Beyond The Sieve: A Survey of Artificial Intelligence Approaches for Prime Number Prediction and Structural Analysis

Abstract

The study of prime numbers, the essential structural blocks of number theory, has been the area of rigorous logical and algebraic methods. While effective for showing general properties, these standard methods often struggle to identify delicate, high-dimensional patterns within the arrangement of primes that may not be simply expressible in closed-form mathematical language. This report studies the developing paradigm of using Artificial Intelligence (AI), mostly deep learning, to balance classical number theory. We survey key methodologies, including the application of recurrent neural networks (RNNs) and transformers for next-prime calculation, and the use of unsupervised learning for uncovering latent structures in prime distributions. We discuss important successes, such as models that complete high truth in expecting subsequent primes within unnatural sequences and the identification of potential novel inter-prime relationships. However, we also address serious limitations, most particularly the models' leaning toward speaking rather than true extrapolation, their absence of interpretability, and the important challenge of moving from pattern credit to mathematically generalizable proof. The deduction suggests that AI serves not as a replacement for standard theory, but as a powerful investigative tool for generating novel hypotheses about prime distribution, possibly guiding mathematicians toward new theorems and a deeper understanding of the primes.