By
Konstantin Kakaes
Contributing Writer
August 3, 2026
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artificial intelligence
combinatorics
Erdős conjecture
features
foundations of mathematics
mathematics
proofs
All topics
O
n May 20, 2026, OpenAI made an announcement that shook the mathematical world. An internal AI model — one not available to the public — had come up with a
counterexample to the “unit distance” problem
, a conjecture made in 1946 by Paul Erdős, the prolific, itinerant Hungarian mathematician.
Erdős posed thousands of questions, but this one was special: It was both simple to explain and mathematically deep. It was the first historically significant proof to come from an AI model. Though the model’s result wasn’t definitive — human mathematicians would substantially improve on it within weeks — it was innovative, bringing in ideas from a distant branch of math that no one had successfully applied to this problem before. And it was influential: Within a few days, related techniques were used to solve other important problems.
Then on August 1, OpenAI
announced
that an unreleased model named Astra made 10 additional mathematical advances, including finding solutions to three more problems posed by Erdős.
Many mathematicians have hailed developments such as these as a phase transition in the mathematical capability of AI models. These models are “changing dramatically the way mathematical research is being done,” said
Noga Alon
of Princeton University, who has solved dozens of Erdős problems over his decades-long career.
Paul Erdős, one of the most prolific mathematicians in history, was deeply whimsical when it came to mathematics, and deeply cynical when it came to authority.
Photo by George Csicsery from the documentary
N is a Number: A Portrait of Paul Erdős
©1993. All Rights Reserved.
Erdős and his conjectures have long fascinated mathematicians. He traveled constantly — living out of a suitcase for years at a time, staying with friends, owning almost nothing. He rattled off problems in published papers and letters to mathematicians around the world, often attaching prize money that he would pay out of pocket to the first person to come up with a solution. The reward might be a token $10 or $25, or, for problems he considered important or difficult, it could range into the thousands. Erdős died of a heart attack in 1996 while attending a math conference in Warsaw, but a nonprofit foundation based in Iowa
has promised
to make good on his bounties.
He was a beloved figure, but also a downright weird one. He only wore silk, and he avoided the touch of other people. Deeply cynical about authority, he gave away most of the money he earned and relied on a friend to manage his finances and other practical affairs. He referred to God as the “Supreme Fascist” and fueled his incessant output of mathematical ideas with a steady diet of amphetamines. It is a strange irony of history that the problems he suggested have now become a central proving ground — and, in effect, a series of PR coups — for the world’s biggest and most powerful technology companies.
But in all likelihood none of this would have happened had it not been for an English mathematician named
Thomas Bloom
.
Many Meetings
Like Erdős, Bloom was interested in both number theory and combinatorics. His focus has been an area called arithmetic combinatorics, which lies at the intersection of the two. After getting his doctorate in 2014, Bloom established himself as a rising star in the field, landing a
prestigious fellowship from Britain’s Royal Society
, which let him work at almost any university he wanted to. (He’s now at the University of Manchester.)
Bloom has liked Erdős’ style for as long as he can remember. But he always found it hard to keep track of which problems had been solved and which had been forgotten entirely. So in early 2023, he decided to gather as many problems as he could into a list.
He intended it for his own use. But “I thought it would be easier if I could access it wherever I was,” he said; he figured he “might as well make a website, kind of with the expectation that maybe nobody would use it.” He gathered a couple hundred problems and launched
erdosproblems.com
. Bloom used ChatGPT to write the Python code that ran the website, which was, at the time, a remarkable thing for a large language model to be able to do. Using one to collaborate on the math itself still seemed like only a distant possibility.
Thomas Bloom’s website of Erdős problems became a home for mathematics at its best. Then AI came on the scene.
Courtesy of Thomas Bloom
His goal was not just to cross items off a list. He wondered if “modern day mathematics, often using techniques unknown by Erdős, could clear up many of these more obscure problems,” he wrote in a
blog post
. “We will then be left with a core of interesting, difficult problems, which can serve to demonstrate the limits of our knowledge.”
Bloom did crucial work in curating the list: Sometimes Erdős stated problems in ambiguous or unclear ways, and Bloom figured out what the most sensible version of each problem should be. He kept adding problems to the site, and gradually its audience grew. Over the course of 2024 and the first eight months of 2025, the statuses of 111 problems on the list were changed from “open” to “solved” (although some of these had been solved years earlier, and their status change reflected the rediscovery or verification of a proof).
Then, in August 2025, some colleagues suggested that Bloom add a commenting function, so that people could talk about problems they were interested in. He was able to do so quickly, using ChatGPT to write the code. By now he’d cataloged nearly 1,000 problems.
Bloom’s timing was good. He made it possible for like-minded people to talk to one another, and that “really let a community build up,” he said. For the most part, comments were sporadic — a problem might attract a single comment pointing out an example or noting how hard the problem looked. But activity steadily grew, and some problems catalyzed nuanced mathematical discussions between strangers.
