Artificial Intelligence in Number Theory: LLMs for Algorithm Generation and Ensemble Methods for Conjecture Verification

Researchers demonstrate that large language models like Qwen2.5-Math achieve 95%+ accuracy on algorithmic number theory problems with optimal hints, and empirically verify a folklore conjecture that Dirichlet character moduli are uniquely determined by L-function zeros using machine learning ensemble methods.

This research bridges artificial intelligence and pure mathematics by demonstrating practical applications of LLMs and machine learning to specialized computational domains previously requiring expert human mathematicians. The work addresses a critical gap in AI evaluation: rather than pursuing the currently intractable goal of automated theorem proving, the authors focus on algorithmic problem-solving and computational verification where AI can deliver measurable value. The Qwen2.5-Math model's near-perfect accuracy on number-theoretic problems suggests that modern LLMs possess genuine mathematical reasoning capabilities beyond pattern matching, particularly when provided with strategic hints that guide rather than spoil solutions.

The conjecture verification component introduces an ensemble approach using LightGBM to analyze statistical properties of Dirichlet L-function zeros. Achieving 93.9% test accuracy in predicting character moduli from zero statistics empirically validates decades-old theoretical speculation, demonstrating that machine learning can serve as a discovery tool in analytic number theory. This approach complements traditional mathematical methods by leveraging pattern recognition across high-dimensional feature spaces.

For the broader AI landscape, these results indicate that language models and classical machine learning excel in specialized mathematical domains where interpretability and accuracy matter more than raw scale. The availability of published code and datasets enables reproducibility and future research. The work validates a hybrid human-AI workflow where models handle computational tasks while mathematicians focus on proof strategies and novel insights. This pattern—AI excelling at narrow, well-defined mathematical problems—may define productive AI applications in academic research for years to come, influencing how institutions allocate resources toward AI-assisted discovery versus pure automation.

• →LLMs achieve 95%+ accuracy on algorithmic number theory problems when given non-spoiling hints, indicating genuine mathematical reasoning capability
• →Machine learning empirically verified a long-standing analytic number theory conjecture about Dirichlet character moduli determination with 93.9% accuracy
• →Ensemble methods using statistical feature extraction from mathematical objects can effectively predict theoretical properties without explicit proofs
• →This research demonstrates AI's value in narrow mathematical domains through computational verification rather than general theorem proving
• →Published code and datasets enable reproducible AI research in pure mathematics, establishing methodologies for future conjecture verification