Erdos Problems Solved by AI Models
Paul Erdős did not own a house. He carried his life in two suitcases, sometimes three, and he showed up at colleagues’ doorsteps unannounced, often at three in the morning. The Hungarian mathematician had no fixed address, no steady job, no bank account of his own. What he had was an endless supply of questions.
Between the 1930s and his death in 1996, Erdős posed thousands of mathematical problems. Some were worth $10. Others carried bounties in the thousands of dollars, payable out of his own pocket to whoever solved them first. He was famously eccentric: he wore only silk, avoided physical contact with other people, and fueled his relentless thinking with amphetamines. He called God the “Supreme Fascist” and gave away most of the money he earned.
His questions were deceptively simple to state and fiendishly hard to answer. The “unit distance” problem, posed in 1946, asks how many times a single unit length can appear among pairs of points in the plane. It sounds like a puzzle a child could understand. Mathematicians wrestled with it for eight decades.
In recent years, AI models have begun making inroads into mathematical research, producing results that surprise even seasoned mathematicians. The unit distance problem, which asks how many times a single unit length can appear among pairs of points in the plane, has seen new approaches emerge from AI-assisted research. While no single AI model has yet produced a definitive proof, the techniques being developed are opening new avenues for human mathematicians to explore.
AI models have borrowed ideas from distant branches of mathematics that no one had successfully applied to these problems before. Other researchers have since used related techniques to make progress on different problems entirely.
AI models have begun contributing to solving several Erdős problems, though the pace of progress remains a subject of debate among mathematicians. Some advances have been achieved through human-AI collaboration, with models suggesting novel approaches that humans then verify and refine.
Noga Alon of Princeton University, who has solved dozens of Erdős problems over a decades-long career, said these models are ‘changing dramatically the way mathematical research is being done.’ [1] That statement carries weight. Alon is not prone to hyperbole.
The strange irony is that Erdős, the ultimate itinerant outsider, has become a central proving ground for the world’s most powerful technology companies. His problems are now a series of public relations victories for OpenAI and others. The man who owned almost nothing has posthumously become a benchmark for artificial intelligence.
But none of this would have happened without an English mathematician named Thomas Bloom. And Bloom’s journey to this moment began with a simple practical problem: he could not keep track of which Erdős problems had been solved and which had been forgotten.
Bloom works at the intersection of number theory and combinatorics, a field called arithmetic combinatorics. After earning his doctorate in 2014, he landed a prestigious fellowship from Britain’s Royal Society, which allowed him to work at nearly any university he chose. [5] He is now at the University of Manchester. [2]
In early 2023, Bloom decided to gather as many Erdős problems as he could into a single list. He intended it for his own use, but he wanted to access it from anywhere. So he built a website.
The remarkable part: Bloom used ChatGPT to write the Python code that ran the site. At the time, that was a stunning thing for a large language model to accomplish. Using AI to collaborate on the mathematics itself still seemed like a distant fantasy.
Bloom launched erdosproblems.com with a couple hundred problems. He did not expect much. He figured ‘maybe nobody would use it.’ The site had the look and feel of an earlier internet era, which turned out to be part of its charm.
His ambition went beyond simply crossing items off a checklist. Bloom wondered whether ‘modern day mathematics, often using techniques unknown by Erdős, could clear up many of these more obscure problems.’ He wrote in a blog post that they would then ‘be left with a core of interesting, difficult problems, which can serve to demonstrate the limits of our knowledge.”
Curating the list required real mathematical judgment. Sometimes Erdős stated problems in ambiguous or unclear ways. Bloom had to figure out the most sensible version of each problem. He kept adding 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 changed from ‘open’ to ‘solved.’ Some of those had actually been solved years earlier; the status change reflected rediscovery or verification of a proof. But the momentum was real.
In August 2025, colleagues suggested Bloom add a commenting function. That would let people discuss problems they found interesting. Bloom implemented it quickly, again using ChatGPT to write the code. By then, he had cataloged nearly 1,000 problems.
His timing turned out to be excellent. The commenting function ‘really let a community build up,’ Bloom said. For the most part, comments were sporadic at first. A problem might attract a single comment pointing out an example or noting how difficult it looked.
But activity grew steadily. Some problems catalyzed nuanced mathematical discussions between strangers. The website was becoming an example of the internet at its democratic best.
“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 the site. Van Doorn is not exactly a professional mathematician. He works ‘for a company that gets hired by other companies to do customer service support.”
But he is not exactly an amateur either. A decade prior, he almost completed a master’s degree in mathematics at KU Leuven in Belgium. [3] He left without finishing, went into the working world, and for years mathematics remained a distant memory.
In 2024, something shifted. Van Doorn watched how capable large language models were getting. He took a six-month leave of absence from work to focus on mathematics. At the time, he did not 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.”
In October 2025, van Doorn, now back at his day job, left the first comment on the page for Problem 1102. The problem, which Erdős posed in 1981, asks about properties of sets of ‘square-free’ integers.
Square-free integers have no repeated prime factors. The number 30 is square-free because it equals 2 × 3 × 5. The number 18 is not square-free because it equals 2 × 3 × 3; the 3 appears twice. This distinction, simple as it sounds, opens a door into deep mathematical structure.
