# Number theory

The number theory is a branch of mathematics that deals with the properties of integers busy. Sub-areas are, for example, elementary or arithmetic number theory - a generalization of arithmetic , the theory of the Diophantine equations , analytic number theory and algebraic number theory .

## Sub-areas

The various sub-areas of number theory are usually differentiated according to the methods with which number theoretic problems are dealt with.

### Elementary or arithmetic number theory

From antiquity to the seventeenth century, number theory asserted itself as a fundamental discipline and managed without any other mathematical sub-areas. Their only aids were the properties of whole numbers , especially prime factorization ( fundamental theorem of arithmetic ), divisibility and calculating with congruences . Such a pure approach is also known as elementary number theory . Important results that can be achieved with the help of elementary methods are Fermat's little theorem and its generalization, Euler's theorem , the Chinese remainder theorem , Wilson's theorem and the Euclidean algorithm .

Even today, individual questions about divisibility, congruences and the like are being researched using elementary number theoretic methods. Attempts are also made to “translate” proofs of number theory that use more extensive methods into elementary terms, from which new insights can arise. One example is the elementary consideration of number theoretic functions such as the Möbius function and Euler's Phi function .

### Analytical number theory

First was Euler points out that methods of analysis and theory of functions can use to solve number theory problems. Such an approach is known as analytic number theory. Important problems that have been solved with analytical methods mostly concern statistical questions about the distribution of prime numbers and their asymptotics. These include, for example, the prime number theorem assumed by Gauss , but only proved at the end of the 19th century, and the Dirichlet theorem on prime numbers in arithmetic progressions . In addition, analytical methods also serve to prove the transcendence of numbers such as the circle number or Euler's number . In connection with the prime number theorem, the Riemann zeta function was examined for the first time , which today, together with its generalizations, is the subject of both analytical and algebraic research and the starting point of the Riemann Hypothesis . ${\ displaystyle \ pi}$ ${\ displaystyle e}$

### Algebraic number theory and arithmetic geometry

One of the great milestones in number theory was the discovery of the law of square reciprocity . The law shows that it is easier to solve questions about the solvability of Diophantine equations in whole numbers by moving to other number ranges ( square number fields , Gaussian numbers ). For this purpose, one considers finite extensions of the rational numbers, so-called algebraic number fields (from which the name algebraic number theory comes from). Elements of number fields are zeros of polynomials with rational coefficients. These number fields contain subsets analogous to the whole numbers, the whole rings . They act like the ring of whole numbers in many ways . The clear decomposition into prime numbers is only valid in number fields of class number 1. However, wholeness rings are Dedekind rings and every broken ideal therefore has a clear decomposition into prime ideals . The analysis of these algebraic number fields is very complicated and requires methods from almost all areas of pure mathematics, especially algebra , topology , analysis , function theory (especially the theory of modular forms ), geometry and representation theory . The algebraic number theory continues to deal with the study of algebraic function fields over finite fields, the theory of which is largely analogous to the theory of number fields. Algebraic number and function fields are summarized under the name "global fields" . It often turns out to be fruitful to ask questions “locally”, i. H. to be considered individually for each prime p . In the case of integers, this process uses the p-adic numbers , generally local fields .

For the formulation of modern algebraic number theory, the language of homological algebra and especially the originally topological concepts of cohomology , homotopy and the derived functors are essential. The highlights of algebraic number theory are the class field theory and the Iwasawa theory .

After the reformulation of algebraic geometry by Grothendieck and especially after the introduction of the schemes , it turned out (in the second half of the twentieth century) that number theory can be viewed as a special case of algebraic geometry. The modern algebraic number theory is therefore also called geometric number theory or arithmetic geometry, in which the concept of the scheme plays a central role.

Each number field has a zeta function whose analytical behavior reflects the arithmetic of the number field. For the Dedekind zeta functions, too, the Riemann Hypothesis is generally unproven. For finite bodies, their statement is contained in the famous Weil conjectures and was solved by Pierre Deligne using algebraic geometry, for which he received the Fields Medal in 1978 .

### Algorithmic number theory

The algorithmic number theory is a branch of number theory, which came with the advent of computers with broad interest. This branch of number theory deals with how number theoretic problems can be implemented algorithmically efficiently. Important questions are whether a large number is prime , the factorization of large numbers and the closely related question of an efficient calculation of the discrete logarithm . In addition, there are now algorithms for calculating class numbers, cohomology groups and for the K-theory of algebraic number fields.

