Bubeck.

The prompt was composed by AI. It described the conjecture and instructed the model that a complete solution must either prove or disprove it. Mathematicians had believed the conjecture was true. Yet the model tried to disprove it instead.

Wood said the AI counterexample built from tools from two of the oldest and most foundational mathematical fields: algebra and number theory. “It seems that these areas shouldn’t have anything to do with this geometry question,” she said. But she added that the result shows how tools from one part of mathematics can apply “really fruitfully in this other area.” She believes the work could push mathematicians to find new ways to reuse those tools.

The AI’s counterexample also doesn’t fit neatly into a quick sketch. Wood said it isn’t easy to visualize because it involved building a complicated grid in a high-dimensional space and then projecting it onto a flat plane. Still. she said a depiction—based on an idea by Will Sawin and created with Kai Williams/ChatGPT—captures the gist of the approach.

Not everyone was impressed with what the result says about AI itself.

Wood said she wasn’t convinced it represented a breakthrough in artificial intelligence. When she read the solution, she thought the latest publicly available AI models could have come up with it. She also noted that one researcher posted on X that he had reproduced the proof using a publicly available model.

Mathematician Thomas Bloom of the University of Manchester in England had a similar reaction. He said in the outside-experts paper on the achievement that it would have been “truly incredible” if the AI had managed to prove the conjecture. because that kind of solution would require creative insight.

Bubeck, for his part, acknowledged a limitation in the way the AI worked. The proof “isn’t exactly the spark of genius that we see sometimes in mathematics.” He said AI still struggles to make leaps of discovery, but can “patiently slog” through a huge number of unlikely strategies.

That patient search is part of what makes the guardrails feel necessary.

The declaration calling for safeguards frames AI not just as a new instrument, but as a new risk. Experts have pointed to how AI technology can threaten the ability to produce responsible, verifiable and ethical mathematics.

One concern is reliability. In this case, Bloom said the proof was relatively easy for a human expert to verify. But he has seen people online claim they have solutions to open problems—using AI to generate hundreds of pages of math that they can’t understand or even read. “It could be right. It could be nonsense. Who’s going to be able to check this?” Bloom asked.

Bloom said that if mathematicians knew the probability that an AI-generated proof was correct, that would help. But Wood said OpenAI does not share all the times their internal model failed to solve an open problem in math. and does not share times it produced an incorrect solution with flawed reasoning.

OpenAI also hasn’t said how much time the model spent working on its solution.

Bubeck said the team ran their prompt on the Erdős conjecture through the same model multiple times. He said it produced the correct solution in 50 percent of those trials. Lijie Chen. a colleague of Bubeck. said the new model is better than current models at generating an “I cannot solve it” response when it runs into difficulty. But data supporting those claims have not been released or peer-reviewed.

There’s another problem—one that sits uncomfortably close to how mathematicians work day to day: credit.

Right now, Wood said, AI generates mathematical reasoning without showing what work inspired the ideas. That clashes with standard mathematical practice, where breakthroughs are tied to the lineage of prior concepts and contributions. “LLMs have read ALL the papers. They have read all the commentary and notes, and everything that’s online…. It’s not clear that there’s a way for [AI] to reasonably attribute the source of the ideas. ” Wood said.

Access is the third pressure point.

Bloom said that if the most powerful tools are expensive and private, mathematics could become less open and democratic. He also said some people may wonder why they should learn math at all if the work can be outsourced to expensive systems.

Even with those concerns, there is still a thread of cautious optimism.

Wood said, “I do think [AI] is going to become an indispensable tool in mathematics.” The question now is whether the math community can adopt that tool without losing the standards that make proof more than persuasion—standards like transparency, verifiability, attribution, and openness.

For now, the unit distance problem has gained a new chapter. Erdős’ conjecture—posed in 1946—has been disproved by a proof posted May 20. The AI that produced it may not be publicly available yet. but the debate it triggered is already out in the open. with 1. 590 signatures and counting behind the push for guardrails by June 5.

unit distance problem Erdős conjecture OpenAI AI in mathematics mathematical proof guardrails Melanie Matchett Wood Thomas Bloom Sébastien Bubeck Lijie Chen