Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated. (See the list of number theory topics. This is a list of Number theory topics, by Wikipedia page See also List of recreational number theory topics Topics in cryptography )

The term "arithmetic" is also used to refer to number theory. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone This is a somewhat older term, which is no longer as popular as it once was. Number theory used to be called the higher arithmetic, but this too is dropping out of use. Nevertheless, it still shows up in the names of mathematical fields (arithmetic functions, arithmetic of elliptic curves, fundamental theorem of arithmetic). In Number theory an arithmetic function or arithmetical function is a Function defined on the set of Natural numbers (i In Mathematics, the arithmetic of abelian varieties is the study of the Number theory of an Abelian variety, or family of those In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written This sense of the term arithmetic should not be confused either with elementary arithmetic, or with the branch of logic which studies Peano arithmetic as a formal system. Elementary arithmetic is the most basic kind of Mathematics: it concerns the operations of Addition, Subtraction, Multiplication, and division Logic is the study of the principles of valid demonstration and Inference. In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist Mathematicians working in the field of number theory are called number theorists. A mathematician is a person whose primary area of study and research is the field of Mathematics.

When arranging the natural numbers in a spiral and emphasizing the prime numbers, an intriguing and not fully explained pattern is observed, called the Ulam spiral. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The Ulam spiral, or prime spiral (in other languages also called the Ulam cloth) is a simple method of graphing the Prime numbers that reveals

Fields

Elementary number theory

In elementary number theory, integers are studied without use of techniques from other mathematical fields. Questions of divisibility, use of the Euclidean algorithm to compute greatest common divisors, integer factorizations into prime numbers, investigation of perfect numbers and congruences belong here. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without In Number theory, the Euclidean algorithm (also called Euclid's algorithm) is an Algorithm to determine the Greatest common divisor (GCD In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In mathematics a perfect number is defined as a positive integer which is the sum of its proper positive Divisors that is the sum of the positive divisors excluding In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers Several important discoveries of this field are Fermat's little theorem, Euler's theorem, the Chinese remainder theorem and the law of quadratic reciprocity. Fermat's little theorem (not to be confused with Fermat's last theorem) states that if p is a Prime number, then for any Integer a In Number theory, Euler's theorem (also known as the Fermat-Euler theorem or Euler's totient theorem) states that if n is a positive Integer The Chinese remainder theorem is a result about congruences in Number theory and its generalizations in Abstract algebra. The law of quadratic reciprocity is a theorem from Modular arithmetic, a branch of Number theory, which shows a remarkable relationship between the solvability The properties of multiplicative functions such as the Möbius function and Euler's φ function, integer sequences, factorials, and Fibonacci numbers all also fall into this area. Outside number theory the term multiplicative function is usually used for Completely multiplicative functions This article discusses number theoretic multiplicative For the rational functions defined on the complex numbers see Möbius transformation. In Number theory, the totient \varphi(n of a Positive integer n is defined to be the number of positive integers less than or equal to In Mathematics, an integer sequence is a Sequence (ie an ordered list of Integers An integer sequence may be specified explicitly by Definition The factorial function is formally defined by n!=\prod_{k=1}^n k In Mathematics, the Fibonacci numbers are a Sequence of numbers named after Leonardo of Pisa, known as Fibonacci

Many questions in number theory can be stated in elementary number theoretic terms, but they may require very deep consideration and new approaches outside the realm of elementary number theory to solve. Examples include:

• The Goldbach conjecture concerning the expression of even numbers as sums of two primes. Goldbach's conjecture is one of the oldest unsolved problems in Number theory and in all of Mathematics. In Mathematics, the parity of an object states whether it is even or odd
• Mihăilescu's theorem (formerly Catalan's conjecture) regarding successive integer powers. Catalan's conjecture (occasionally now referred to as Mihăilescu's theorem) is a Theorem in Number theory that was conjectured by the mathematician
• The twin prime conjecture about the infinitude of prime pairs. The twin prime conjecture is a famous unsolved problem in Number theory that involves Prime numbers It states There are infinitely many primes A twin prime is a Prime number that differs from another prime number by Two.
• The Collatz conjecture concerning a simple iteration. The Collatz conjecture is an unsolved Conjecture in Mathematics.
• Fermat's last theorem (stated in 1637, but not proved until 1994) concerning the impossibility of finding nonzero integers x, y, z such that xⁿ + yⁿ = zⁿ for some integer n greater than 2. Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like

