The math problem that has troubled humanity for 80 years has been solved by AI! Altman: My feelings are complicated.
An AI has achieved something that human mathematicians failed to do for nearly 80 years.
On May 21, OpenAI officially announced that an internal general reasoning model had independently overturned the “Planar Unit Distance Conjecture” (Erdős Unit Distance Problem)—a geometric challenge proposed by Hungarian mathematician Paul Erdős in 1946, which has troubled the mathematical community for 79 years.
The news shocked both the mathematics and AI communities.
OpenAI CEO Sam Altman reposted the news on X, writing a meaningful comment: “Feelings are complicated.”
Fields Medalist and Cambridge University professor Timothy Gowers called it a “milestone for AI mathematics.” To express his astonishment, he posted: “If you are a mathematician, you might need to make sure you are sitting down before you read further.”
This time, OpenAI didn’t mess up again
This is not the first time OpenAI claimed that AI solved the “Planar Unit Distance Conjecture,” also known as the Erdős problem.
Seven months ago, OpenAI’s former Vice President Kevin Weil had posted on X in a high-profile manner: “GPT-5 found solutions to ten previously unsolved Erdős problems!”
But very quickly this was debunked—GPT-5 had merely located existing answers in the literature, without actually solving the problems. Google DeepMind CEO Demis Hassabis and Meta Chief AI Scientist Yann LeCun both mocked the claim, and Weil promptly deleted his post.
This time, OpenAI was clearly more cautious.
The company released the proof along with a joint “companion paper” signed by several top mathematicians, including: Fields Medalist Timothy Gowers, Princeton mathematician Noga Alon, Toronto University number theorist Arul Shankar (of Indian descent), and mathematician Thomas Bloom, who maintains the Erdős problem website—Bloom was the one who earlier deemed Weil's post as “grossly misleading.”
These mathematicians independently reviewed the AI’s proof and confirmed its validity.
What mathematicians say
Fields Medalist and Cambridge University professor Timothy Gowers called this result in the explanatory paper:
A milestone for AI mathematics.
Princeton University renowned number theorist Arul Shankar stated:
In my opinion, this paper demonstrates that current AI models are not just assistants for mathematicians—they can generate genuinely original, creative ideas and carry them through to completion.
Princeton combinatorial mathematician Noga Alon said:
"Erdős himself mentioned this problem several times in lectures, and I heard him speak about it. It’s fair to say every mathematician studying combinatorial geometry has thought about this problem… In my view, the solution to this problem by OpenAI’s internal model is an outstanding achievement. The correct answer was not in the previously conjectured form, which is surprising, and its construction and analysis skillfully and elegantly used rather complex tools from algebraic number theory."
Mathematician Thomas Bloom—the one who criticized OpenAI as “grossly misleading” seven months ago—wrote in the explanatory paper:
AI is helping us explore more fully the mathematical cathedral we have built over centuries. What unseen wonders are still waiting?
Alexander Wei, an OpenAI core research scientist active in both AI and mathematics circles, posted five tweets expressing his shock:
Ten months ago, I was ecstatic because AI could win IMO gold medals. Today, that excitement feels insignificant: an internal OpenAI model has overturned the Erdős Unit Distance Conjecture—a result that could be accepted for publication in the Annals of Mathematics without hesitation.
Mathematics is a precursor of what’s to come. Very soon—perhaps quicker than any of us imagine—AI will begin autonomously producing milestone results in computer science, physics, economics, biology, and other fields. We should prepare for a new world where the nature and methods of science will change.
Why is this problem so hard?
The question itself is not complicated:
On a plane, randomly place n points. What is the maximum number of pairs whose distance is exactly 1?
This is the “planar unit distance problem.”
It sounds simple, but in nearly 80 years no one has found the exact answer.
Mathematicians long believed the optimal solution looks like this: arrange the points in a square lattice, then scale accordingly, yielding about n1+C/loglogn pairs of unit distance. This growth is just slightly faster than linear.
Erdős conjectured this was the ceiling—no configuration could significantly outperform the lattice.
For 79 years, no one could refute this conjecture, nor prove it correct.
How did AI do it?
OpenAI’s model provided a completely new family of point configurations, for infinitely many values of n, with the number of unit distance pairs reaching n1+δ, where δ is a fixed positive constant.
In other words, it not only surpassed the lattice, but directly overturned Erdős’s conjecture.
Princeton math professor Will Sawin further refined this result, giving the specific δ value: δ = 0.014.
The tools used in the proof were even more surprising.
The breakthrough came from a completely different mathematical field: algebraic number theory—an abstract algebraic theory studying integer extensions and factorization. Specifically, the model used the “infinite class field tower” and “Golod–Shafarevich theory.”
These two tools are familiar to algebraic number theorists, but no one expected they could solve a planar geometry problem.
Princeton mathematician Noga Alon commented: “The fact that the correct answer isn’t n1+o(1) is itself surprising, and the construction uses quite complex tools from algebraic number theory in an elegant and clever way.”
Why does this matter?
This is not just about solving a math problem.
OpenAI emphasized in the announcement that the proof was completed by a general reasoning model, not a system specifically trained for mathematics or this problem.
This means the same reasoning ability—to chain together complex arguments, connect knowledge across domains, and find paths experts may overlook—also applies to biology, physics, materials science, engineering, and medicine.
OpenAI stated: “AI is about to play a very important role in the creative parts of research, most importantly in AI research itself.”
But the announcement also notes that human judgment remains indispensable: “Expertise becomes more valuable, not less. AI can help with searching, suggesting, and verifying. People choose the important questions, interpret results, and decide what to pursue next.”
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