Summary
Language models have recently shown they can sometimes solve famous math problems, but it's unclear how well they do on a broader set of math questions. The authors collected a large set of open math problems from recent papers and tested powerful language models on a random sample of 200 problems. They found the models suggested solutions for about 60% of problems, more often finding counterexamples than full proofs. Their study shows the models sometimes give answers that differ from what the original problem meant and often rely on existing math knowledge, highlighting the need for careful evaluation beyond just checking if the answer is correct.
What this means in practice
- •For mathematics software developers: Incorporate language models to assist in proposing candidate proofs or counterexamples for new math conjectures without requiring tailored mathematical input.
- •For automated theorem proving teams: Use insights from model behavior on random conjectures to improve verification techniques and better distinguish trivial or unintended resolutions.
Authors
Helen Jenne, Davis Brown, Jesse He, Henry Kvinge
Abstract
The last year has seen a shift in AI-assisted mathematics, with frontier language models producing proofs of and counterexamples to long-standing open conjectures. However, studying only the current set of major breakthroughs provides an incomplete picture of LLM capabilities and the computational resources needed to achieve reported results. To address this, we introduce a new dataset, OpenConjecture, which collects conjectures from mathematics papers published on the arXiv since February 2026. We use these conjectures as a source of open mathematical problems, and apply frontier GPT models to a random sample of 200 of these conjectures over three passes using a reproducible protocol without problem-specific mathematical guidance. Rather than attempt expert verification of these candidate solutions, we analyze the types of problems for which LLMs produce solutions and the characteristics of these solutions. The models we test produce candidate solutions for roughly 60% of the conjectures, with counterexamples about 1.3 times more common than proofs. We estimate solution novelty, assess how solutions cite and relate to existing literature, and analyze whether the proposed resolution relies on either an error in or underspecification of the conjecture. We find that models sometimes propose resolutions that are outside the authors' intent (e.g., proposing the trivial group as a counterexample) and that a number of proofs follow closely from existing results, but that some resolutions exhibit at least moderate novelty. Our findings underscore the need for evaluations that go beyond correctness to assess fidelity to the intended statement, attribution, and the nature of the mathematical contribution.