“Erdos problem #728 was solved more or less autonomously by AI”
TL;DR Highlight
Terence Tao used AI tools (Aristotle + LLM) to formally prove unsolved math problem Erdős #728 in Lean.
Who Should Read
Mathematicians and CS researchers interested in AI-assisted formal verification, and ML researchers tracking AI's progress on math reasoning.
Core Mechanics
- Terence Tao (Fields Medal winner) used an AI tool called Aristotle in combination with an LLM to construct a formal Lean proof for Erdős conjecture #728 — a long-standing open problem.
- This is significant because it's not a toy problem — Erdős conjectures are real mathematical research-level challenges, and #728 had resisted proof for decades.
- The proof process was collaborative: Tao provided the high-level mathematical insight and strategy, while Aristotle/LLM handled the Lean proof formalization.
- The result was machine-verified — not just plausible but formally checked by Lean's type system, guaranteeing correctness.
- This demonstrates the emerging pattern: world-class mathematicians can now prove harder theorems faster with AI assistance than without.
- Lean proof assistants become a trust anchor — even if the AI makes reasoning mistakes, the proof checker catches them before they're accepted.
Evidence
- The Lean proof was made publicly available and verified by the community — not just a claim but checkable artifact.
- Tao himself described the experience as significantly faster than manual formalization, even with his deep mathematical expertise.
- HN discussion noted the importance of the 'Aristotle' tool specifically — it bridges between natural language mathematical reasoning and Lean's formal syntax.
- Researchers in the Lean community confirmed this is a meaningful result — Erdős #728 is a legitimate research-level theorem, not a textbook exercise.
- Comparison was made to AlphaProof's work on IMO problems — a pattern of AI enabling mathematical progress rather than replacing mathematicians.
How to Apply
- If you work in software verification, track Aristotle and similar tools — the ability to bridge natural language specs to formal proofs is the missing piece for wider adoption.
- For researchers: AI-assisted Lean proof writing is now at the stage where it's worth attempting on your own open problems, not just educational exercises.
- The Tao collaboration model (human provides insight, AI handles formalization) suggests the near-term workflow: focus your effort on the mathematical ideas, delegate the proof mechanics.
- Monitor the Lean/Mathlib ecosystem for AI tooling improvements — this is moving fast and the tools available in 6 months will be significantly better.
Terminology
Erdős conjecturesA large collection of open mathematical problems posed by prolific mathematician Paul Erdős, ranging from elementary to research-level difficulty.
Lean proof assistantA software tool where mathematical proofs are written in a formal language and machine-checked for logical correctness.
Aristotle (tool)An AI tool that helps bridge natural language mathematical reasoning and formal Lean proof syntax, used in Tao's collaboration.
Formal verificationMathematically proving that a statement (or program) satisfies a specification, with machine-checked guarantees of correctness.