Number Theory for Artificial Intelligence

Number theory studies integers, primes and divisibility, including modular arithmetic. AI uses it in cryptography that protects models and data, in hashing and fast transforms inside ML systems, in integer quantization, and as a domain where AI systems learn and prove mathematics.

Homomorphic encryption schemes such as BFV and CKKS compute on encrypted data using arithmetic modulo large integers, secure aggregation in federated learning uses modular masking, and lattice problems such as learning with errors underpin both FHE and post-quantum standards like ML-KEM.

Yes. Small transformers trained on tasks like modular addition show grokking, where they generalize long after memorizing the training set. Mechanistic interpretability studies found some learn Fourier-like features to solve it, and different networks can use different algorithms.

AI has helped find patterns, such as murmurations in elliptic-curve data, and systems like AlphaProof reached olympiad level. Research-level benchmarks such as FrontierMath remain hard, and formal proof in Lean is a key route to verifying results.

Links were checked in October 2026. This is an introductory map, not a textbook: security-relevant choices (for example cryptographic parameters) should follow current standards, and AI-for-mathematics results are changing quickly. Sites that block automated checks are kept and may need a manual click.