Modern developments in number theory: Insights into prime distribution

Abstract

The distribution of prime numbers has long intrigued mathematicians, shaping the evolution of number theory. From 2010 to 2024, significant breakthroughs have expanded our understanding of prime distribution through analytic, algebraic, and computational innovations. This review examines advances in the Riemann Hypothesis, bounded prime gaps, sieve methods, and machine learning applications in prime prediction. Progress in computational tools, such as GIMPS and distributed algorithms, has facilitated the discovery of large primes and refined testing mechanisms. These developments have profound implications for cryptography, quantum computing, and random matrix theory, offering insights into both theoretical challenges and practical applications. By synthesizing recent advancements, this review highlights unresolved questions and sets the stage for future interdisciplinary research in number theory and its applications.