The theory of Diophantine equations has even been shown to be undecidable (see Hilbert's tenth problem). In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only In Computability theory and Computational complexity theory, a decision problem is a question in some Formal system with a yes-or-no answer depending on Hilbert's tenth problem is the tenth on the list of Hilbert's problems of 1900

Analytic number theory

Analytic number theory employs the machinery of calculus and complex analysis to tackle questions about integers. In Mathematics, analytic number theory is a branch of Number theory that uses methods from Mathematical analysis to solve number-theoretical problems Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex The prime number theorem (PNT) and the related Riemann hypothesis are examples. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved Waring's problem (representing a given integer as a sum of squares, cubes etc. In Number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every Natural number k there exists an associated positive In Mathematics, a square number, sometimes also called a Perfect square, is an Integer that can be written as the square of some other In Arithmetic and Algebra, the cube of a number n is its third power &mdash the result of multiplying it by itself three times ), 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. The twin prime conjecture is a famous unsolved problem in Number theory that involves Prime numbers It states There are infinitely many primes Goldbach's conjecture is one of the oldest unsolved problems in Number theory and in all of Mathematics. Proofs of the transcendence of mathematical constants, such as π or e, are also classified as analytical number theory. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true In Mathematics, transcendence refers to the property of not being algebraic. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line While statements about transcendental numbers may seem to be removed from the study of integers, they really study the possible values of polynomials with integer coefficients evaluated at, say, e; they are also closely linked to the field of Diophantine approximation, where one investigates "how well" a given real number may be approximated by a rational one. In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of Real numbers by Rational In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions

Algebraic number theory

In algebraic number theory, the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or This article is about the zeros of a function which should not be confused with the value at zero. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions These domains contain elements analogous to the integers, the so-called algebraic integers. This article deals with the ring of complex numbers integral over Z. In this setting, the familiar features of the integers (e. g. unique factorization) need not hold. The virtue of the machinery employed—Galois theory, group cohomology, class field theory, group representations and L-functions—is that it allows to recover that order partly for this new class of numbers. In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory In Abstract algebra, Homological algebra, Algebraic topology and Algebraic number theory, as well as in applications to Group theory proper In Mathematics, class field theory is a major branch of Algebraic number theory. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of The theory of L -functions has become a very substantial and still largely Conjectural, part of contemporary Number theory.

Many number theoretic questions are best attacked by studying them modulo p for all primes p (see finite fields). In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements 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. In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 In Mathematics, the term local analysis has at least two meanings - both derived from the idea of looking at a problem relative to each Prime number p first

Geometry of numbers

The geometry of numbers) incorporates some basic geometric concepts, such as lattices, into number-theoretic questions. In Number theory, the geometry of numbers is a topic and method arising from the work of Hermann Minkowski, on the relationship between Convex sets and It starts with Minkowski's theorem about lattice points in convex sets, and leads to basic proofs of the finiteness of the class number and Dirichlet's unit theorem, two fundamental theorems in algebraic number theory. In Mathematics, Minkowski's theorem is the statement that any Convex set in R n which is symmetric with respect to the origin and with In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the In Mathematics, the extent to which Unique factorization fails in the ring of integers of an Algebraic number field (or more generally any Dedekind domain In Algebraic number theory, Dirichlet 's unit theorem determines the rank of the Group of units in the ring O K

Combinatorial number theory

Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects Paul Erdős is the main founder of this branch of number theory. Paul Erdős ( Hungarian: Erdős Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 &ndash Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a set of integers. In Mathematics, a covering system is a collection \{a_1(\mathrm{mod}\ {n_1}\ \ldots\ a_k(\mathrm{mod}\ {n_k}\} of finitely manyresidue In Number theory, zero-sum problems are a certain class of Combinatorial questions In Additive number theory and Combinatorics, a restricted sumset has the form S=\{a_1+\cdots+a_n\ a_1\in A_1\ldotsa_n\in A_n \ \mathrm{and}\ In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members Algebraic or analytic methods are powerful in this field.

