Artificial intelligence is rapidly changing the landscape of mathematical research, enabling even amateur enthusiasts to tackle long-standing, unsolved problems. Recent developments show that AI models, particularly large language models like ChatGPT, have crossed a critical threshold in mathematical reasoning, surprising professional mathematicians and hinting at a fundamental shift in how mathematical progress is made.
The Erdős Problem Legacy
The focus of this progress lies with problems posed by the legendary Hungarian mathematician Paul Erdős. Erdős, prolific in his six-decade career, specialized in deceptively simple yet exceptionally difficult questions across combinatorics, number theory, and other fields. Over 1,000 of his unsolved conjectures remained open until recently, serving as benchmarks for advancement in their respective disciplines.
These problems, while elementary to state, often require novel insights to resolve. Mathematicians have begun feeding these challenges to AI tools like ChatGPT, initially as an experiment. Researchers have observed a marked change in AI performance since October, with models now capable of identifying relevant literature and even generating partial or entirely new solutions.
From Hallucinations to Valid Proofs
Thomas Bloom, from the University of Manchester, who maintains a catalog of Erdős problems, recalls that AI initially struggled with basic mathematical tasks. “Before, ChatGPT just made up papers, completely hallucinating,” he says. However, recent improvements have enabled it to retrieve and analyze existing research effectively.
Undergraduate student Kevin Barreto and amateur mathematician Liam Price exemplify this shift. They fed Erdős problem #728 to ChatGPT-5.2 Pro, which produced a proof considered “quite nice and sophisticated.” They then employed Aristotle, an AI tool created by Harmonic, to verify the proof using Lean, a formal mathematical programming language. This automated verification process saves researchers valuable time.
Limited Gains, but Significant Implications
As of mid-January, AI tools had fully solved six Erdős problems, though five were later found to be previously solved. The only verified new solution came from Barreto and Price for problem #205. Additionally, AI has contributed partial solutions to seven other problems, some of which appear to be novel.
The debate now centers on whether AI is genuinely proving new ideas or simply rediscovering forgotten solutions. Bloom argues that AI’s ability to translate problems into new forms and uncover obscure papers is valuable. “A lot of these papers, I wouldn’t have found… maybe nobody would have found for a lot longer without this sort of tool,” he emphasizes.
The Future of Mathematical Research
While the current progress focuses on relatively straightforward problems, experts agree that AI’s impact will extend beyond simple solutions. Terence Tao, at the University of California, Los Angeles, suggests AI could enable a more empirical, large-scale approach to mathematics.
“We are just so resource-limited by how much expert attention we have, that we don’t look at 99 per cent of all the problems that we could be studying,” Tao explains. AI could allow mathematicians to survey hundreds of problems, test different methods, and identify promising areas for further research – something previously impossible due to human limitations.
This shift could democratize mathematical exploration, allowing researchers to draw on broader knowledge bases and accelerate discovery. The current capabilities of AI are still modest compared to the most difficult open problems, but even these “green shoots” represent a fundamental change in how mathematics is done.
