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What you'll discover in this article
- An 80-year-old open mathematical problem was recently solved by generative AI.
- Princeton University's Noga Alon told IFLScience the result is "really challenging the way mathematical research looks."
- The news has sparked deep questions about the future of mathematics and how the field will adapt to the rise in AI.
- “I truly think that there is a place for AI use in research mathematics,” said Thomas Chen, a mathematical physicist at the University of Texas at Austin. “But […] it needs to be clarified what that role should be.”
On May 20, 2026, a group of researchers made a surprising announcement: they had, they claimed – and reviewers would later agree – disproved a longstanding open problem originally set by one of the most famous names in modern math. After 80 years, the Erdős unit distance conjecture was finally closed.
But that wasn’t what caught people’s eye. What made this result really special wasn’t so much what was proven, but who proved it: an AI.
“This is […] the most spectacular single result,” says Noga Alon, a combinatorist at Princeton University who reviewed the proof in a companion paper for OpenAI. “But you see now almost every few days there is some new AI proof. It’s really changing the way mathematical research looks.”
Whether that change is a good thing, though, is less clear. For some mathematicians, the result is something of a warning sign: a reminder that, for all its perceived usefulness, AI is still mostly an unknown quantity in math – and one which, if left unchecked, could feasibly shake the discipline on a fundamental level.
“I truly think that there is a place for AI use in research mathematics,” says Thomas Chen, a mathematical physicist at the University of Texas at Austin. “But […] it needs to be clarified what that role should be.”
“Involving a black box into the work stream of rigorous mathematics is something that has never happened before,” he tells IFLScience. “And I think that it probably shouldn't. This really scratches at the very foundation of what mathematics as a human endeavor should be.”
A little leap
Objectively speaking, the unit distance conjecture isn’t much of a big deal. It’s small, self-contained; “it's not like the Riemann hypothesis, where people think that a solution would revolutionize mathematics,” Alon tells IFLScience. “I don't think that the result itself would have really far-reaching consequences. But still, it's very nice.”
It goes like this: given some arbitrary number of points in a plane, what’s the maximum number of pairs that sit the same distance away from each other?
Like so many of the problems posed by Paul Erdős, it sounds easy. “It's very, very simple to state,” Alon says, “so you get the impression – which maybe is a wrong impression – that you understand what's going on.”
Add to that Erdős’s penchant for offering monetary rewards for problems he was particularly interested in seeing resolved, and it’s almost surprising the unit distance conjecture hasn’t been proven yet. Erdős even did some of the legwork himself, offering up a potential solution: for n distinct points, he conjectured, there would be at most n1+O(1/log logn) pairs that were an equal distance apart. All anybody had to do was confirm it.
“The feeling was that it would probably be solved,” says Alon. And yet, after 80 years of mathematicians attacking the problem from various angles, a proof still eluded the world.
As it turns out, there was a very good reason nobody could prove the conjecture. When OpenAI was tasked with resolving the problem, it instead produced something nobody expected: a counterexample.
Even more surprisingly, it wasn’t even that difficult to find – it was, University of Manchester arithmetic combinatorist Thomas Bloom wrote in his section of the companion paper reviewing the result, “a natural, albeit highly non-trivial, generalisation of the original lattice-based construction of Erdős.”
So why did it take the rise of generative AI for it to be found?
The strengths of AI
In context, the answer to that question is not so mysterious.
For a human to have come up with the counterexample would have required a few unlikely things to happen at once: they would have to be not just familiar with the problem, but dedicated to it; they would need a background in class field theory, a fairly unrelated area of math unless you look at the entire problem slightly sideways.
“The AI was using some pretty sophisticated tools from algebraic number theory,” Alon explains. “Maybe in retrospect, it's not completely unexpected that it's related, but it's not the first thing you think about.”
They would also need the right temperament: perseverant long past the point at which most people would give up – “human authors, we often have some ideas,” Alon points out, “but then you try, and if after a month you don't succeed, then you kind of convince yourself that it's not going to work.” Perhaps most importantly, they’d have to be contrary enough to be trying to disprove a conjecture that most everybody agreed was all but certain to be correct.
But it was precisely these requirements that made an AI so well-suited to solving the problem. “There aren't so many mathematicians that know discrete geometry and enough algebraic number theory to try this approach,” Alon says. “And even if you decide to try it, there are still lots of things that can go wrong.”
“But AI has an advantage,” he jokes. “It knows everything.”
AI has no ego; no time to waste. It can follow a “thought”, for want of a better term, until it’s proven to be a dead end, rather than quitting on a hunch and a rumbling stomach. It does, in a way, know everything: trained, often controversially, on vast tracts of human output, it can source and summarize pretty much any result or paper out there in mere seconds.
“It is very helpful in retrieving information very, very quickly,” admits Chen. “What used to be days spent in the library is now two seconds using one of the AIs. That is, I think, a good aspect of how AI informs research.”
For problems like the unit distance conjecture, then, AI is invaluable.
“The Erdős-type problems, you know, coming from […] combinatorics and discrete probability and things like this,” says Chen; those are “probably more exposed [because] if you can formulate your question in such a way that you don't need to spend 10 minutes explaining the conceptual background, you can just start [the AI].”