Computational number theory

Computational number theory studies algorithms relevant in number theory. In Mathematics, computational number theory, also known as algorithmic number theory, is the study of Algorithms for performing number theoretic In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Fast algorithms for prime testing and integer factorization have important applications in cryptography. A primality test is an Algorithm for determining whether an input number is prime. Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write"

History

Greek number theory

Number theory was a favorite study among the Greek mathematicians of the late Hellenistic period (3rd century AD) in Alexandria, Egypt, who were aware of the Diophantine equation concept in numerous special cases. Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century Alexandria ( Egyptian Arabic: اسكندريه Eskendereyya; Standard Arabic: ar الإسكندرية Al-Iskandariyya; Ἀλεξάνδρεια This article is about the country of Egypt For a topic outline on this subject see List of basic Egypt topics. In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only The first Greek mathematician to study these equations was Diophantus. Diophantus of Alexandria ( Greek: b between 200 and 214 d between 284 and 298 AD sometimes called "the father of Algebra " a title some claim should

Diophantus also looked for a method of finding integer solutions to linear indeterminate equations, equations that lack sufficient information to produce a single discrete set of answers. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable An indeterminate equation, in Mathematics, is an equation for which there is an infinite set of solutions for example 2x = y is a simple indeterminate equation The equation x + y = 5 is such an equation. Diophantus discovered that many indeterminate equations can be reduced to a form where a certain category of answers is known even though a specific answer is not.

Classical Indian number theory

Diophantine equations were extensively studied by mathematicians in medieval India, who were the first to systematically investigate methods for the determination of integral solutions of Diophantine equations. In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only Aryabhata (499) gave the first explicit description of the general integral solution of the linear Diophantine equation ay + bx = c, which occurs in his text Aryabhatiya. Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics This kuttaka algorithm is considered to be one of the most significant contributions of Aryabhata in pure mathematics, which found solutions to Diophantine equations by means of continued fractions. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} The technique was applied by Aryabhata to give integral solutions of simulataneous linear Diophantine equations, a problem with important applications in astronomy. He also found the general solution to the indeterminate linear equation using this method. An indeterminate equation, in Mathematics, is an equation for which there is an infinite set of solutions for example 2x = y is a simple indeterminate equation A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable

Brahmagupta in 628 handled more difficult Diophantine equations. Brahmagupta ( (598–668 was an Indian mathematician and astronomer. He used the chakravala method to solve quadratic Diophantine equations, including forms of Pell's equation, such as 61x² + 1 = y². The chakravala method is a cyclic Algorithm to solve indeterminate Quadratic equations including Pell's equation. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. Pell's equation is any Diophantine equation of the form x^2-ny^2=1\ where n is a nonsquare integer and x His Brahma Sphuta Siddhanta was translated into Arabic in 773 and was subsequently translated into Latin in 1126. The main work of Brahmagupta, Brahmasphuta-siddhanta (The Opening of the Universe, written in the year 628, contains some remarkably advanced ideas including Arabic (ar الْعَرَبيّة (informally ar عَرَبيْ) in terms of the number of speakers is the largest living member of the Semitic language Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. The equation 61x² + 1 = y² was later posed as a problem in 1657 by the French mathematician Pierre de Fermat. This article is about the country For a topic outline on this subject see List of basic France topics. Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the The general solution to this particular form of Pell's equation was found over 70 years later by Leonhard Euler, while the general solution to Pell's equation was found over 100 years later by Joseph Louis Lagrange in 1767. Meanwhile, many centuries ago, the general solution to Pell's equation was recorded by Bhaskara II in 1150, using a modified version of Brahmagupta's chakravala method, which he also used to find the general solution to other indeterminate quadratic equations and quadratic Diophantine equations. Bhaskara (1114 &ndash 1185 also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher" was an Indian mathematician Bhaskara's chakravala method for finding the general solution to Pell's equation was much simpler than the method used by Lagrange over 600 years later. Bhaskara also found solutions to other indeterminate quadratic, cubic, quartic, and higher-order polynomial equations. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. In Mathematics, a quartic equation is one which can be expressed as a Quartic function equalling zero In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations Narayana Pandit further improved on the chakravala method and found more general solutions to other indeterminate quadratic and higher-order polynomial equations. Narayana Pandit (नारायण पण्डित (1340 &ndash 1400 was a major Mathematician of the Kerala school.

