Learning to Discover Interesting Mathematics

2026-09-23

AI Summary

This study addresses the limited interestingness and evaluation difficulty of mathematical theorems generated by large language models. It proposes an intrinsic interestingness metric based on the proof-to-statement length ratio, enabling a goal-free autonomous discovery paradigm independent of human-specified objectives. Methodologically, a 27B-parameter model is employed to predict proof difficulty for search optimization, while formal verification and iterative generation techniques are combined to construct a self-expanding, machine-verified library. Experimental results demonstrate that this approach significantly reduces overlap with Mathlib to 30.6%, successfully generating highly interesting, out-of-distribution novel theorem candidates and effectively enhancing the efficiency of autonomous knowledge base expansion.

Abstract

Recently, Large Language Models (LLMs) have been increasingly able to solve advanced mathematical problems, including many that have been open for decades. This opens the door to expansion of mathematical knowledge at unprecedented scale. Yet, while LLMs may be able to conjecture and prove more and more theorems, it remains open whether this new mathematical knowledge is interesting or useful. We define intrinsic interestingness of a theorem as the ratio between the length of its proof and the length of its statement. We show that this correlates strongly with an extrinsic measure of the downstream utility of a theorem. We identify the difficulty of a proof conditioned on a set of premises as a useful primitive for computing these metrics, and train a 27B model that predicts proof difficulty more accurately than frontier general-purpose models. Optimizing for our metric creates a model capable of producing more interesting theorems, while also reducing substantial or full overlap with Mathlib from 91.9% to 30.6%, showcasing the creation of more out-of-distribution math. We show that our system can generate candidate theorems, select the most interesting among them, and iteratively build on a self-expanding mathematical library. These metrics provide a practical and quantifiable signal for ranking conjectures and guiding proof search within formal mathematical libraries. Our framework provides a path towards self-expanding, machine-verified mathematical libraries that can choose worthwhile statements without relying on human-supplied targets.

Problem

• Large Language Models
• Mathematical Theorem Discovery
• Intrinsic Interestingness
• Formal Mathematical Libraries
• Automated Theorem Proving

Innovation

• Intrinsic Interestingness
• Proof Difficulty Prediction
• Large Language Models
• Automated Theorem Proving
• Self-expanding Mathematical Library