Amateur mathematicians solve long-standing maths problems with AI

Amateur mathematicians are using artificial intelligence chatbots to solve long-standing problems, which has surprised professionals. Although the problems in question are not the most advanced in the mathematical canon, the success of AI models in solving them shows that their mathematical performance has passed an important threshold, the researchers say, and could fundamentally change the way we do mathematics.

The questions solved by AI come from Hungarian mathematician Paul Erdős, famous for his ability to ask useful but difficult questions over a career spanning more than six decades. “The questions were generally very simple, but very difficult,” says Thomas Bloom of the University of Manchester, UK.

At his death in 1996, there were more than 1,000 of these unsolved Erdő problems, covering a wide range of mathematical disciplines, from combinatorics (the study of combinations) to number theory. Today, they are considered indicators of progress in these areas, says Bloom, who runs a website that lists problems and tracks mathematicians’ progress in solving them.

Because Erdő problems are often simple to state, mathematicians have started experimenting with feeding them to AI tools like ChatGPT. Bloom says that in October last year, he started seeing people using AI models to find relevant references in mathematical literature that helped them find solutions.

Soon after, AI tools began seeing partial improvements in results, some of which had been found in previous papers, while others seemed new.

“I was surprised then,” Bloom says. “Before, when I tried ChatGPT, he would make up articles, it was completely mind-blowing, and so I had given up on using it. But clearly there was some sort of change around October. I actually found genuine articles because he had read them all, and often in a non-trivial way.”

Inspired by this progress, Kevin Barreto, an undergraduate mathematics student at the University of Cambridge, and Liam Price, an amateur mathematician, began looking for simple, understudied Erdős problems that they could solve with AI. After finding one of these problems, number 728, a conjecture in number theory, they passed it to ChatGPT-5.2 Pro to solve it.

“I looked at the statement and thought, ‘Maybe this problem could be solved by ChatGPT, so let’s try it,'” says Barreto. “Indeed, it comes back with a pretty cool argument that a lot of people would agree is pretty sophisticated.”

After ChatGPT produced a proof, Barreto and Price used another AI tool called Aristotle, created by AI company Harmonic, to verify their work. Aristotle converted the proof in conventional language into a proof written in Lean, a mathematical programming language. Its accuracy can then be instantly verified by a computer. According to Bloom, this is an important step because it saves researchers limited time to verify whether a result is correct or not.

As of mid-January, six Erdős problems had been fully solved by AI tools, although a later review by professional mathematicians found that five of these problems had already been solved in the mathematical literature. Only one issue, issue 205, was fully resolved by Barreto and Price without any pre-existing solutions. AI tools also enabled small improvements and partial solutions to seven other problems that do not appear to pre-exist in the literature.

As a result, there is an ongoing debate about whether these tools actually prove new ideas or whether they just unearth old, forgotten solutions. Bloom points out that AI models often need to translate problems into new forms and discovers articles that make no mention of the Erdős. “I wouldn’t have found many of these papers, and maybe no one would have found them for much longer without these kinds of documents. [use of] the AI ​​tool,” he says.

Another question is how far this approach can go. Not all of these problems are the most demanding in mathematics and could perhaps be solved by a first-year doctoral student, but it’s still impressive, Bloom says. “To me, it’s amazing that AI is capable of this, because it requires non-trivial effort.”

Barreto also claims that the problems solved are relatively simple, even when compared to the more difficult Erdős problems, which current AI models fail to solve. “Once [AI] Some of the hardest problems have prize money reserved for anyone who can solve them, but Barreto thinks that’s probably not going to happen soon: “Some people are trying to solve problems with bounties, and to me that’s kind of crazy. I don’t think the models are there yet.

Solving the Erdő problems using AI is promising progress, says Kevin Buzzard of Imperial College London, but because most of the problems it solves are either relatively simple or have received little attention, it’s difficult to determine whether it’s a significant achievement – ​​or something that should worry professionals. “It’s progress, but mathematicians aren’t going to look over their shoulders yet,” Buzzard says. “These are green shoots.”

But even if the models’ capacity remains static, their ability to handle relatively complex mathematics could fundamentally change the way researchers search for and write proofs, Bloom says, because it will allow mathematicians with limited knowledge of areas outside their particular discipline to draw on other areas.

“Almost no one knows all aspects of mathematics, which means we are quite limited in the set of tools we can use,” Bloom says. “The fact that you can get an answer instantly, without having to bother another human, without having to waste months learning potentially useless knowledge, opens up a lot of connections. That’s going to be a huge change that we’ll see, just increasing the scope of research being done.”

It could also allow mathematicians to practice a whole new way of working, says Terence Tao of the University of California, Los Angeles, who helped validate some of the AI-assisted solutions to Erdő’s problems.

Mathematicians often focus on a small number of difficult problems due to limited time, while many less difficult but still important problems do not receive much attention. If AI tools can be applied to all of these at once, it could lead to a more empirical and scientific way of doing mathematics, Tao says, where different ways of solving a problem could be tested on a large scale.

“Our resources are so limited by the number of experts we have that we don’t look at 99 percent of all the problems we could study,” Tao says. “So we don’t do things like investigate hundreds of problems, trying to find one or two really interesting ones, or do statistical studies like: we have two different methods, which one is better?

“It’s a type of math that just isn’t practiced,” he says. “We don’t do math at scale because we don’t have the intellectual resources, but AI shows it’s possible.”