LLM Framework for Discovering Major Mathematical Conjectures: AI's Quest for the Next Riemann Hypothesis

Introduction

Major mathematical conjectures remain among the most challenging and intellectually significant pursuits in science. Their discovery has traditionally relied on deep expert intuition, often developed over years or decades of immersion in a specific field. However, a unified, systematic method for generating and validating conjectures with substantial mathematical potential has remained elusive. This paper introduces a novel three-stage pipeline designed to bridge that gap, leveraging large language models (LLMs) to assist in the discovery of conjectures that could reshape mathematical landscapes.

Methodology

Our proposed framework operates in three distinct stages, each addressing a critical component of the conjecture discovery process:

1. Region Search – This initial stage uses explicit local evidence modules to identify promising mathematical regions—areas where patterns or anomalies suggest the existence of an underlying structure. By focusing on concrete, localized examples, the model can propose conjectures grounded in observable data rather than abstract speculation.


1. Reflective Validation – The second stage subjects each candidate conjecture to rigorous reflection against three criteria: foundationality (whether it connects to core principles), novelty (whether it introduces genuinely new ideas), and potential significance (whether it could influence broader research). This step is crucial for filtering out trivial or non-impactful propositions early in the process.


1. Formal Validation in Lean 4 and Mathlib – The final stage translates the natural-language conjecture into a formal proof environment using Lean 4 and its mathematical library, Mathlib. This ensures that the conjecture is not only plausible but also well-defined and mechanically checkable, providing a level of rigor that is often absent in purely informal reasoning.



The ultimate goal of this pipeline is the discovery of mathematical problems with high problem taste—problems whose solutions could reorganize the language of an entire research area, offering durable benefits to human mathematicians.




Results




We evaluated our framework on twenty candidate conjectures spanning diverse mathematical domains. The results were remarkably consistent:



• Full parsing and type checking: All twenty candidates (20/20) passed Lean 4 parsing and type checking, confirming that each conjecture could be formally expressed without syntax or type errors.
• Non-triviality checks: Every candidate (20/20) was deemed non-trivial, as none could be directly absorbed by exact? or automatically discharged by aesop—tools that typically handle straightforward proofs. This indicates that all conjectures possess substantive depth.
• Originality: The framework identified no explicit duplicates or near-duplicates among the candidates, underscoring the novelty of the generated conjectures.



These results demonstrate a stable transition from natural-language insight to formal representation, a key enabler for scalable AI-assisted mathematical discovery.




Discussion and Outlook




This work represents a significant step toward automating the earliest, most creative phase of mathematical research: conjecture generation. By combining LLMs with formal verification, we offer a pipeline that not only proposes but also validates potential breakthroughs, aligning with recent trends in AI for mathematics that emphasize rigorous, machine-checkable outputs.




Looking ahead to 2026 and beyond, we anticipate further integration with interactive theorem provers and richer mathematical databases, which could expand the scope of detectable patterns. The balance between human intuition and machine-driven exploration will remain central—our framework is designed to augment, not replace, the mathematician's role, serving as a collaborative partner in the quest for the next Riemann Hypothesis.




Conclusion




We have presented a three-stage LLM-based framework for discovering major mathematical conjectures, validated across twenty candidates. Its ability to produce original, formally checkable, and substantive conjectures suggests a promising path forward for AI-assisted mathematical research. By providing a systematic method for conjecture generation, we hope to unlock new avenues for discovery that were previously inaccessible due to the reliance on individual expertise.




References




[Full reference list available in the original paper, arXiv:2607.28632 [cs.AI], submitted 19 April 2026.]

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