number theory

number theory

Number theory

is the branch of math concerned with the study of the

integers

, and of the objects and

structures

that naturally arise from their study.

It is one of the oldest parts of mathematics, alongside geometry, and has been studied at least since the ancient Mesopotamians and Egyptians. Perhaps because of its purely mathematical nature (at least until the development of

http://en.wikipedia.org/wiki/

Cryptanalysis

cryptography and cryptanalysis in the twentieth century, number theory was thought to be devoid of practical applications), number theory has often been considered as a central and particularly beautiful part of mathematics. Carl Friedrich Gauss, arguably the

greatest number

theorist of all time, has called mathematics the “Queen of science” and he referred to Number Theory as the “Queen of Mathematics”. Number theory has attracted many of the most outstanding mathematicians in history: Euclid, Diophantus, Fermat, Legendre, Euler, Gauss, Dedekind, Jacobi, Eisenstein and Hilbert all made immense contribution to its development. Great twentieth century number theorists include Artin, Hardy, Ramanujan, André Weil, Alexandre Grothendieck, Jean-Pierre Serre, Pierre Deligne, Gerd Faltings, John Tate and Andrew Wiles.

Number theory is also remarkable because small, easy-to-understand conjectures abound alongside far-reaching problems.
One can mention the twin-prime problem, the

Goldbach conjecture

, and the odd perfect number problem. Recent progress includes the

resolution

of Catalan’s conjecture (Mihailescu, 2002) and Fermat’s last

theorem

(Taylor and Wiles, 1994) .

The most important unsolved problem in number theory is probably the

Riemann Hypothesis

, which states that all non-trivial zeros of the

Riemann zeta function

have

real part

equal to one-half. Though formulated in terms of complex analysis, this problem is central to understanding the distribution of prime numbers, the distribution of power residues modulo an integer and the size of the

class group

of a

number field

(to give just a few examples).

The Greeks, notably Euclid and Pythagoras, were the first to elucidate the basic theory of

irrational numbers

. Greeks improved a 1,500 year older Egyptian

rational number

system. Greeks wrote 1/p as p’ (in ciphered letters) and generally converted rational numbers to exact unit fraction series by selecting optimizing

aliquot parts

(of

denominators

). A proof of the irrationality of

2

. Euclid also understood and proved some basic properties of prime and

composite numbers

. Euler, Liebniz, Liouville, and Lindemann, among many others, defined and refined the basic

theory of algebraic and transcendental numbers

. These four ideas recur over and over in number theory.

Four of the greatest achievements of number theorists in the twentieth century were

abelian

class field theory

(the theory of

abelian extensions

of number fields, which extends the law of quadratic reciprocity) by Hilbert, Takagi, Artin, Tate and others; the proof of the

Weil conjectures

(which include the Riemann hypothesis for

function fields

and are intimately linked to geometry over

finite fields

) by Dwork, Grothendieck and Deligne using methods of

ℓ

-adic

cohomology

; the proof of the

Mordell conjecture

(which asserts that a curve of genus

greater

than one has only finitely many

rational points

) by Faltings; and the proof of the

Taniyama-Shimura-Weil conjecture

(that states that every rational

elliptic curve

is modular) by Wiles, Taylor,

Diamond

, Conrad and Breuil.

Number theory, long regarded as the purest of the pure sciences, has recently begun to find applications in cryptography (

http://planetmath.org/CryptographyAndNumberTheory

). The recent invention of public-key cryptosystems, which are usually based on the difficulty of a particular number-theoretic computation, has encouraged research in number theory which is essentially applied.

Problems in number theory are often solved using sophisticated techniques from different branches of mathematics. Number theory itself can be loosely divided (not partitioned!) as follows

•

Analytic number theory

uses the machinery of

analysis

to tackle questions related to integers and transcendence. One of its most famous achievements is the proof of the

prime number theorem

.

•

Algebraic number theory

can either be defined as the study of

algebraic numbers

or as an

algebraic

study of number theory (depending on how you

associate

(

http://planetmath.org/

Associative

) in English). By the definition of algebraic numbers, these two are

equivalent

. In the theory of abelian extensions of number fields, which extends the reciprocity laws of Gauss, Legendre, Hilbert et al. of the first half of the past century, class field theory constituted the bulk of algebraic number theory research. In the latter half, algebraic number theory has been subsumed under

arithmetic geometry

(see below) largely due to the efforts of Grothendieck, Serre, Deligne, et al.

•

Arithmetic geometry seeks to bring methods that have been spectacularly successful in “classical” geometric theories (such as

complex manifold

theory) to have bearing on problems in number theory. An illustrative example is the identification of the

absolute Galois group

of a field with the

fundamental group

of the associated

affine scheme

. Arithmetic geometry is often referred to as

arithmetic

algebraic geometry

or Diophantine geometry.

•

Finally,

computational number theory

is the study of computations with numbers, developing algorithms to calculate things such as factorizations,

discrete logarithms

, numbers of points on curves, class groups and

cohomology groups

.

The attached

bibliography for number theory

contains many additional references for these topics.

Title

number theory

Canonical name

NumberTheory

Date of creation

2013-03-22 14:07:53

Last modified on

2013-03-22 14:07:53

Owner

olivierfouquetx (2421)

Last modified by

olivierfouquetx (2421)

Numerical id

37

Author

olivierfouquetx (2421)

Entry type

Topic

Classification

msc 11-01

Related topic

BibliographyForNumberTheory

Related topic

Divisibility

Related topic

Congruences

Related topic

FundamentalTheoremOfArithmetic

Related topic

NumberField

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