Number theory

Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. It contains many results and open problems that are easily understood, even by non-mathematicians. More generally, the field has come to be concerned with wider classes of problems that have arisen naturally from the study of integers. Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated. See for example the list of number theory topics. Mathematicians working in the field of number theory are called number theorists.

Fields

Elementary number theory

In elementary number theory, the integers are studied without use of techniques from other mathematical fields. Questions of divisibility, the Euclidean algorithm to compute greatest common divisors, factorization of integers into prime numbers, investigation of perfect numbers and congruences belong here. Typical statements are Fermat's little theorem and Euler's theorem extending it, the Chinese remainder theorem and the law of quadratic reciprocity. The properties of multiplicative functions such as the Möbius function and Euler's φ function are investigated; so are integer sequences such as factorials and Fibonacci numbers.

Related Topics:
Divisibility - Euclidean algorithm - Greatest common divisor - Prime number - Perfect number - Congruences - Fermat's little theorem - Euler's theorem - Chinese remainder theorem - Quadratic reciprocity - Multiplicative function - Möbius function - Euler's φ function - Integer sequence - Factorial - Fibonacci number

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Many questions in elementary number theory appear simple but may require very deep consideration and new approaches. Examples are

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• The Goldbach conjecture concerning the expression of even numbers as sums of two primes,
• Catalan's conjecture regarding successive integer powers,
• The twin prime conjecture about the infinitude of prime pairs, and
• The Collatz conjecture concerning a simple iteration.

The theory of Diophantine equations has even been shown to be undecidable

Related Topics:
Diophantine equation - Undecidable

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(see Hilbert's tenth problem).

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Analytic number theory

Analytic number theory employs the machinery of calculus and complex analysis to tackle questions about integers. The prime number theorem and the related Riemann hypothesis are examples. Waring's problem (representing a given integer as a sum of squares, cubes etc.), the Twin Prime Conjecture (finding infinitely many prime pairs with difference 2) and Goldbach's conjecture (writing even integers as sums of two primes) are being attacked with analytical methods as well. Proofs of the transcendence of mathematical constants, such as π or e, are also classified as analytical number theory. While statements about transcendental numbers may seem to be removed from the study of integers, they really study the possible values of polynomials with integer coefficients evaluated at, say, e; they are also closely linked to the field of Diophantine approximation, where one investigates "how well" a given real number may be approximated by a rational one.

Related Topics:
Analytic number theory - Calculus - Complex analysis - Prime number theorem - Riemann hypothesis - Waring's problem - Squares - Cubes - Twin Prime Conjecture - Goldbach's conjecture - Proofs - Transcendence - π - E - Transcendental numbers - Polynomials - Diophantine approximation - Rational

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Algebraic number theory

In algebraic number theory, the concept of number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. These domains contain elements analogous to the integers, the so-called algebraic integers.

Related Topics:
Algebraic number - Rational - Algebraic integer

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In this setting, the familiar features of the integers (e.g. unique factorization) need not hold.

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The virtue of the machinery employed -- Galois theory, group cohomology, class field theory, group representations and L-functions -- is that it allows to recover

Related Topics:
Galois theory - Group cohomology - Class field theory - Group representation - L-function

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that order partly for this new class of numbers.

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Many number theoretical questions are best attacked by studying them modulo p for all primes p (see finite fields). This is called localization and it leads to the construction of the p-adic numbers; this field of study is called local analysis and it arises from algebraic number theory.

Related Topics:
Finite field - P-adic number - Local analysis

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Geometric number theory

Geometric number theory (traditionally called geometry of numbers) incorporates all forms of geometry. It starts with Minkowski's theorem about lattice points in convex sets and investigations of sphere packings. Algebraic geometry, especially the theory of elliptic curves, may also be employed. The famous Fermat's last theorem was proved with these techniques.

Related Topics:
Geometric number theory - Geometry of numbers - Minkowski's theorem - Convex - Sphere packing - Elliptic curve - Fermat's last theorem

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Combinatorial number theory

Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdős is the main founder of this branch of number theory. Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a set of integers. Algebraic or analytic methods are powerful in this field.

Related Topics:
Combinatorial number theory - Paul Erdős - Covering system - Zero-sum problems - Restricted sumsets

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Computational number theory

Computational number theory studies algorithms relevant in number theory. Fast algorithms for prime testing and integer factorization have important applications in cryptography.

Related Topics:
Computational number theory - Prime testing - Integer factorization - Cryptography

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