AI Systems Solve Erdős Problem Without Specialized Training

Liam Price, who has no formal mathematics training, used ChatGPT last month to solve Erdős problem #1196. The problem, posed in 1966, concerns primitive sets of whole numbers in which no number divides another. Price collaborated with Cambridge undergraduate Kevin Barreto on earlier Erdős problems.

Mathematician Jared Duker Lichtman at Stanford University posted that the solution used a strategy no prior human solver had considered. Terence Tao at the University of California, Los Angeles, noted that GPT solved the problem in its original formulation rather than converting it to probability language.

Daniel Litt at the University of Toronto described the result as reasonably interesting.

Bubeck at OpenAI stated that a year earlier researchers expected large language models to remain limited to their training data. Thang Luong, who leads the Superhuman Reasoning team at Google DeepMind, said models tested internally can now produce proofs up to ten pages.

Current public models remain limited to proofs of three or four pages. Lauren Williams at Harvard University said human referees already face heavy workloads evaluating mathematics papers and that AI-generated submissions are increasing that burden.

She added that models can produce outputs that appear convincing but require substantial time to verify. Luong said scaling compute and improving algorithmic efficiency are expected to extend proof length further. He noted that reaching one-hundred-page proofs is not currently possible but remains a stated goal.