Artificial intelligence solved a mathematical problem that stumped scientists for 80 years

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The history of this problem began more than 80 years ago and is associated with the name of Pál Erdős — one of the most prolific and eccentric mathematicians of the XX century, who traveled the world with a single suitcase. The scientist left behind hundreds of scientific works and a whole list of complex questions. Among his favorite problems was the so-called unit distance problem.

The essence of this problem seems simple at first glance: if you place a certain number of points on a sheet of paper, what is the maximum number of pairs of these points that can be exactly one unit distance from each other?

In 1946, Erdős proposed an arrangement of n points in the form of special grids and hypothesized that for any other configuration, the number of pairs of points at a unit distance would not be significantly greater. Erdős once offered a $500 reward for solving this problem. For decades, mathematicians tried to confirm his conjecture, but all attempts ended in failure.

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Pál Erdős. Photo taken in 1992. Photo: Wikimedia Commons

The situation changed when artificial intelligence from OpenAI took up the challenge. Researchers gave this problem to the new model as a test to find out if it was better than previous models, not expecting a sensation.

However, the result stunned even skeptics: AI not only solved the problem, it disproved Erdős's hypothesis. The model showed that the number of pairs of points at a unit distance from each other can grow faster than Erdős had assumed. Thus, his long-standing hypothesis turned out to be false.

Why did people miss this solution for so many years?

Scientists identify several main reasons. One possible reason is that most researchers spent decades trying to prove Erdős's hypothesis. The direction of the search itself was set by the previous history of the problem. The model, however, had no such intellectual limitations and was ready to consider even unlikely paths.

Another reason is related to how modern artificial intelligence works. Humans usually specialize in relatively narrow fields. A model can simultaneously use knowledge from different branches of mathematics. In this case, the solution combined ideas from algebraic number theory and discrete geometry — areas traditionally considered quite distant from each other.

Finally, artificial intelligence possesses virtually unlimited patience. It can spend hours and days checking directions that a human would consider hopeless and abandon.

According to researchers, the model generated a line of reasoning about 75,000 words long — roughly the same as the first Harry Potter book. The AI worked without breaks for sleep, food, or other activities. According to estimates, the entire solution required about 32 hours of work from a powerful server and approximately $1000 in computational costs.

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Schematic representation of the unit distance problem, which mathematicians worked on for almost 80 years. Image created by AI

Artificial Intelligence Changes the Future of Science

Such radical changes in the scientific world can evoke various feelings — from anxiety to excitement. However, the developers at OpenAI themselves look at the future of mathematicians with unexpected optimism. They are confident: artificial intelligence will not destroy the profession, but, on the contrary, will help people reach new heights.

As an example, they cite chess. It was once feared that computer victories over champions would kill interest in the game. In fact, the opposite happened: thanks to machine analysis, people started playing better and understanding strategy more deeply.

AI for a mathematician is like a super-powerful calculator: it doesn't kill curiosity, but expands boundaries. Scientists have already started using the methods found by the model to tackle other complex problems that previously seemed insurmountable.

Of course, one must maintain a sober perspective. Solving one, albeit famous, problem by Erdős is not yet the creation of "superhuman intelligence". But ignoring this fact is no longer possible.

This story became further proof that artificial intelligence is capable of advancing fundamental science in any field where there are complex problems. The hypothesis about the usefulness of AI for major science is already a proven fact. Or, as in the case of Erdős's problem, a brilliant refutation of old doubts.

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