“Tom probably never really realized this, but for me it’s honestly changed my life,” said Wouter van Doorn, the
fourth-most-prolific commenter
on Bloom’s website. Like many people who became active on the site in the autumn of 2025, van Doorn isn’t exactly a professional mathematician. He works “for a company that gets hired by other companies to do customer service support,” as he put it. But he isn’t exactly an amateur either — a decade prior, he almost completed a master’s degree in math at KU Leuven in Belgium. In 2024, spurred in part by how capable he saw LLMs getting, he took a six-month leave of absence from work to focus on math. At the time, while he didn’t particularly want to use AI, he remembers thinking, “Right now I’m still better at mathematics than an AI is, but who knows what it’ll be in a year, two years, five years? If I want to finish these projects, and I want them to be mine, now is the time.”
Problem 1102
We say that $latex A \subseteq \mathbb{N}$ has property $latex P$ if, for all $latex n \geq 1$, there are only finitely many $latex a \in A$ such that $latex n + a$ is square-free. We say that $latex A$ has property $latex Q$ if there are infinitely many $latex n$ such that $latex n + a$ is square-free for all $latex a < n$. How fast must sequences $latex A = \{a_1 < a_2 0$ and $latex \epsilon > 0$ be sufficiently small. Are there infinitely many integers $latex a, b, n$ with $latex a \geq \epsilon n$ and $latex b \geq \epsilon n$ such that $latex a!b! \mid n!(a + b – n)!$ and $latex a + b > n + C \log n$?
Undeterred, he and Price kept at it, and by January 4, 2026, they’d used GPT-5.2 Pro to find a solution to
Erdős 728
, a problem about when certain numbers are divisible by other numbers. This time nobody could find prior work already proving it. Barreto used another AI tool called Aristotle (developed by a startup called Harmonic) to certify that the proof held together logically. Nat Sothanaphan, a software engineer and the only forum participant more prolific than Bloom, Tao, and van Doorn, had
ChatGPT write up the formalized result
and posted it online.
Price developed a methodology for how to ask LLMs to solve open questions. First, he would ask a chatbot for a solution. Then he would feed that solution into a fresh instance of the chatbot, asking it to check the previous chatbot’s work. He’d repeat this process until he had what looked like a workable solution. (This echoes some of the work that companies have been doing internally to create what they call harnesses or scaffolds, which automate the sort of iteration that Price does by hand.)
Barreto and Price’s papers represent just a fraction of the many Erdős problems solved at least in part by AI over the past few months. There are multiple reasons why these problems in particular have become such a fertile test bed for LLMs. The primary one is that, by and large, Erdős problems are in number theory, combinatorics, and graph theory, all areas of math that have proved more accessible than others to large language models. The problems also vary widely in difficulty and mathematical significance. This variation makes them appropriate for a nascent technology whose abilities also vary widely.
Many of Erdős’ problems had a monetary value attached to them from their moment of inception, a playful incentive from a wandering eccentric. But now, as the problems have become an informal benchmark for AI, their solutions are being discussed in terms of their “per-problem cost” — the price of the tokens needed to solve them.
Photo by George Csicsery from the documentary
N is a Number: A Portrait of Paul Erdős
©1993. All Rights Reserved.
“A lot of my recent papers should be mostly credited to AI,” van Doorn said. “The ideas involved were ideas I did not come up with myself.” Like many people active on the Erdős site, van Doorn is excited about the way LLMs are allowing him to do more things more quickly. “If I read an idea by an LLM, I digest it, try to understand it, simplify it, and generalize it,” he said. He uses AI to better understand the math.
Not everyone holds themselves to this standard. “A big problem is AI is being used a lot by people who aren’t mathematicians, who don’t have a huge mathematical background and are not capable of verifying the output,” Bloom said. “They like to move fast, ask their AI to check it, it grows and grows. We’re seeing a lot more of these 100- to 200-page papers that people are posting. ‘I solved this theorem; I got AI to generate the proof and check the proof and write the paper.’ But no human has read it, and no human is going to read it. It’s a huge challenge now.”
Problem 1196
Is it true that, for any $latex x$, if $latex A \subset [x, \infty)$ is a primitive set of integers (so that no distinct elements of $latex A$ divide each other) then $latex \displaystyle\sum_{a \in A} \frac{1}{a \log a} < 1 + o(1),$ where the $latex o(1)$ term $latex \to 0$ as $latex x \to \infty$?
By Price’s own assessment, he doesn’t have enough mathematical understanding to verify the solutions he ultimately coaxes from the LLMs. But with Barreto’s help, he’s been able to find mathematicians knowledgeable and willing enough to check the results. Both Price and Barreto are co-authors with Tao,
Jared Duker Lichtman
of Stanford University, and other accomplished mathematicians on a May 2026 paper
resolving Erdős Problem 1196
, one of their more significant results. (1196 asks about the possible size of so-called primitive sets — collections of integers, such as {2, 5, 9, 21}, in which no number divides any other.)
Bloom was surprised that despite lots of attention from OpenAI, Google DeepMind, and several startups, most of the new results had come from hobbyists and undergraduates using publicly available LLMs, not from corporate labs using more advanced internal models.
But that would change a few weeks later, on May 20, 2026, when OpenAI announced that they had solved one of the most well known Erdős problems of all, the unit distance problem.
