AI models crack computational number theory with near-perfect accuracy on algorithmic tasks

A new arXiv paper explores two distinct applications of artificial intelligence to number theory problems, moving beyond the often-ambitious goal of automated general theorem proving into more targeted computational domains.

LLM performance on algorithmic number theory#

The first part of the study evaluates Qwen2.5-Math-7B-Instruct, an open-source language model, on a curated set of computational and algorithmic problems drawn from classical number theory texts and Mathematics Stack Exchange. The benchmark consisted of sixty total items—half purely algorithmic, half computational in nature.

When the model received appropriately framed guidance (hints that did not directly spoil solutions), it achieved at least 95% accuracy across all test cases. This level of performance suggests that specialized large language models can be reliable tools for computational tasks in algorithmic number theory, even without reaching toward the harder problem of formal theorem proof.

Machine learning verifies a number theory conjecture#

In the second part of the research, the team used machine learning to empirically test a folk conjecture about Dirichlet characters and their L-functions. The conjecture holds that the modulus of a Dirichlet character should be uniquely recoverable from just the first few nontrivial zeros of its corresponding L-function—something long suspected but never systematically verified at scale.

The researchers trained a LightGBM multiclass classifier on statistical features extracted from zeros—including moments, finite-difference measurements, and FFT magnitudes—to predict the conductor (modulus) for 214 Dirichlet L-functions. The model achieved over 93.9% test accuracy, providing empirical confirmation that the conjecture holds for small moduli values when sufficient statistical properties are available.

Broader implications#

These experiments illustrate how AI can bridge computational mathematics and data-driven discovery. Rather than attempting general-purpose theorem proving, the work shows that both language models and classical machine learning excel when focused on well-defined, specialized mathematical problems. The researchers have made code and datasets publicly available for the second study, supporting further investigation into conjecture verification and computational number theory.