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This mathematical problem, which seems like child's play, is called the planar unit distances problem. Posed in 1946 by the famous Hungarian mathematician Paul Erdős, it turned out to be a real headache. Since 1984, no human had managed to make significant progress in solving it.
That is now a thing of the past: OpenAI announced that one of its internal artificial intelligence models had just cracked this 80-year-old secret, marking a historic milestone for scientific research.
How many unit distances can fit on a single sheet of paper?
Image credit: OpenAI
The feat: an AI that wasn't even a math specialist
The most surprising aspect of this discovery lies not only in the result, but in how it was obtained. The AI model used by OpenAI is a general reasoning system. This means it was not specifically trained for mathematics, let alone this specific problem.
To overcome the stagnation humans had faced since the 80s, the AI took completely unprecedented paths by combining geometry (the study of shapes and spaces) with algebraic number theory (the study of properties of integers). OpenAI researchers even admitted their surprise, because while these algebraic concepts were well known to specialists, no one had imagined they could apply to a planar geometry question together.
Company representatives thus emphasized that this proof constituted a historic milestone for the mathematics and artificial intelligence communities, noting that it is the very first time such an important open problem has been solved autonomously by a machine.
Human and Machine: a team effort
Faced with fears of AI replacing researchers, OpenAI and the scientific community are playing it cool. The algorithm found the key to the problem, but it was human mathematicians who took over to analyze, clean up, and improve the initial proof in order to explore all its consequences.
The enthusiasm is shared by top experts. Tim Gowers, professor at the University of Cambridge and Fields medalist, stated bluntly that the solution to this problem was a historic step for mathematical AI. He added that if a human had written this paper and had been asked for his opinion for publication in the prestigious journal Annals of Mathematics, he would have recommended immediate acceptance without the slightest hesitation.
A comeback after false alarms
This major announcement also sounds like a consecration for OpenAI, which now proceeds with more caution. In October 2025, company executives had claimed a bit too quickly that their model (GPT-5) had solved about ten other Erdős problems, before experts demonstrated that these questions had already been solved by humans in the past.
This time, the discovery was meticulously verified and validated by independent peers. It proves that AI is no longer just regurgitating existing knowledge: it is becoming capable of advancing science at the frontier of the unknown.