AI-Powered Agents Autonomously Solve Open Erdős Problems via Formal Proof Search
Key Takeaways
- ▸Autonomous agent resolved 9 of 353 open Erdős problems—long-standing unsolved mathematical challenges—at costs of a few hundred dollars per problem
- ▸Proved 44 of 492 OEIS conjectures, demonstrating broad applicability beyond a single problem class
- ▸System is actively deployed in combinatorics, optimization, graph theory, algebraic geometry, and quantum optics research
Summary
Researchers have demonstrated the first large-scale evaluation of using large language models (LLMs) to generate formal proofs for open mathematical problems. The work addresses a key limitation of LLMs in mathematics—their unreliability—by having them generate proofs in Lean, a formal verification language that ensures correctness. An autonomous agent successfully resolved 9 of 353 open Erdős problems at a cost of a few hundred dollars per problem and proved 44 of 492 OEIS conjectures, with simpler alternating approaches showing comparable results on some problems.
The research demonstrates practical viability of AI-aided formal proof search across multiple mathematical disciplines. The system is already being deployed in active research spanning combinatorics, optimization, graph theory, algebraic geometry, and quantum optics. By combining LLM-based proof generation with Lean-based verification, the approach eliminates the risk of hallucinated or incorrect proofs that plague LLM-only mathematics applications.
This breakthrough suggests that LLMs are becoming capable partners in mathematical discovery when augmented with formal verification systems. The work sheds light on agent designs that enable solving notoriously difficult open problems and opens new possibilities for AI-assisted research across pure mathematics.
- Formal verification in Lean ensures proof correctness, eliminating unreliability of LLM-only approaches to mathematics
- Agent design insights provide a roadmap for combining generative AI with formal verification in other domains
Editorial Opinion
This research represents a genuine inflection point for AI in mathematics. For decades, the field resisted automation due to the need for absolute correctness—a single faulty step invalidates an entire proof. By coupling LLMs with formal verification, this work finally solves that problem, unlocking computational power that human mathematicians simply cannot match. The fact that these systems are already being deployed in active research labs, not just academia, signals that AI-assisted mathematical discovery is transitioning from proof-of-concept to practical tool.