## Applications of number theory

Applications of number theory can be found in cryptography , particularly when it comes to the security of data transmission on the Internet . Both elementary methods of number theory (prime factorization, for example in RSA or Elgamal ) and advanced methods of algebraic number theory such as encryption using elliptic curves ( ECC ) are widely used.

Another area of ​​application is coding theory , which in its modern form is based on the theory of algebraic function fields.

## Historical development

### Number theory in ancient times and in the Middle Ages

The first written evidence of number theory reaches back to approx. 2000 BC. BC back. The Babylonians and Egyptians already knew the numbers less than a million, the square numbers and some Pythagorean triples .

However, the systematic development of number theory did not begin until the first millennium BC. In ancient Greece . The most outstanding representative is Euclid (approx. 300 BC), who introduced the method of mathematical proof invented by Pythagoras into number theory. His most famous work, Euclid's Elements , was used as the standard textbook on geometry and number theory well into the eighteenth century. Volumes 7, 8 and 9 deal with number-theoretical questions, including the definition of the prime number , a method for calculating the greatest common divisor ( Euclidean algorithm ) and the proof of the existence of an infinite number of prime numbers ( Euclid's theorem ).

In the third century AD , the Greek mathematician Diophantos of Alexandria was the first to deal with the equations later named after him, which he tried to reduce to known cases with linear substitutions. With it, he was actually able to solve some simple equations. Diophant's main work is the arithmetic .

The Greeks raised many important arithmetic questions - some of which have not yet been resolved (such as the problem of the twins of prime numbers and the problem of perfect numbers ), or the solutions of which took many centuries, and which are exemplary for the development of the Number theory stand.

With the fall of the Greek states, the heyday of number theory in Europe also died out . From this time only the name of Leonardo di Pisa ( Fibonacci , circa 1200 AD) is noteworthy, who in addition to number sequences and the solution of equations by radicals also dealt with Diophantine equations .

### Number theory in the early modern period

The first important proponent of modern number theory was Pierre de Fermat (1607–1665). He proved Fermat's little theorem , examined the representability of a number as the sum of two squares, and invented the method of infinite descent , with which he could solve Fermat's great theorem he established in the case . The attempt at a general solution of the large theorem inspired the methods of number theory over the next centuries and into modern times . ${\ displaystyle n = 4}$

The 18th century of number theory is dominated by three mathematicians: Leonhard Euler (1707–1783), Joseph-Louis Lagrange (1736–1813) and Adrien-Marie Legendre (1752–1833).

Euler's complete work is very extensive, and only a small part of his work on number theory can be mentioned here. He introduced the analytical methods into number theory and in this way found a new proof for the infinity of the set of prime numbers. He invented number theoretic functions , especially Euler's φ function , examined partitions and considered the Riemann zeta function 100 years before Bernhard Riemann . He discovered the quadratic reciprocity law (but couldn't prove it), showed that Euler's number is irrational , and solved Fermat's great theorem in the case . ${\ displaystyle e}$ ${\ displaystyle n = 3}$

Lagrange proved Wilson's theorem , established the systematic theory of Pell's equation and the theory of quadratic forms , which only found its conclusion in the first half of the twentieth century.

Legendre introduced the Legendre symbol into number theory and formulated the law of square reciprocity in its current form. His proof, however, uses the infinity of the set of prime numbers in arithmetic progressions, which was only proven in 1832 by Peter Gustav Lejeune Dirichlet .

The next major turning point in the history of number theory will be determined by the work of Carl Friedrich Gauß (1777–1855). Gauss was the first to give (six different) complete proofs for the quadratic reciprocity law. He developed Legendre's theory of quadratic forms and expanded it into a complete theory. He created the arithmetic of the quadratic number fields, although he remained rooted in the conceptual formations of the quadratic forms. In this way he found the law of decomposition of prime numbers into , the Gaussian numbers . Likewise, he first examined the solids of the circle dividing , i.e. H. the solutions of the equation , and developed the calculus of Gaussian sums , which is still very important today. He also discovered the Gaussian prime number theorem , but could not prove it. Overall, one can say that number theory only became an independent and systematically ordered discipline through Gauss. ${\ displaystyle \ mathbb {Z} [i]}$${\ displaystyle x ^ {p-1} = 1}$

### 19th century

The 19th century in particular was a heyday of analytical number theory. Under Niels Henrik Abel (1802–1829), Carl Gustav Jacobi (1804–1851), Gotthold Eisenstein (1823–1852) and Peter Gustav Lejeune Dirichlet (1805–1859) the theory of elliptical functions was developed, which eventually became the theory of elliptical functions Puts curves on a completely new foundation. Dirichlet invents the concept of the L-series and thus proves the prime number theorem in arithmetic progressions . Dirichlet and Eisenstein use the theory of modular shapes to study the number of times a number is represented as the sum of four and five squares, respectively. The uniform rate of Dirichlet (who also excelled in a purely algebraic area) is now one of the pillars of algebraic number theory.

