Published Paper Details:

Number Theory, often termed the "Queen of Mathematics," is a profound field that explores the properties and relationships of integers, particularly prime numbers. This paper investigates the historical development, fundamental theorems, and modern applications of number theory, with a special focus on prime numbers and their role in cryptography, computational mathematics, and theoretical computer science. We examine key results such as the Prime Number Theorem, Euclid's proof of infinite primes, and the Riemann Hypothesis, while also discussing unsolved problems and interdisciplinary impacts. Number theory, the study of integers and their properties, stands as one of the oldest and most captivating branches of mathematics. At its heart lie prime numbers, the fundamental building blocks of all integers, whose distribution and characteristics have puzzled mathematicians for millennia. This paper offers a comprehensive investigation into the rich landscape of number theory, with a particular focus on the profound significance and ongoing mysteries surrounding prime numbers. We delve into the historical evolution of these fields, from ancient Greek inquiries to modern computational algorithms. Key concepts such as divisibility, congruences, arithmetic functions, and the various forms of prime numbers are explored through rigorous mathematical formulation. Furthermore, we examine the indispensable role of mathematical logic and set theory in establishing the foundational rigor and formalization necessary for advanced number-theoretic research. The paper highlights the diverse applications of prime numbers, particularly in cryptography and computer science, and discusses the persistent challenges and open problems that continue to drive contemporary research, such as the Riemann Hypothesis and the Twin Prime Conjecture. Our aim is to illustrate how the seemingly simple realm of integers harbors some of mathematics' most profound and impactful questions.