OpenAI made a breakthrough on 80 year old maths problem

Over the past year Artificial Intelligence (AI) has been getting much better at maths and even won some Maths Olympiads. Now OpenAI has claimed a further advance in AI reasoning after its technology successfully tackled an 80-year-old maths problem.

The company behind ChatGPT said it had with a challenge first posed by Hungarian mathematician Paul Erdős in 1946: the planar unit distance problem.

The question posed by Erdős is simple to explain. If you take a sheet of paper and add some dots, how many pairs can be the same distance apart? Erdős proposed the number would rise only slightly faster than the number of dots themselves.

OpenAI’s model concluded otherwise by drawing on different branches of mathematics to uncover a family of arrangements that break the limit in Erdős’s conjecture.

“For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids,” OpenAI wrote on X. “An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better.”

While the work has excited mathematicians, the broader problem remains unsolved because the AI did not come up with a new answer for how fast the pairs of dots rise, but merely showed that the limit Erdős proposed was too low.

OpenAI, which is preparing to float on the US stock market, said the calculations had been made by a general-purpose reasoning model – which breaks down problems into smaller steps – rather than a system trained specifically for mathematics.

The startup has been tripped up before by its attempts to solve Erdős’s problems, having hailed a supposed breakthrough last year that was in fact based on already existing literature absorbed by the model. This time, OpenAI’s work has been validated by mathematicians, including Thomas Bloom, a mathematician who maintains the website and criticised OpenAI’s prior Erdős claims.

Bloom co-authored a companion paper to OpenAI’s blog post flagging the Erdős achievement. Bloom wrote that the AI system had attained its results by “persevering down paths that a human may have dismissed as not worth their time to explore.”

However, he added that humans had been involved in the AI’s work.

“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 in discussing, digesting and improving this proof, and exploring its consequences,” he wrote.

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

Andrew Rogoyski, of the Institute for People-Centred AI at the University of Surrey, said the announcement showed AIs were giving humans new ways to look at problems.

“It’s becoming clear that AI is impacting the world of creative thought and will become a fundamental tool of future scientific research,” he said.

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