Bernhard Riemann (1826–1866) discovered and proved the functional equation of the Riemannian zeta function and made profound conjectures that linked the analytical properties of this function with arithmetic.

The brief work of Évariste Galois (1811–1832), who developed the Galois theory and with it many old questions, such as the squaring of the circle , the construction of n-vertices using compasses and rulers and the solvability of polynomial equations, was very significant for all of mathematics clarified by root expressions. The Galois theory plays an exposed role in number theory today.

In the algebraic school of the 19th century, Ernst Eduard Kummer (1810–1893), Leopold Kronecker (1823–1891) and Richard Dedekind (1831–1916) should be mentioned. Together these established the cornerstones of the modern structural conception of algebra, in particular the theory of groups , rings and ideals as well as the algebraic number fields . Kronecker introduced the concept of a divisor and discovered the formula known today as the Kronecker-Weber theorem , according to which every Abelian extension of the rational number field is contained in a circle division field . Kummer proved Fermat's great theorem for all regular primes , and Dedekind showed the existence of wholeness bases in number fields.

### The 20th century and the modern

The 20th century finally brought some solutions to number theory that have been researched for so long, namely:

• The complete solution of the simplest (non-trivial) type of the Diophantine equation: the equation belonging to a quadratic form .
• With class field theory and Iwasawa theory a by no means complete, but structurally satisfactory description of the Abelian and cyclic number fields, which led to a general reciprocity law for arbitrary power residues, Artin's reciprocity law .
• The (as yet unproven) solution of the second simplest type of Diophantine equation: the equations associated with elliptic curves .

The discovery of p-adic numbers by Kurt Hensel was groundbreaking for number theory in the 20th century . Building on his work, the mathematicians Hermann Minkowski and Helmut Hasse were able to solve the problem of square shapes: A square shape has a rational zero if and only if it has a zero in every body . This famous Hasse-Minkowski theorem thus provides a first example of a local-global principle that became very important for modern number theory. ${\ displaystyle f (x, y) \ in \ mathbb {Q} [X, Y]}$${\ displaystyle (x, y) \ in \ mathbb {Q} ^ {2}}$${\ displaystyle \ mathbb {Q} _ {p}}$

Building on the work of Kummer, class field theory was developed by a number of mathematicians in the early twentieth century. Among them are especially David Hilbert , Helmut Hasse , Philipp Furtwängler , Teiji Takagi and Emil Artin , whereby Takagi proved the important existential proposition from which Artin derived his famous reciprocity law. However, a complete calculation of the Hilbert symbol and thus the practical application of the reciprocity law was only given by the mathematician Helmut Brückner in the second half of the twentieth century. Class field theory has been translated into the modern language of group cohomology , abstract harmonic analysis, and representation theory by mathematicians such as John Tate and Robert Langlands . Langlands suspected extensive generalizations of class field theory and thus laid the foundation for the Langlands program , which is an important part of current number theory research.

For cyclotomic bodies , Kenkichi Iwasawa finally developed the Iwasawa theory , which could explain these bodies even better. Certain p-adic L-series are linked to these bodies . The main conjecture of the Iwasawa theory, which explains the various possibilities of defining these L-series as equivalent, was proven for totally real number fields by Barry Mazur and Andrew Wiles at the end of the 1980s.

Number theorists also made great strides in the field of elliptic curves . Louis Mordell investigated the group law for elliptic curves and showed that the group of its rational points is always finitely generated, a simple version of Mordell-Weil's theorem . Carl Ludwig Siegel was finally able to show that every elliptic curve only has a finite number of complete solutions ( Siegel's theorem ). This made the problem of whole and rational points on elliptic curves open to attack.

Mordell assumed that for curves of sex> 1 (which are no longer elliptical curves) the set of rational points is always finite ( Mordell conjecture ). This was proven by the German mathematician Gerd Faltings , for which he received the Fields Medal in 1986 . This showed that the equation of Fermat's great theorem could at most have a finite number of solutions (the theorem says that there are none).

