TL;DR
Terence Tao reported that AI tools — including ChatGPT, Aristotle and the Lean proof assistant — produced a solution to Erdős Problem #728 after iterative feedback and formal verification. Tao highlights not only the mathematical result but the emerging capability of AI to rapidly generate and rewrite expository writeups of proofs.
What happened
In a Mathstodon thread, Terence Tao described how a community effort using several AI tools produced a solution to Erdős Problem #728. The problem, originating from questions posed in the 1970s about prime factors of binomial coefficients, had been vaguely formulated; recent reconstructions of the intended statement clarified constraints that ruled out trivial AI-found solutions. An initial AI system named AlphaProof found trivial solutions that were deemed out of the problem’s spirit, prompting an added upper bound constraint on parameters. On Jan. 4, ChatGPT produced a proof under a small-constant interpretation; that argument was formalized in Lean by a tool called Aristotle. Further prompting led ChatGPT to adapt the argument for the large-constant interpretation, and Aristotle repaired minor errors to yield a Lean-verified proof. Participants then used iterative AI-assisted editing to generate multiple natural-language expositions of the argument, culminating in a writeup Tao judged close to acceptable research-paper standard, albeit with room for improvement.
Why it matters
- Demonstrates recent AI capability to produce nontrivial mathematical arguments and to interact with formal proof assistants for verification.
- Highlights the growing role of AI in drafting, rewriting and polishing mathematical exposition, not just generating raw proofs.
- Suggests a potential shift in scholarly workflow: a single authoritative human-authored paper could be accompanied by many AI-generated alternate expositions.
- Raises questions about novelty assessment, literature overlap, and how to evaluate AI-derived mathematical results against existing work.
Key facts
- The reported event concerns Erdős Problem #728 as listed on erdosproblems.com.
- Tao states the solution was produced "more or less autonomously by AI" after human feedback and iterations.
- The original problem statement contained vagueness; participants reconstructed the statement in the intended spirit.
- AlphaProof first produced trivial solutions that were ruled out by adding the constraint a,b ≤ (1−ε)n.
- On Jan. 4, ChatGPT produced a proof under a small-C interpretation; that proof was formalized in Lean by Aristotle.
- ChatGPT later adapted the argument to handle a large-C interpretation; Aristotle automatically repaired minor errors and produced a Lean-verified proof.
- Multiple participants used AI tools to rewrite and improve natural-language expositions; Tao reviewed these and found one close to research-paper quality but improvable.
- Tao notes that, to the best of their knowledge, this particular result has not been replicated in existing literature, though similar results proven by related methods were located.
What to watch next
- Development of objective measures to assess how novel an AI-derived result is relative to existing literature, an idea raised in the thread.
- Not confirmed in the source: whether journals or publishers will adopt consistent policies for authorship, verification, and citation of AI-assisted mathematical proofs.
- Adoption of workflows combining AI text generation with formal proof assistants (e.g., Lean) for drafting, verifying, and iteratively revising mathematical papers.
Quick glossary
- Erdős problem: A problem originating from questions or conjectures associated with the mathematician Paul Erdős; typically posed in number theory or combinatorics.
- Lean: A formal proof assistant software system used to write machine-verifiable mathematical proofs.
- ChatGPT: A large-language-model-based conversational AI that can generate natural-language text and, with prompting, outline or produce mathematical arguments.
- Formal verification: The use of formal logic and software tools to check that a proof satisfies the rules of a formal system without informal gaps.
Reader FAQ
Was the proof produced entirely by AI without human involvement?
Tao describes the process as largely autonomous but involving human feedback, iterative prompting, and community participants; not fully independent AI discovery.
Had this result appeared previously in the literature?
Tao says the result, as reconstructed and solved, was not replicated in existing literature to the best of their knowledge, though similar results exist.
Which AI tools were involved?
The thread names ChatGPT for natural-language proofs, Aristotle for formalization and error repair, AlphaProof for earlier exploration, and the Lean proof assistant for verification.
Will the final research paper be AI-generated?
Tao expressed a preference for the essential portions of the final writeup to be human-generated, while allowing AI/Lean to handle some routine proofs.
Back Terence Tao @tao@mathstodon.xyz Recently, the application of AI tools to Erdos problems passed a milestone: an Erdos problem (#728 https://www. erdosproblems.com/728 ) was solved more or less autonomously by…
Sources
- AI solves Erdos problem #728 (Terence Tao mathstodon post)
- Examples for the use of AI and especially LLMs in notable mathematical …
- AI solved a 30-year math problem in just 6 hours. Terence …
- The story of Erdős problem #1026 | What's new – Terence Tao
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