Recent Advances and Applications in Modern Number Theory: A Comprehensive Review

Abstract

Number theory investigates the natural numbers and their properties. The classical disciplines of number theory treat natural numbers either as objects of investigation or as tools to solve diophantine equations. These and other ancillary theories are the staples of number theory and include prime numbers, divisibility, modular arithmetic, quadratic reciprocity, and the construction of number systems.

The study of divisibility in the integers led to the introduction of the Euclidean algorithm of finding the greatest common divisor, which in turn produced the concept of a prime number. The work of Euclid develops in an elementary setting the idea of studying natural numbers and certain properties like divisibility.

The modern study of number theory introduces a spiral of ideas that diverge from that of Euclid. Euclid’s original investigation looked exclusively at the integers while the more modern approach examines the integers as part of a wider algebraic setting consisting of rational, real, complex, -adic, and other algebraic integers. Moreover, instead of relying on elementary arithmetic, analysis techniques are used to illuminate the subject. The combination of analytic as well as algebraic methods culminates in the foundation of modern number theory.