Many Partings
In the first months of 2026, the major tech companies began to see opportunity in erdosproblems.com. As Lichtman explained, “Erdős had over 1,000 papers. They were scattered.” An institute in Hungary had collected scanned images of many of the papers, but nobody had collected all the problems. “This kind of single repository that anyone can access — labs realized that this could effectively be a benchmark.”
In January, a team of 24 researchers led by Google DeepMind
shared a paper
solving four problems and finding old, forgotten solutions to nine more, after “using Gemini to systematically evaluate 700 conjectures labeled ‘Open’ in Bloom’s Erdős Problems database.” In May, a separate DeepMind team of 21 researchers
announced
that “our most capable agent autonomously resolved 9 of 353 open Erdős problems at the per-problem cost of a few hundred dollars.” (As of this article’s publication, Bloom’s database contains 565 solved problems and 652 open ones, but the DeepMind team narrowed their search to problems that have been written in formal logic.)
And, on May 20, OpenAI
shared a solution
to the unit distance problem, along with a
blog post
explaining the work and a
companion paper
that featured nine world-class mathematicians commenting on the correctness of the proof and the importance of what had been done (as well as presenting a streamlined human version of the result). Mathematicians had generally believed that Erdős’ conjecture — about how many evenly spaced points can be placed on a plane — was correct. To general surprise, OpenAI’s internal model found a counterexample. To do so, it had found a sophisticated way to use tools from an area of math called algebraic number theory. As
Jacob Tsimerman
of the University of Toronto wrote in the companion article, “This is a really impressive piece of work. … It is definitely an intimidating construction.”
In the same article,
Tim Gowers
of Cambridge and the Collège de France wrote that “if a human had written the paper and submitted it to the
Annals of Mathematics
and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation. No previous AI-generated proof has come close to that.”
The author on the paper that presented the original solution was given simply as “OpenAI.”
Terence Tao was 10 years old when he met Erdős.
Billy Grace Tao
Later, using techniques related to the ones the AI model had applied to the unit distance problem, a group of four mathematicians, including Bloom,
disproved
a version of another long-standing Erdős conjecture. The
“sum-product” conjecture
proposed that if you have sets of numbers, either their sum or their product must grow quickly. The mathematicians found a set of real numbers for which both the sum and the product grow more slowly than expected. The conjecture for integers remains open.
Figuring out what impact AI will have on math and mathematicians means not only looking to its most important results, but also examining how it changes the everyday practice of solving quotidian problems. Noga Alon, the Princeton mathematician, estimates that he has solved a few dozen Erdős problems over his career. He has now stopped trying. “Once AI started to solve them, there is no point anymore,” he said. Terry Tao has stepped away from the Erdős problem community to focus on getting work done.
Van Doorn, who for now still has his day job at a customer service company, said that LLMs “are clearly better at thinking and doing math than I am. I don’t hold a candle to current AI systems.” However, he added, “the eventual proofs that I write are simpler, more general, and easier to read for other people than the thing that ChatGPT came up with.” AI has indisputably boosted his productivity, and he’s still having fun. “If you want to play piano, you aren’t going to hire a piano-playing machine that does it better than you. You will play the piano because you like playing the piano. I enjoy thinking about numbers, doing math, writing papers. I’m not going to hire a paper-making machine that does it for me.”
Related:
The AI Revolution in Math Has Arrived
Mathematical Beauty, Truth and Proof in the Age of AI
When ChatGPT Broke an Entire Field: An Oral History
For van Doorn, there is joy to be found in digesting the responses he gets from LLMs. “I’ve been doing a lot of math recently thanks to the Erdős-problems community. It used to be the case I just did everything all by myself, struggled alone in my room. I don’t know how it happened, but nowadays people contact me saying, ‘I have this idea. Do you want to join me thinking about this?’”
The increasing capability of AI has made it easier for people like Price or van Doorn, with less mathematical training, to solve puzzles, while making those puzzles less interesting to people like Alon who have devoted a lifetime to understanding them.
Nonetheless, “many and maybe most good mathematicians will use AI,” Alon said. He notes that a number of first-rate mathematicians have left academia to work at AI companies, not only because they are well paid to do so but because “maybe this is where the action now is.” In July 2026, on the same day that
Tsimerman was awarded the Fields Medal
, the highest honor in math, he announced that he was leaving academia for a job at OpenAI.
What Is the Unit Distance Problem?
Say you want to arrange points in a plane to create pairs of points that are the same distance apart. How can you create the most pairs?
If you arrange your points so that they form polygons like squares and pentagons, you will get the same number of pairs as you have points.
Mark Belan/
Quanta Magazine
But if you put points in a lattice, you can create substantially more equidistant pairs.
Erdős conjectured that it wasn’t possible to do substantially better than this. For 80 years, mathematicians generally believed he was correct.
But Erdős was mistaken. OpenAI found a pattern (similar to the one shown below) that, for a given number of points, produces more pairs than a lattice can.
Mark Belan/
Quanta Magazine
; source:
Kai Williams
Close
Mark Belan/
Quanta Magazine
; source:
Kai Williams
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