Islamic number theory

From the 9th century, Islamic mathematics had a keen interest in number theory. The first of these mathematicians was Thabit ibn Qurra, who discovered an algorithm which allowed pairs of amicable numbers to be found, that is two numbers such that each is the sum of the proper divisors of the other. (836 in Harran, Mesopotamia &ndash February 18, 901 in Baghdad) was an Arab astronomer, mathematician Amicable numbers are two different Numbers so related that the sum of the Proper divisors of the one is equal to the other one being considered In the 10th century, Al-Baghdadi looked at a slight variant of Thabit ibn Qurra's method.

In the 10th century, al-Haitham seems to have been the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors) as those of the form 2^{k − 1}(2^k − 1) where 2^k − 1 is prime. TemplateInfobox Muslim scholars --> ( Arabic: ابو علی، حسن بن حسن بن هيثم Latinized In mathematics a perfect number is defined as a positive integer which is the sum of its proper positive Divisors that is the sum of the positive divisors excluding Al-Haytham is also the first person to state Wilson's theorem, namely that if p is prime then 1 + (p − 1)! is divisible by p. In Mathematics, Wilson's theorem states that p > 1 is a Prime number If and only if (p-1!\ \equiv\ -1\ (\mbox{mod}\ p It is unclear whether he knew how to prove this result. It is called Wilson's theorem because of a comment made by Edward Waring in 1770 that John Wilson had noticed the result. Edward Waring (1736 – August 15, 1798) was an English Mathematician who was born in Old Heath (near Shrewsbury) John Wilson (1741&ndash1793 was an English Mathematician, born in Applethwaite, Westmorland. There is no evidence that John Wilson knew how to prove it and most certainly Waring did not. Lagrange gave the first proof in 1771.

Amicable numbers played a large role in Islamic mathematics. In the 13th century, Persian mathematician Al-Farisi gave a new proof of Thabit ibn Qurra's theorem, introducing important new ideas concerning factorisation and combinatorial methods. layout and formatting it should ensure no clashes with the top of the infobox Kamal al-Din Abu'l-Hasan Muhammad Al-Farisi (1267-ca1319/1320 (كمال‌الدين ابوالحسن محمد فارسی was a prominent Persian Muslim physicist He also gave the pair of amicable numbers 17296, 18416 which have been attributed to Euler, but we know that these were known earlier than al-Farisi, perhaps even by Thabit ibn Qurra himself. In the 17th century, Muhammad Baqir Yazdi gave the pair of amicable numbers 9,363,584 and 9,437,056 still many years before Euler's contribution. Muhammad Baqir Yazdi is an Iranian mathematician living 16th century

Early European number theory

Number theory began in Europe in the 16th and 17th centuries, with François Viète, Bachet de Meziriac, and especially Fermat, whose infinite descent method was the first general proof of diophantine questions. François Viète (or Vieta) seigneur de la Bigotière ( 1540 - February 13, 1603) generally known as Franciscus Vieta, Claude Gaspard Bachet de Méziriac ( October 9, 1581 - February 26, 1638) was a French Mathematician born in Bourg-en-Bresse Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the In Mathematics, a proof by infinite descent is a particular kind of proof by Mathematical induction. Fermat's last theorem was posed as a problem in 1637, a proof of which wasn't found until 1994. Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like Fermat also posed the equation 61x² + 1 = y² as a problem in 1657.

In the eighteenth century, Euler and Lagrange made important contributions to number theory. Euler did some work on analytic number theory, and found a general solution to the equation 61x² + 1 = y². In Mathematics, analytic number theory is a branch of Number theory that uses methods from Mathematical analysis to solve number-theoretical problems Lagrange found a solution to the more general Pell's equation. Euler and Lagrange solved these Pell equations by means of continued fractions, though this was more difficult than the Indian chakravala method. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country The chakravala method is a cyclic Algorithm to solve indeterminate Quadratic equations including Pell's equation.

