An AI system solved nine open math problems with every step checked

The system also proved dozens of conjectures, while most of the problems it attempted remained unsolved.

Editorial illustration — not from the study.

Researchers built an AI agent that searches for mathematical proofs and submits them to Lean, a system that automatically checks each logical step. A more advanced version solved nine of 353 formalized open Erdős problems, including two that had been open for 56 years, and proved 44 of 492 formalized Online Encyclopedia of Integer Sequences conjectures.

The agent combined language-model-generated proof attempts with compiler feedback and, in some versions, additional proof tools and an evolutionary search over candidate arguments. The researchers report that a simpler agent using repeated generation and Lean verification also reproduced the Erdős results, although it was more costly on the hardest problems.

The problems it solved

The full-featured agent solved 9 of 353 open Erdős problems that had been translated into Lean. Researchers checked that the formal statements matched the original problems, and Lean checked the proof steps. The solved set included two questions that had remained open for 56 years. The agent also proved 44 of 492 Online Encyclopedia of Integer Sequences conjectures. The reported cost was a few hundred dollars per Erdős problem on average. A basic agent that alternated language-model attempts with Lean verification reproduced the Erdős successes, but cost more on the hardest problems.

Why checked proofs matter

Formal checking addresses a central problem with AI-generated mathematical arguments: a fluent-looking proof can contain a subtle logical error. By requiring a proof system to verify each step, the approach can identify arguments that are ready for expert review and can let mathematicians focus on unresolved parts rather than rechecking an entire informal proof. The researchers also report that formal proof attempts helped their mathematician collaborators understand problems and identify incorrect formal statements.

Evidence and limits

This was a large-scale evaluation of AI agents on formalized open problems and conjectures, rather than a demonstration that AI can solve mathematics generally. The results depend on which problems had been formalized and on the capabilities of the Lean mathematics library. The agents' successes were concentrated in areas such as combinatorics, convex optimization and number theory, where problems often break into manageable subgoals. Most Erdős problems remained unsolved, and the researchers say the systems are still limited on problems requiring substantial new theory. They also report variability between searches and biases inherited from the underlying language models.

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