On November 1, 2025, van Doorn shared progress toward an answer in a comment on the problem page. He had figured it out without relying on AI. The argument was his own.
Later that day, another commenter replied, claiming to have found a flaw in van Doorn’s reasoning. The two traded remarks in rapid succession. Van Doorn convinced his interlocutor that the argument was correct.
“I see how your argument works now. Nice!’ the other mathematician replied.
That other mathematician was Terence Tao, a professor at the University of California, Los Angeles. [4] Tao is arguably the best-known mathematician alive today, and inarguably one of the most influential. When Tao was just 10 years old, he had crossed paths with Erdős himself.
The encounter on Bloom’s website was a collision of worlds. A customer service worker from Belgium and a Fields Medal winner from UCLA, trading comments on a website built with AI-generated code, solving a problem posed by a man who died in 1996.
“This entire collaboration would not have been possible without Tom’s website and the comments section there,’ van Doorn said. It did not matter whether one had tenure or not. It did not matter if one was young or old. It did not matter if one was at a fancy university or working a day job in customer support.
The website flattened the hierarchy of mathematics. On erdosproblems.com, Tao’s comment carried exactly the same weight as van Doorn’s. The argument either worked or it did not.
Bloom’s curation had created something more valuable than a list. He had built a meeting place. And he had done it at precisely the moment when AI models were becoming capable enough to contribute meaningfully to mathematical research.
The timing was not accidental. Bloom had used ChatGPT to build the site in the first place, and the same technology that made the website possible was now helping solve the problems cataloged there.
Mathematicians have described these developments as a phase transition in the mathematical capability of AI models. The term, borrowed from physics, describes a moment when a substance changes state — ice becoming water, water becoming steam.
The claim is that mathematical research itself is changing state. The old model was solitary: a mathematician alone in an office, scratching at a problem for years. The new model involves human mathematicians, AI systems, and online communities all interacting in real time.
The Erdős problems have become the test bed for this transition because they are so well-suited to it. They are clearly stated, historically significant, and varied in difficulty. Some yield to a clever undergraduate. Others have resisted the best minds for half a century.
Some AI models have produced results that were innovative and influential, even when those results were not definitive. The key contribution has been opening doors to new mathematical ideas.
Human mathematicians have improved on AI-generated results, but the improvements were often only possible because the AI had opened a door. It had brought in ideas from distant branches of mathematics that no one had successfully applied to these problems before.
The pattern emerging from recent research is that AI models are not just assisting human mathematicians — they are suggesting approaches that humans would not have considered. This collaborative dynamic is reshaping how mathematical research is conducted.
Noga Alon has spent decades solving Erdős problems. He has seen the field evolve through multiple technological shifts. His assessment that AI is ‘changing dramatically the way mathematical research is being done’ is not a casual observation. It is a judgment from someone who knows exactly what the old way looked like.
The Erdős problems were always a peculiar institution. Erdős attached prize money to them, sometimes as little as $10, sometimes thousands. A nonprofit foundation based in Iowa has promised to make good on his bounties, even after his death.
The prizes were never really about the money. They were about attention. Erdős wanted people to work on his problems, and the prizes were a way of signaling which ones he considered important.
Now the problems serve a different kind of attention economy. They are a proving ground for AI companies seeking to demonstrate their models’ capabilities. Each solved problem is a headline. Each counterexample is a press release.
The irony would not be lost on Erdős, who was deeply cynical about authority and gave away most of the money he earned. He relied on a friend to manage his finances and other practical affairs. He owned almost nothing.
Now his problems are being solved by the world’s most powerful technology companies, which own vast data centers, enormous computing clusters, and billions of dollars in capital. The man who lived out of a suitcase has become a benchmark for corporate AI research.
Thomas Bloom’s website remains a modest affair. It has the look and feel of an earlier time, before the internet became dominated by platforms and algorithms. It is a simple list of problems with a comments section.
But that simplicity is precisely what made it work. There are no paywalls, no login requirements, no reputation systems. Anyone can read the problems. Anyone can comment. The only thing that matters is whether one’s mathematical argument is correct.
Van Doorn’s experience illustrates the power of this approach. He is not a professional mathematician. He works in customer service support. But he had the training, the persistence, and the timing to make a real contribution.
The contribution he made to Problem 1102 was his own. He did not use AI to figure it out. But the website that made the collaboration possible was built with AI-generated code. The tools are entangled in ways that would have been hard to predict even a few years ago.
Bloom’s original question was whether modern mathematics, using techniques unknown to Erdős, could clear up many of these more obscure problems. The answer, so far, appears to be yes. And the clearing is happening faster than anyone expected.
The remaining problems, Bloom predicted, would form ‘a core of interesting, difficult problems, which can serve to demonstrate the limits of our knowledge.’ That core is still there. But the limits are shifting.
What comes next is not entirely clear, but the direction points toward continued interaction between human mathematicians, AI models, and online communities, all focused on the Erdős problems. The website has become a central meeting point for this work.
Bloom keeps adding problems. The community keeps commenting. The AI models keep producing results. Whatever this phase transition turns out to be, it is happening in real time on a website that looks like it was built in 2003.
Sources
3. KU Leuven