Overview
Number theory is a branch of mathematics devoted to the study of whole numbers and related structures. At its core it asks what integers are like, how they factor, and what patterns appear among them. Basic concepts include divisibility, greatest common divisors, congruences, and the special role of prime numbers; many classical questions ask for integer solutions to polynomial equations, known as Diophantine problems. The subject often focuses on the set of integers but also considers rational, real and complex numbers in contexts where they illuminate integer behavior.
Main topics and branches
Number theory contains several overlapping areas. Elementary number theory treats basic properties of integers and congruences. Analytic number theory uses tools from analysis to study distribution questions such as the occurrence of primes. Algebraic number theory replaces the integers by larger rings to study factorization and field extensions. Computational or algorithmic number theory develops methods for primality testing and factorization that are important in practice.
- Prime numbers and their distribution
- Factorization and unique factorization phenomena
- Modular arithmetic, congruences and residues
- Diophantine equations and integer solutions
- Algebraic integers and class groups
Techniques and tools
Number theorists use a wide array of methods. Elementary proofs rely on clever manipulations of divisibility and residues; analytic approaches invoke complex analysis and estimates for sums; algebraic methods use field theory and Galois groups. Linear algebra and matrices also appear, for example in the study of lattices and modular forms, while computational experiments guide conjectures and verify large cases.
History and development
The subject has ancient roots: records of divisibility and special numbers appear in early civilizations, and Greek and Hellenistic writers began systematic treatments of integers. Important milestones include work of Diophantus on equations, Fermat's statements and conjectures about primes, and later systematic foundations by Euler and Gauss. In modern times, analytic methods and algebraic frameworks expanded the field, and some famous conjectures — notably the Riemann hypothesis — remain central open problems, while other longstanding problems such as Fermat's Last Theorem were resolved in the late twentieth century.
Applications and notable results
Number theory is both pure and applied. Its results underpin public-key cryptography (where large primes and factorization are essential), error-correcting codes, pseudorandom number generators, and algorithms in computer science. Among celebrated theorems and principles are the fundamental theorem of arithmetic, the Chinese remainder theorem, Fermat's little theorem, quadratic reciprocity, and the prime number theorem; texts collecting such results offer a compact view of the field's core ideas and proofs (survey of theorems).