Beginnings of modern number theory

Around the beginning of the nineteenth century books of Legendre (1798), and Gauss put together the first systematic theories in Europe. Adrien-Marie Legendre ( September 18 1752 – January 10 1833) was a French Mathematician. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Gauss's Disquisitiones Arithmeticae (1801) may be said to begin the modern theory of numbers. The Disquisitiones Arithmeticae is a textbook of Number theory written by German Mathematician Carl Friedrich Gauss in 1798

The formulation of the theory of congruences starts with Gauss's Disquisitiones. See Congruence (geometry for the term as used in elementary geometry He introduced the symbolism

and explored most of the field. Chebyshev published in 1847 a work in Russian on the subject, and in France Serret popularised it. Pafnuty Lvovich Chebyshev (Пафну́тий Льво́вич Чебышёв ( –) was a Russian Mathematician. Joseph Alfred Serret ( August 30[[ 819]] - March 2, 1885) was a French Mathematician who was born in Paris France

Besides summarizing previous work, Legendre stated the law of quadratic reciprocity. Adrien-Marie Legendre ( September 18 1752 – January 10 1833) was a French Mathematician. The law of quadratic reciprocity is a theorem from Modular arithmetic, a branch of Number theory, which shows a remarkable relationship between the solvability This law, discovered by induction and enunciated by Euler, was first proved by Legendre in his Théorie des Nombres (1798) for special cases. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that Independently of Euler and Legendre, Gauss discovered the law about 1795, and was the first to give a general proof. The following have also contributed to the subject: Cauchy; Dirichlet whose Vorlesungen über Zahlentheorie is a classic; Jacobi, who introduced the Jacobi symbol; Liouville, Zeller(?), Eisenstein, Kummer, and Kronecker. Johann Peter Gustav Lejeune Dirichlet (ləʒœn diʀiçle February 13, 1805 &ndash May 5, 1859) was a German Mathematician de ''[[Vorlesungen über Zahlentheorie]]'' ( German for Lectures on Number Theory) is a textbook of Number theory written by German mathematicians Carl Gustav Jacob Jacobi ( December 10, 1804 - February 18, 1851) was a Prussian Mathematician, widely considered to be The Jacobi symbol is a generalization of the Legendre symbol introduced by Jacobi in 1837 Joseph Liouville ( March 24 1809 &ndash September 8 1882) was a French Mathematician. Ferdinand Gotthold Max Eisenstein ( 16 April, 1823 – 11 October, 1852) was a German Mathematician. Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued The theory extends to include cubic and quartic reciprocity, (Gauss, Jacobi who first proved the law of cubic reciprocity, and Kummer). In Mathematics, cubic reciprocity is any of various results connecting the solvability of two related Cubic equations in Modular arithmetic.

To Gauss is also due the representation of numbers by binary quadratic forms. In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables

Prime number theory

A recurring and productive theme in number theory is the study of the distribution of prime numbers. Carl Friedrich Gauss conjectured the limit of the number of primes not exceeding a given number (the prime number theorem) as a teenager. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German

Chebyshev (1850) gave useful bounds for the number of primes between two given limits. Pafnuty Lvovich Chebyshev (Пафну́тий Льво́вич Чебышёв ( –) was a Russian Mathematician. Riemann introduced complex analysis into the theory of the Riemann zeta function. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in This led to a relation between the zeros of the zeta function and the distribution of primes, eventually leading to a proof of prime number theorem independently by Hadamard and de la Vallée Poussin in 1896. Jacques Salomon Hadamard ( December 8, 1865 – October 17, 1963) was a French Mathematician best known for his proof of Charles-Jean Étienne Gustave Nicolas Baron de la Vallée Poussin ( August 14[[ 866]] - March 2[[ 962]] was a Belgian Mathematician. However, an elementary proof was given later by Paul Erdős and Atle Selberg in 1949. Paul Erdős ( Hungarian: Erdős Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 &ndash Atle Selberg ( June 14, 1917 &ndash August 6, 2007) was a Norwegian Mathematician known for his work in Analytic number Here elementary means that it does not use techniques of complex analysis; however, the proof is still very ingenious and difficult. The Riemann hypothesis, which would give much more accurate information, is still an open question. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved

