An artificial intelligence (AI) model has just achieved something that for decades many mathematicians considered practically impossible.

OpenAI announced that one of its systems managed to refute a conjecture widely accepted for nearly 80 years about a famous geometric problem associated with the famous Hungarian mathematician Paul Erdős, whose intuition had guided much of the research on this topic for generations.

The question in question, on the surface, is simple. If points are placed on a plane, how many pairs can be exactly the same distance away? Erdős formulated the so-called “unitary distance problem” in 1946 with the suspicion that the best way to maximize these connections was to organize the points on a kind of grid. For decades, many mathematicians believed that their intuition was correct.

An unexpected path: algebraic number theory

The path that AI took was as surprising as the result itself. As the company explained, the system discovered much more efficient configurations by turning to algebraic number theory, an extremely abstract branch of mathematics that until now had not played a central role in this geometric problem.

Thus, instead of following traditional grid-based strategies, the model worked with much more complex mathematical structures and managed to transform them into new arrangements of points within the plane. The result was a construction so intricate that, according to researchers linked to the project, it would be extremely difficult to represent it clearly on a sheet of paper.

The reaction among mathematicians was immediate. “It's a problem I didn't expect to see solved in my lifetime,” Misha Rudnev of the University of Bristol told New Scientist. “It's absolutely a bomb.”

Fields Medalist Timothy Gowers – recipient of the most prestigious prize in mathematics, considered by many to be the equivalent of the Nobel Prize in this discipline – wrote that if a human researcher had submitted this work to the Annals of Mathematics, one of the world's leading mathematical journals, he would have recommended it for publication without hesitation.

Along the same lines, mathematician Will Sawin of Princeton University told New Scientist that it was the “most significant achievement of AI” in mathematics to date.

A general purpose model

Most surprising is that OpenAI claims that the model was not specifically trained to investigate advanced mathematics. It was, according to the company, a general reasoning system. Still, he managed to produce hundreds of pages of arguments and calculations that were later reviewed and validated by outside specialists tasked with examining the proof.

The progress, however, does not mean that the problem has been completely closed. The AI ​​did not find the definitive answer as to what the exact maximum number of possible pairs is, but rather showed that the limit proposed by Erdős was too low. That is, the mathematician's famous intuition was wrong.

Why didn't any mathematician try it before?

Several experts stressed that the breakthrough was not about inventing completely new mathematics, but about combining existing ideas in ways that humans had never explored. That seems to be one of the great current strengths of AI.

For decades, many mathematicians invested enormous efforts in trying to prove that Erdős was right, not in disproving him. And those who did look for counterexamples would hardly have followed such a complex and unpromising path for too long.

AI, in contrast, can explore lines of mathematical reasoning for much longer and continue investigating paths that many human researchers would likely have abandoned before.

Thomas Bloom, a mathematician specializing in Erdős problems, explained in The Guardian that the system obtained results “by persevering in paths that a human could have ruled out.” For his part, Jacob Tsimerman summarized the difference to Scientific American with a striking image: AIs are capable of exploring “more dangerous waters” for much longer without exhausting themselves.

That doesn't mean human mathematicians are out of the game. In fact, several researchers stressed that the original proof had to be reviewed, reorganized and refined by specialists before being made public.

Melanie Matchett Wood also warned of a potential problem in the way the system operates. As he explained to Scientific American, AI tended to present ideas already existing in mathematical literature as if they were completely original. “If a human had been familiar with those results and had not accredited them, it would be professional negligence,” he said.

Can AI replace human mathematical creativity?

The general consensus seems clear. Many consider this episode as the first major mathematical result obtained autonomously by an artificial intelligence in an open problem of great importance. And although no one yet believes that current models can replace the deepest human creativity, the announcement leaves a feeling that is difficult to ignore: AI no longer just helps calculate. Now you also begin to discover.

Daniel Litt, a mathematician at the University of Toronto who participated in the external review of the proof and was consulted by Scientific American, summed up the moment with a mix of excitement and discomfort. “This is the only interesting result produced autonomously by AI so far,” he said. But then he added an even more disturbing phrase: “My suspicion is that we are about to discover that, in reality, they are not that rare.”