Numbertheory

Number theory is a branch of mathematics focused on the integers and on functions that take integer values. It studies questions of divisibility, primality, congruences, and the arithmetic structure of numbers. The field draws methods from algebra, analysis, geometry, and computation and ranges from concrete problems to highly abstract theories.

Core topics include the study of primes, factorization and multiplicative functions, Diophantine equations, and modular arithmetic. The distribution of primes, arithmetic progressions, and congruence relations are central themes. Subfields include algebraic number theory, analytic number theory, arithmetic geometry, and computational number theory, each using different tools to understand integers and their properties.

Historical milestones include Euclid’s proof that there are infinitely many primes, the fundamental theorem of arithmetic, and Fermat’s little theorem. The 20th century saw major advances by Gauss, Dirichlet, and Riemann, while Andrew Wiles proved Fermat’s Last Theorem. The prime number theorem describes the asymptotic distribution of primes, roughly stating that primes occur with density about 1/log n.

Applications include cryptography, where algorithms such as RSA and elliptic-curve cryptography rely on number-theoretic facts about factorization and discrete logs. Other uses appear in coding theory, random number generation, and algorithm design, as well as in mathematical research and modeling.

Today number theory remains active and diverse, with ongoing work in problems about primes, congruences, and Diophantine equations, and connections to algebraic geometry, representation theory, and the mathematics of computation.