Formal Conjectures: An Open and Evolving Benchmark for Verified Discovery in Mathematics

Large language models (LLMs) increasingly excel at mathematical reasoning, but their unreliability limits their utility in mathematics research. A mitigation is using LLMs to generate formal proofs in languages like Lean. We perform the first large-scale evaluation of this method's ability to solve open problems. Our most capable agent autonomously resolved 9 of 353 open Erdős problems at the per-problem cost of a few hundred dollars, proved 44/492 OEIS conjectures, and is being deployed in combinatorics, optimization, graph theory, algebraic geometry, and quantum optics research. A basic agent alternating LLM-based generation with Lean-based verification replicated the Erdős successes but proved costlier on the hardest problems. These findings demonstrate the power of AI-aided formal proof search and shed light on the agent designs that enable it.

Can AI make progress on important, unsolved mathematical problems? Large language models are now capable of sophisticated mathematical and scientific reasoning, but whether they can perform novel research is still widely debated and underexplored. We introduce HorizonMath, a benchmark of 113 predominantly unsolved problems spanning eight domains in mathematics and the mathematical sciences, paired with an open-source evaluation framework for automated verification. Our benchmark targets the generator-verifier gap: problems where discovery is hard and requires meaningful mathematical insight, but verification is computationally straightforward. This contrasts with most existing research-level benchmarks, which instead rely on formal proof verification or manual review, both of which are expensive to scale. Because these solutions are unknown, HorizonMath is resistant to data contamination, and most state-of-the-art models score under 10%. Using this framework, we identify six novel solutions to research problems that either resolve previously open questions or improve on the best-known published results, with GPT-5.4 Pro and GPT-5.6 Sol each discovering three of these solutions. Across seven frontier model families, reasoning efficiency and behavior also vary substantially. We release HorizonMath as an open challenge and a growing community resource, where each verified solution is a candidate contribution to the mathematical literature.

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.