Nineteenth-century developments

Cauchy, Poinsot (1845), Lebesgue (1859, 1868), and notably Hermite have added to the subject. Louis Poinsot (1777 - 1859 was a French Mathematician and Physicist. Henri Léon Lebesgue leɔ̃ ləˈbɛg ( June 28, 1875, Beauvais &ndash July 26, 1941, Paris) was a French Charles Hermite (ʃaʁl ɛʁˈmit ( December 24, 1822 &ndash January 14, 1901) was a French Mathematician who did In the theory of ternary forms, Eisenstein has been a leader, and to him and H. J. S. Smith is also due a noteworthy advance in the theory of forms in general. Ferdinand Gotthold Max Eisenstein ( 16 April, 1823 – 11 October, 1852) was a German Mathematician. Henry John Stephen Smith ( November 2 1826 Dublin, Ireland &ndash February 9 1883 Oxford, Oxfordshire Smith gave a complete classification of ternary quadratic forms, and extended Gauss's researches concerning real quadratic forms to complex forms. In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables The investigations concerning the representation of numbers by the sum of 4, 5, 6, 7, 8 squares were advanced by Eisenstein and the theory was completed by Smith.

Dirichlet was the first to lecture upon the subject in a German university. Johann Peter Gustav Lejeune Dirichlet (ləʒœn diʀiçle February 13, 1805 &ndash May 5, 1859) was a German Mathematician Among his contributions is the extension of Fermat's last theorem:

which Euler and Legendre had proven for n = 3,4 (and therefore by implication, all multiples of 3 and 4), Dirichlet showing that . Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like Among the later French writers are Borel; Poincaré, whose memoirs are numerous and valuable; Tannery, and Stieltjes. Félix Édouard Justin Émile Borel ( January 7, 1871 in Saint-Affrique, France &ndash February 3, 1956 in Paris Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician Jules Tannery ( March 24, 1848 – December 11, 1910) was a French Mathematician who notably studied under Charles This article is about Thomas Joannes Stieltjes (pronounced 'stiltʃəs the mathematician Among the leading contributors in Germany were Kronecker, Kummer, Schering, Bachmann, and Dedekind. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician. Ernst Christian Friedrich Schering ( May 31, 1824 &ndash December 27, 1889) was a German apothecary and industrialist who created Paul Gustav Heinrich Bachmann ( June 22 1837 &ndash March 31 1920) was a German Mathematician. Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important In Austria Stolz's Vorlesungen über allgemeine Arithmetik (1885-86), and in England Mathews' Theory of Numbers (Part I, 1892) were scholarly general works. Otto Stolz ( May 3, 1842 – October 25, 1905) was an Austrian mathematician noted for his work on Mathematical analysis and Genocchi, Sylvester, and J. W. L. Glaisher have also added to the theory. Angelo Genocchi was an Italian Mathematician who specialized in Number theory. James Joseph Sylvester ( September 3, 1814 London – March 15, 1897 Oxford) was an English Mathematician James Whitbread Lee Glaisher ( 5 November 1848 - 7 December 1928) son of James Glaisher, the meteorologist was a prolific English

