Amateur mathematicians have stunned professionals by using AI tools like ChatGPT to tackle long-standing problems posed by Paul Erdős. While most solutions rediscover existing results, one new proof highlights AI's potential to transform mathematical research. Experts see this as an early step toward broader applications in the field.
Paul Erdős, a renowned Hungarian mathematician, left behind over 1,000 unsolved problems upon his death in 1996, covering areas from combinatorics to number theory. These problems, often simple to state but difficult to solve, serve as benchmarks for progress in mathematics. Thomas Bloom at the University of Manchester maintains a website tracking these challenges.
Starting in October 2023, enthusiasts began feeding Erdős problems into AI chatbots like ChatGPT. Initially used to locate relevant literature, the tools soon generated partial improvements and proofs. Undergraduate Kevin Barreto at Cambridge University and amateur Liam Price targeted problem 728, a number theory conjecture. Using ChatGPT-5.2 Pro, they obtained a sophisticated argument, which they verified with Aristotle, an AI from Harmonic that translates proofs into the Lean programming language for automated checking.
By mid-January 2024, AI had fully resolved six Erdős problems, though professionals later found five had prior solutions in the literature. Barreto and Price's work on problem 205 stands as the sole novel resolution. Additionally, AI contributed fresh partial solutions or enhancements to seven others, some linking to overlooked papers.
This raises questions about novelty versus rediscovery. Bloom notes that AI often reformulates problems to uncover hidden connections: “A lot of these papers, I wouldn’t have found... without this sort of [use of] the AI tool.” Barreto acknowledges the problems as relatively straightforward, predicting tougher ones, including those with prizes, remain beyond current AI capabilities.
Kevin Buzzard at Imperial College London calls it “green shoots” of progress, not yet a threat to experts. Terence Tao at UCLA suggests AI could enable a more empirical approach: “We don’t do large-scale mathematics because we don’t have the intellectual resources, but AI is showing that you can.” Bloom envisions expanded research breadth, allowing mathematicians to draw instantly from unfamiliar fields without extensive learning.
