Number theory

Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. More generally, it has come to be concerned with a wider class of problems that are "easily understood by non-mathematicians" - this expansion has occurred as the techniques are used to attack wider varieties of problems. Number theory may be subdivided into several fields according to the methods used and the questions investigated.

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.

Many questions in elementary number theory are exceptionally deep and require completely new approaches. Examples are

• 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 (see Hilbert's tenth problem).

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. Warings 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. Statements about transcendental numbers may seem to moved away from the study of integers. On the other hand they really study the possible values of polynomials with integer coefficients evaluated at, say, e; and are also closely linked to the field of Diophantine approximation.

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. In this setting, the familiar features of the integers (e.g. unique factorization) need not hold. The virtue of the machinery employed -- Galois theory, field cohomology[?], class field theory, group representations and L-functions -- is that it allows to recover that order partly for this new class of numbers.

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.

Geometric number theory 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.

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