LLM Framework Discovers Major Mathematical Conjectures

The discovery of major mathematical conjectures has historically relied heavily on expert intuition, lacking a unified, systematic method for generation and validation. This research presents an innovative three-stage pipeline powered by large language models (LLMs) designed to address this gap. The framework's objective is to uncover mathematical problems possessing "high problem taste"—those whose eventual proofs could fundamentally reshape a research area and provide lasting value to human mathematicians. The pipeline begins with a region search, utilizing explicit local evidence modules to identify potential conjectures. This is followed by a reflective validation stage, where candidates are assessed for their foundationality, novelty, and potential significance. The final stage involves formal validation within Lean 4 and Mathlib, ensuring the conjectures are syntactically correct and logically sound within a formal proof assistant environment. Experiments conducted on twenty candidate conjectures demonstrated the system's robust capability to transition from natural language descriptions to formal checks. All twenty candidates successfully passed Lean parsing and type checking, were not directly absorbed by existing proofs, and were not automatically discharged by automated theorem provers, indicating their novelty. The absence of explicit duplicates or near duplicates further highlights the framework's ability to generate unique and potentially significant mathematical problems.