Late nineteenth- and early twentieth-century developments

It was the time of major advancements in number theory due to the work of Axel Thue on diophantine equations, of David Hilbert in algebraic number theory (he also proved the Waring's prime number conjecture), and to the creation of geometric number theory by Hermann Minkowski, but also thanks to Adolf Hurwitz, Georgy F. Voronoy, Waclaw Sierpinski, Derrick Norman Lehmer and several others. Axel Thue ( 19 February 1863 – 7 March 1922) was a Norwegian Mathematician, known for highly original work in Diophantine In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers In Mathematics, Waring's prime number conjecture is a Conjecture in Number theory, closely related to Vinogradov's theorem. In Number theory, the geometry of numbers is a topic and method arising from the work of Hermann Minkowski, on the relationship between Convex sets and Hermann Minkowski ( June 22 1864 – January 12 1909) was a Russian born German Mathematician, of Jewish Adolf Hurwitz ( 26 March 1859 - 18 November 1919) (ˈadɒlf ˈhurvits was a German mathematician and was described by Jean-Pierre Georgy Feodosevich Voronoy (Георгий Феодосьевич Вороной 28 April 1868 &ndash 20 November 1908) was a famous Russian Wacław Franciszek Sierpiński ( March 14 1882 — October 21 1969) (ˈvaʦwaf fraɲˈʨiʂɛk ɕɛrˈpʲiɲskʲi a Polish Mathematician Derrick Norman Lehmer ( 27 July 1867, Somerset Indiana, USA &mdash 8 September 1938 in Berkeley California, USA

Twentieth-century developments

Major figures in twentieth-century number theory include Hermann Weyl, Nikolai Chebotaryov, Emil Artin, Erich Hecke, Helmut Hasse, Alexander Gelfond, Yuri Linnik, Paul Erdős, Gerd Faltings, G. H. Hardy, Edmund Landau, Louis Mordell, John Edensor Littlewood, Srinivasa Ramanujan, André Weil, Ivan Vinogradov, Atle Selberg, Carl Ludwig Siegel, Igor Shafarevich, John Tate, Robert Langlands, Goro Shimura, Kenkichi Iwasawa, Jean-Pierre Serre, Pierre Deligne, Enrico Bombieri, Alan Baker, Peter Swinnerton-Dyer, Bryan Birch, Vladimir Drinfeld, Laurent Lafforgue, Andrew Wiles, and Richard Taylor. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. Nikolai Chebotaryov (often spelled Chebotarov or Chebotarev) (Николай Григорьевич Чеботарёв Микола Григорович Чоботарьов Emil Artin ( March 3, 1898, in Vienna – December 20, 1962, in Hamburg) was an Austrian Mathematician Erich Hecke ( September 20, 1887 &ndash February 13, 1947) was a German Mathematician. Helmut Hasse (ˈhasə ( 25 August 1898 – 26 December 1979) was a German Mathematician working in Algebraic Alexander Osipovich Gelfond (Александр Осипович Гельфонд October 24 1906, St Petersburg — November 7 1968 Yuri Vladimirovich Linnik (Юрий Владимирович Линник January 8 1915 – June 30 1972 was a Russian Mathematician active in Number theory Paul Erdős ( Hungarian: Erdős Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 &ndash Gerd Faltings (born July 28, 1954 in Gelsenkirchen -Buer is a German Mathematician known for his work in arithmetic Algebraic Godfrey Harold Hardy FRS ( February 7, 1877 Cranleigh, Surrey, England &ndash December 1, 1947 Edmund Georg Hermann (Yehezkel Landau ( February 14, 1877 – February 19, 1938) was a German Jewish Mathematician Louis Joel Mordell ( 28 January 1888 - 12 March 1972) was a British mathematician known for pioneering research in Number theory. John Edensor Littlewood ( 9 June 1885 &ndash 6 September 1977) was a British Mathematician, best known for his long collaboration André Weil should not be confused with two other mathematicians with similar names Hermann Weyl (1885-1955 who made substantial contributions Ivan Matveevich Vinogradov (Иван Матвеевич Виноградов September 14, 1891 &ndash March 20, 1983) was a Russian Atle Selberg ( June 14, 1917 &ndash August 6, 2007) was a Norwegian Mathematician known for his work in Analytic number Carl Ludwig Siegel ( December 31 1896 &ndash April 4 1981) was a Mathematician specialising in Number theory. Igor Rostislavovich Shafarevich ( Russian: Игорь Ростиславович Шафаревич born June 3, 1923 in Zhytomyr) is a John Torrence Tate Jr born March 13, 1925 in Minneapolis Minnesota, is an American Mathematician, distinguished for many fundamental Robert Phelan Langlands (born October 6, 1936 in New Westminster, British Columbia, Canada) was one of the most influential Mathematicians Goro Shimura ( Japanese: 志村 五郎 Shimura Gorō; born 1930 in Hamamatsu Japan) is a Japanese Mathematician, and currently Kenkichi Iwasawa (岩澤 健吉 Iwasawa Kenkichi, September 11 1917 – October 26 1998 Pierre René Viscount Deligne (born 3 October 1944 in Brussels) is a Belgian Mathematician. Enrico Bombieri (born November 26, 1940) is an Italian Mathematician, born in Milan. Alan Baker (born on August 19 1939) is an English Mathematician. Sir Henry Peter Francis Swinnerton-Dyer 16th Baronet KBE FRS (b Bryan John Birch FRS (born 1931 is a British mathematician His name has been given to the Birch and Swinnerton-Dyer conjecture. Vladimir Gershonovich Drinfel'd (born February 4, 1954 in the Ukrainian SSR; Владимир Гершонович Дринфельд is a Russian Laurent Lafforgue (born November 6, 1966, in Antony Hauts-de-Seine, France) is a French Mathematician. Sir Andrew John Wiles KBE FRS (born 11 April 1953 is a British Mathematician and a professor at Princeton University Richard Taylor (born Richard Lawrence Taylor 19 May 1962) is a British Mathematician working in the field of Number theory

