# number theory

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/Cryptanalysiscryptography 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 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 $\sqrt{2}$. 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 $\ell$-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

The attached bibliography for number theory contains many additional references for these topics.

Title number theory NumberTheory 2013-03-22 14:07:53 2013-03-22 14:07:53 olivierfouquetx (2421) olivierfouquetx (2421) 37 olivierfouquetx (2421) Topic msc 11-01 BibliographyForNumberTheory Divisibility Congruences     FundamentalTheoremOfArithmetic NumberField