The work of Bryan Birch and Peter Swinnerton-Dyer marked a major breakthrough in the second half of the twentieth century. They hypothesized that an elliptic curve has an infinite number of rational solutions if and only if its L-series takes on a value not equal to zero at the point . This is a very weak form of what is known as the Birch and Swinnerton-Dyer conjecture . Although it is in principle unproven, there are strong theoretical and numerical arguments for its correctness. Recently, Don Zagier and Benedict Gross have proven their validity for a variety of elliptical curves. ${\ displaystyle s = 1}$

The proof of the modularity theorem by Christophe Breuil , Brian Conrad , Fred Diamond and Richard Taylor in 2001 should not go unmentioned , after Andrew Wiles had already proven it for most of the elliptic curves (1995). From the part of the modularity theorem (proved by Wiles) it follows in particular that Fermat's great theorem is true.

## Important number theorists (alphabetical)

 Emil Artin (1898–1962), Austrian mathematician Alan Baker (1939-2018), British mathematician Harold Davenport (1907–1969), English mathematician Richard Dedekind (1831-1916), German mathematician Diophantos of Alexandria , Greek mathematician Peter Gustav Lejeune Dirichlet (1805-1859), German mathematician Gotthold Eisenstein (1823-1852), German mathematician Paul Erdős (1913–1996), Hungarian mathematician Euclid , Greek mathematician Leonhard Euler (1707–1783), Swiss mathematician and physicist Gerd Faltings (* 1954), German mathematician Pierre de Fermat (1607–1665), French mathematician and lawyer Gerhard Frey (* 1944), German mathematician Philipp Furtwängler (1869–1940), German mathematician Évariste Galois (1811-1832), French mathematician Carl Friedrich Gauß (1777–1855), German mathematician, statistician, astronomer, geodesist and physicist Sophie Germain (1776–1831), French mathematician Alexander Grothendieck (1928–2014), French mathematician of German descent Jacques Salomon Hadamard (1865–1963), French mathematician Godfrey Harold Hardy (1877-1947), British mathematician Helmut Hasse (1898–1979), German mathematician Erich Hecke (1887–1947), German mathematician Kurt Hensel (1861–1941), German mathematician Charles Hermite (1822–1901), French mathematician David Hilbert (1862–1943), German mathematician Kenkichi Iwasawa (1917–1998), Japanese mathematician Carl Gustav Jacob Jacobi (1804-1851), German mathematician Leopold Kronecker (1823-1891), German mathematician. Ernst Eduard Kummer (1810–1893), German mathematician and university professor Joseph-Louis Lagrange (1736–1813), Italian mathematician and astronomer Serge Lang (1927–2005), French-American mathematician Adrien-Marie Legendre (1752-1833), French mathematician. Franz Lemmermeyer (* 1962), German mathematician, mathematics historian and mathematics teacher John Edensor Littlewood (1885–1977), English mathematician Yuri Manin (* 1937), Russian mathematician Yuri Wladimirowitsch Matijassewitsch (* 1947), Russian mathematician and computer scientist Barry Mazur (* 1937), American mathematician Hermann Minkowski (1864–1909), German mathematician and physicist Louis Mordell (1888–1972), American-British mathematician Srinivasa Ramanujan (1887-1920), Indian mathematician Bernhard Riemann (1826–1866), German mathematician Klaus Friedrich Roth (1925–2015), British mathematician Igor Rostislawowitsch Schafarewitsch (1923–2017), Russian mathematician Atle Selberg (1917–2007), Norwegian-American mathematician Jean-Pierre Serre (* 1926), French mathematician Gorō Shimura (1930–2019), Japanese-American mathematician Carl Ludwig Siegel (1896–1981), German mathematician Peter Swinnerton-Dyer (1927–2018), English mathematician Teiji Takagi (1875–1960), Japanese number theorist Yutaka Taniyama (1927-1958), Japanese mathematician John T. Tate (1925-2019), American mathematician Pafnuti Lwowitsch Tschebyschow (1821-1894), Russian mathematician Charles-Jean de La Vallée Poussin (1866–1962), Belgian mathematician André Weil (1906–1998), French mathematician Hermann Weyl (1885–1955), German mathematician, physicist and philosopher Andrew Wiles (* 1953), British mathematician Ivan Matwejewitsch Winogradow (1891–1983), Russian mathematician Don Zagier (* 1951), American mathematician