number theory

Number theoryMathworldPlanetmath is the branch of math concerned with the study of the integers, and of the objects and structuresMathworldPlanetmath 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 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 conjectureMathworldPlanetmath, 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 HypothesisMathworldPlanetmath, which states that all non-trivial zeros of the Riemann zeta function have real partMathworldPlanetmath 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 groupMathworldPlanetmath of a number fieldMathworldPlanetmath (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 numbersMathworldPlanetmath. 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 abelianMathworldPlanetmath class field theory (the theory of abelian extensionsMathworldPlanetmathPlanetmath of number fields, which extends the law of quadratic reciprocity) by Hilbert, Takagi, Artin, Tate and others; the proof of the Weil conjecturesMathworldPlanetmath (which include the Riemann hypothesis for function fieldsMathworldPlanetmath and are intimately linked to geometry over finite fieldsMathworldPlanetmath) by Dwork, Grothendieck and Deligne using methods of -adic cohomologyMathworldPlanetmath; the proof of the Mordell conjectureMathworldPlanetmath (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 curveMathworldPlanetmath 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 ( 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 theoryMathworldPlanetmath uses the machinery of analysisMathworldPlanetmath to tackle questions related to integers and transcendence. One of its most famous achievements is the proof of the prime number theoremMathworldPlanetmath.

  • Algebraic number theoryMathworldPlanetmath can either be defined as the study of algebraic numbersMathworldPlanetmath or as an algebraicMathworldPlanetmath study of number theory (depending on how you associate ( in English). By the definition of algebraic numbers, these two are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. 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 groupMathworldPlanetmathPlanetmath of the associated affine schemeMathworldPlanetmath. Arithmetic geometry is often referred to as arithmetic algebraic geometryMathworldPlanetmathPlanetmath or Diophantine geometry.

  • Finally, computational number theory is the study of computations with numbers, developing algorithms to calculate things such as factorizations, discrete logarithmsMathworldPlanetmath, numbers of points on curves, class groups and cohomology groupsPlanetmathPlanetmath.

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 CongruencesMathworldPlanetmathPlanetmathPlanetmathPlanetmath
Related topic FundamentalTheoremOfArithmetic
Related topic NumberField