Milestones in twentieth-century number theory include the proof of Fermat's Last Theorem by Andrew Wiles in 1994 and the proof of the related Taniyama–Shimura conjecture in 1999. Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like Sir Andrew John Wiles KBE FRS (born 11 April 1953 is a British Mathematician and a professor at Princeton University In Mathematics, the modularity theorem establishes an important connection between Elliptic curves over the field of Rational numbers and Modular forms

Quotations

• Mathematics is the queen of the sciences and number theory is the queen of mathematics. —Gauss^[1]
• God invented the integers; all else is the work of man. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German —Kronecker^[2]

References

1. ^ Quoted in Gauss zum Gedächtniss (1856) by Wolfgang Sartorius von Waltershausen
2. ^ "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" Heinrich Weber: Leopold Kronecker. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued Jahresberichte D. M. V 2 (1893) 5-31

• Apostol, T. M. (1986). Introduction to Analytic Number Theory. Springer-Verlag. ISBN 0-387-90163-9.  
• Dedekind, Richard (1963). Essays on the Theory of Numbers. Cambridge University Press. ISBN 0-486-21010-3.  
• Davenport, Harold (1999). The Higher Arithmetic: An Introduction to the Theory of Numbers (7th ed. ). Cambridge University Press. ISBN 0-521-63446-6.  
• Guy, Richard K. (1981). Unsolved Problems in Number Theory. Springer-Verlag. ISBN 0-387-90593-6.  
• Hardy, G. H. and Wright, E. M. (1980). An Introduction to the Theory of Numbers (5th ed. ). Oxford University Press. ISBN 0-19-853171-0.  
• Niven, Ivan, Zuckerman, Herbert S. and Montgomery, Hugh L. (1991). An Introduction to the Theory of Numbers (5th ed. ). Wiley Text Books. ISBN 0-471-62546-9.  
• Ore, Oystein (1948). Number Theory and Its History. Dover Publications, Inc. . ISBN 0-486-65620-9.  
• Smith, David. History of Modern Mathematics (1906) (adapted public domain text)
• Dutta, Amartya Kumar (2002). 'Diophantine equations: The Kuttaka', Resonance - Journal of Science Education.
• O'Connor, John J. and Robertson, Edmund F. (2004). 'Arabic/Islamic mathematics', MacTutor History of Mathematics archive. The MacTutor History of Mathematics archive is an award-winning website maintained by John J
• O'Connor, John J. and Robertson, Edmund F. (2004). 'Index of Ancient Indian mathematics', MacTutor History of Mathematics archive. The MacTutor History of Mathematics archive is an award-winning website maintained by John J
• O'Connor, John J. and Robertson, Edmund F. (2004). 'Numbers and Number Theory Index', MacTutor History of Mathematics archive. The MacTutor History of Mathematics archive is an award-winning website maintained by John J
• Important publications in number theory

External links

• Number Theory Web