Number Theory is the branch of pure mathematics concerned with the properties of integers and related structures, including divisibility, prime numbers, congruences and Diophantine equations. It ranges from elementary results, such as the fundamental theorem of arithmetic, to deep areas like analytic and algebraic number theory. Beyond its theoretical importance, it underpins much of modern cryptography, where the difficulty of certain number-theoretic problems provides security.
Semantic Classification
Content
- Elementary number theory studies the integers directly, establishing results such as unique prime factorisation, properties of greatest common divisors and the behaviour of congruences in modular arithmetic. These tools answer questions about divisibility and the distribution of solutions to equations over the integers.
- More advanced branches bring in other areas of mathematics. Analytic number theory uses techniques from analysis to study the distribution of primes, exemplified by the prime number theorem, while algebraic number theory generalises arithmetic to rings of algebraic integers and number fields.
- Number theory has become central to applied cryptography, since the presumed hardness of problems such as integer factorisation and discrete logarithms secures widely used public-key systems. This connection has turned a historically pure subject into a foundation for digital security and, by extension, for blockchain and digital-asset systems.