The AI models are out there, people can use them, and they seem to perform very well. But the creators of these models don't actually really know, or seem to know, why.
Thomas Chen
Of course, it’s not perfect – AIs can miss references, or misattribute them, or just plain make them up; they can misinterpret results, or produce “complete nonsense,” Alon points out. “You have to be very suspicious.”
AIs are not all that great at abstract reasoning; they struggle with geometry; they don’t exactly understand what a “proof” really is, even.
“It's sometimes okay checking a calculation, or performing a reality check,” says Chen. “But in many of the problems that I've been thinking about, it's been producing an output that's maybe 60 percent correct.” Even then, he tells IFLScience, you need to take that 60 percent “with a grain of salt”.
A bigger issue
For all its flaws, AI is likely here to stay. That’s not a bad thing, exactly – computer-aided proofs have been an established, if controversial, part of math ever since the four color theorem was first solved back in 1976. “But in those uses of computers, the mathematicians involved them for proofs with full control,” Chen points out.
“What the algorithm looks like, why it works, why it converges, and what the error estimates are – it’s all in the papers,” he explains. “So, from the beginning to the end of the conclusion, the computer part is completely under control.”
Not so with the new generation of AI-aided or generated proofs, he says. When you type a problem into a chatbot, an answer pops up very quickly – but what actually went on in the 0.2 seconds in between is a mystery, both to you and, more worryingly, everybody else.
“The individual mathematician [does not] have access to the code, nor do the programmers of the code know why it works, nor does anyone claim to know why and how it works,” Chen points out.
“The AI models are out there, people can use them, and they seem to perform very well,” he says. “But the creators of these models don't actually really know, or seem to know, why.”
See, the problem with AI in math isn’t simply that it sometimes produces incorrect answers. It’s that, for a discipline defined by its reliance on transparency and logic, ceding the nitty-gritty of a calculation like this is pretty much anathema.
Proofs are rarely valuable only for the confirmation of a particular proposition: they’re a record of the mathematician’s thought processes; their intuition; the conceptual leaps they made, and the clever methods they invent to get around knotty problems.
Put more simply: give Fermat’s Last Theorem to an AI, and you maybe, eventually, get a confirmation that it’s true. Give the same problem to a mathematician, and you get the modularity theorem, elliptic curve cryptography, breakthroughs into the Langlands program, and that confirmation. It may sound cheesy, but often, the real proofs are the friends we make along the way.
“In my personal experience, AI performs very well with relatively basic calculations, or proofs for questions which are very well defined,” Chen says. But often, “it’s not about just solving the equation […] it requires a lot of human intuition [to tackle] conceptually complex questions.”
“That’s where the art of mathematics lies,” he says.
A movement for the future
Both mathematicians agree: the increasing use of AI in their discipline is likely unavoidable. “I'm not sure if I'm ‘pro-AI’ or not,” Alon says. “But I think it doesn't matter now if you are pro or against, because it's pretty clear that basically everybody would have to use it, I think.”
But moving with the times is one thing. Giving in completely is very different – and math isn’t going down without a fight.
[Mathematics is] also about asking questions, deciding what is important, what is not, developing some taste, and maybe making the right definitions.
Noga Alon
The Leiden Declaration, a statement launched only last month and signed by over 3,000 scientists and mathematicians at time of writing, advocates against adopting AI into math uncritically: to do so would “threaten [mathematical] values” such as clarity, transparency, evaluation, and autonomy, it warns, replacing them with a black box and the incentives of techno-capitalism.
It's a gloomy prognosis – but it doesn’t have to be. Math has dealt with the rise of pocket calculators, and smartphones, and Mathematica, and while each has reshaped to discipline to a certain extent, none has managed to destroy it.
“The future is unavoidable,” says Chen. “So mathematicians have to accept the fact that AI will be part of their workflow one day. But I think – and this is why also I got interested in AI explainability and safety – is that mathematicians thereby inherit the responsibility to figure out how it works.”
If computer and machine learning scientists can’t illuminate the black box, he argues, then it’s up to mathematicians to do so – at least if they’re going to rely on it for their work. “And I think there's a lot that can be done,” he says. “It's a very young field, and nobody should be blamed for it not being completely transparent yet. I mean, it's a work in progress.”
With the rise of AI, mathematicians are having to grapple with some pretty tough questions about their chosen discipline.
In practical terms, they will need to personally check every theorem referenced; every proposition; every citation. They will need to ensure any new results make sense, as well as all the working along the way. They will, as Chen advises, need to figure out what exactly is going on inside that black box.
But those are just the surface problems. A more existential question looms: just how much can math really be ceded to AI before we lose something fundamental along the way?
“Mathematics is not only about proving theorems,” says Alon. “It's also about asking questions, deciding what is important, what is not, developing some taste, and maybe making the right definitions. And at the moment, these AIs are especially good in finding proofs and not in these other parts.”
“But I think that this is not something fundamental,” he adds. “I think it's likely that they would be able to do that [too]. It's only that it's harder to measure it.”
No doubt we’ll see quicker than we expect whether that’s true. But one thing’s for sure: whatever happens, Alon says, “it's going to be interesting.”
The proof is available via OpenAI.
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