Exploring the Properties and Applications of Prime Numbers in Modern Number Theory

Number theory, often regarded as the "queen of mathematics," has evolved from the study of integers and their properties into a vibrant field with deep theoretical insights and numerous practical applications. This paper focuses on the fundamental role of prime numbers, their distribution, and their significance in both pure and applied mathematics. We review classical results such as the Prime Number Theorem and Euclid’s theorem, along with modern approaches involving modular arithmetic, primality testing algorithms, and elliptic curves. Emphasis is also placed on the role of prime numbers in cryptography, especially in the RSA algorithm, showcasing the practical value of number theoretic principles. Furthermore, we explore unsolved problems like the Riemann Hypothesis and Goldbach’s Conjecture, which continue to drive research in the field. This study aims to provide a comprehensive understanding of prime numbers, encouraging further exploration of their complex and beautiful structure.