A team of researchers has published what they're calling the first large-scale evaluation of using large language models to solve open problems in pure mathematics, and the results are striking. According to a paper posted on arXiv (2605.22763), their most capable AI agent autonomously resolved 9 out of 353 open Erdős problems—a collection of conjectures named after legendary mathematician Paul Erdős that have resisted human mathematicians for decades.

The Lean Approach

Rather than relying on LLMs' notoriously unreliable natural language reasoning, the team focused on formal proof generation in Lean, a statically typed programming language designed specifically for mathematical proofs. This approach forces AI systems to produce machine-verifiable logic chains rather than plausible-sounding but potentially incorrect prose. The research demonstrates that when you constrain AI to verifiable formal systems, it can actually contribute meaningfully to frontier mathematics.

Breaking Down the Numbers

The agent resolved those 9 open Erdős problems at a cost of "a few hundred dollars" per problem—not cheap, but orders of magnitude less than the years of effort human mathematicians might expend on such challenges. Beyond the Erdős collection, the system proved 44 out of 492 OEIS (Online Encyclopedia of Integer Sequences) conjectures. For context, the OEIS contains thousands of unproven hypotheses about integer sequences that underpin much of discrete mathematics and computer science.

Real-World Deployment

The researchers aren't stopping at benchmarks. The paper notes that this AI proof-search technology is already being deployed across multiple research domains including combinatorics, optimization, graph theory, algebraic geometry, and quantum optics. This suggests the approach has moved beyond academic exercise into genuine scientific tooling—a significant milestone for formal methods in AI.

Comparing Agent Architectures

The team also tested a "basic agent" that alternated between LLM-based proof generation and Lean-based verification, replicating many of the same Erdős successes but at higher cost on the hardest problems. This comparison reveals that agent design matters enormously—smarter reasoning strategies can achieve better results while spending less compute budget. The gap between capable and basic agents underscores how much room remains for architectural innovation in formal AI reasoning.

Key Takeaways

  • First large-scale demonstration of LLM-based formal proof search on genuine open problems
  • Resolved 9/353 open Erdős problems at "a few hundred dollars" per problem cost
  • Proved 44/492 OEIS conjectures using automated Lean proof generation
  • Already deployed in combinatorics, graph theory, algebraic geometry, and quantum optics research

The Bottom Line

This isn't just another benchmark-hunting exercise—these are real mathematical problems that humans couldn't crack. When your AI system starts publishing novel proofs in peer-reviewed venues, the game has changed. The question now isn't whether formal AI reasoning will transform mathematics, but who's going to build the infrastructure first.