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. ... For other uses, see Number (disambiguation). ... The integers are commonly denoted by the above symbol. ...

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. ...

The term "arithmetic" is also used to refer to number theory. 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). 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. Mathematicians working in the field of number theory are called number theorists. Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ... In number theory and computability theory, subfields of mathematics, a number-theoretic function is any function whose domain is the set of natural numbers. ... 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 factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ... Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. ... Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of the natural numbers. ... In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

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.

Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... 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 a pattern which has never been fully explained. ...

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. Several important discoveries of this field are Fermat's little theorem, Euler's theorem, the Chinese remainder theorem and the law of quadratic reciprocity. The properties of multiplicative functions such as the Möbius function, Euler's φ function, integer sequences, factorials, and Fibonacci numbers all also fall into this area. In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In number theory, the Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers). ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... Prime decomposition redirects here. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... Fermats little theorem (not to be confused with Fermats last theorem) states that if p is a prime number, then for any integer a, This means that if you start with a number, initialized to 1, and repeatedly multiply, for a total of p multiplications, that number by... In number theory, Eulers theorem (also known as the Fermat-Euler theorem or Eulers totient theorem) states that if n is a positive integer and a is coprime to n, then aφ(n) ≡ 1 (mod n) where φ(n) is Eulers totient function and mod denotes the congruence... Several related results in number theory and abstract algebra are known under the name Chinese remainder theorem. ... In number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. ... In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ... The classical Möbius function is an important multiplicative function in number theory and combinatorics. ... In number theory, the totient φ(n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ... In mathematics, an integer sequence is a sequence (i. ... For factorial rings in mathematics, see unique factorisation domain. ... A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above – see golden spiral In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation: That is...

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.
• Catalan's conjecture (now Mihăilescu's theorem) regarding successive integer powers.
• The twin prime conjecture about the infinitude of prime pairs.
• The Collatz conjecture concerning a simple iteration.
• 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.

The theory of Diophantine equations has even been shown to be undecidable (see Hilbert's tenth problem). Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ... In mathematics, the parity of an object refers to whether it is even or odd. ... Mihăilescus theorem (formerly Catalans conjecture) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proved in 2002 by Preda Mihăilescu. ... The twin prime conjecture is a famous problem in number theory that involves prime numbers. ... A twin prime is a prime number that differs from another prime number by two. ... The Collatz conjecture is an unsolved conjecture in mathematics. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... 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. ... Hilberts tenth problem is the tenth on the list of Hilberts problems of 1900. ...

Analytic number theory

Analytic number theory employs the machinery of calculus and complex analysis to tackle questions about integers. The prime number theorem (PNT) and the related Riemann hypothesis are examples. Waring's problem (representing a given integer as a sum of squares, cubes etc.), 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. Proofs of the transcendence of mathematical constants, such as π or e, are also classified as analytical number theory. 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. Analytic number theory is the branch of number theory that uses methods from mathematical analysis. ... For other uses, see Calculus (disambiguation). ... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ... There is also the Riemann hypothesis for curves over finite fields. ... In number theory, Warings problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers. ... In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. ... In arithmetic and algebra, the cube of a number n is its third power — the result of multiplying it by itself two times: n3 = n × n × n. ... The twin prime conjecture is a famous problem in number theory that involves prime numbers. ... Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... In mathematics, a transcendental function is a function which is not expressible as a composition of a finite number of elementary operations, or inverses of functions so constructible, where the elementary operations consist of addition, multiplication, taking additive or multiplicative inverses, and integer root extraction. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... In mathematics, a transcendental number is any irrational number that is not an algebraic number, i. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...

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. These domains contain elements analogous to the integers, the so-called algebraic integers. 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. This article or section does not cite its references or sources. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ... This article is about homology and cohomology of a group. ... In mathematics, class field theory is a major branch of algebraic number theory. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... 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). 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 abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... 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, and then later trying to integrate the information gained at each prime into a global picture. ...

Geometric number theory

Geometric number theory (traditionally called the geometry of numbers) incorporates some basic geometric concepts, such as lattices, into number-theoretic questions. 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 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 lattices in n-dimensional space. ... 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 lattices in n-dimensional space. ... In mathematics, Minkowskis theorem in the geometry of numbers applies to convex symmetric sets and lattices; it relates the number of contained lattice points to the volume of such a set. ... See lattice for other meanings of this term, both within and without mathematics. ... Look up Convex set in Wiktionary, the free dictionary. ... In mathematics the theory of algebraic number fields gives rise to a finite abelian group constructed from each such field, its ideal class group. ... In algebraic number theory, Dirichlets unit theorem determines the rank of the group of units in the ring OK of algebraic integers of a number field K. The statement is that the rank is r + s − 1 where r is the number of real embeddings and 2s the number...

Combinatorial number theory

Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdős is the main founder of this branch of number theory. Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a set of integers. Algebraic or analytic methods are powerful in this field. Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ... Paul ErdÅ‘s (Hungarian: ErdÅ‘s Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian-born mathematician. ... In mathematics, a covering system is a collection of finitely many residue classes whose union covers all integers. ... In number theory, zero-sum problems are a certain class of combinatorial questions. ... In combinatorial number theory, a restricted sumset has the form where are finite nonempty subsets of a field and is a polynomial over . ... In mathematics, an arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. ...

Computational number theory

Computational number theory studies algorithms relevant in number theory. Fast algorithms for prime testing and integer factorization have important applications in cryptography. In mathematics, computational number theory is a study of number theory with the aid of computer powers. ... Flowcharts are often used to represent algorithms. ... A primality test is an algorithm for determining whether an input number is prime. ... Prime decomposition redirects here. ... The German Lorenz cipher machine, used in World War II for encryption of very high-level general staff messages Cryptography (or cryptology; derived from Greek κρυπτός kryptós hidden, and the verb γράφω gráfo write or λεγειν legein to speak) is the study of message secrecy. ...

History

Vedic number theory

Mathematicians in India were interested in finding integral solutions of Diophantine equations since the Vedic era. The earliest geometric use of Diophantine equations can be traced back to the Sulba Sutras, which were written between the 8th and 6th centuries BC. Baudhayana (c. 800 BC) found two sets of positive integral solutions to a set of simultaneous Diophantine equations, and also used simultaneous Diophantine equations with up to four unknowns. Apastamba (c. 600 BC) used simultaneous Diophantine equations with up to five unknowns^{[citation needed]}. In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. ... This article or section does not cite its references or sources. ... The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... Baudhāyana, (fl. ... Apastamba (c. ...

Jaina number theory

In India, Jaina mathematicians developed the earliest systematic theory of numbers from the 4th century BC to the 2nd century CE. The Jaina text Surya Prajinapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. Each of these was further subdivided into three orders: JAIN is an activity within the Java Community Process, developing APIs for the creation of telephony (voice and data) services. ...

• Enumerable: lowest, intermediate and highest.
• Innumerable: nearly innumerable, truly innumerable and innumerably innumerable.
• Infinite: nearly infinite, truly infinite, infinitely infinite.

The Jains were the first to discard the idea that all infinites were the same or equal. They recognized five different types of infinity: infinite in one and two directions (one dimension), infinite in area (two dimensions), infinite everywhere (three dimensions), and infinite perpetually (infinite number of dimensions). 2-dimensional renderings (ie. ...

The highest enumerable number N of the Jains corresponds to the modern concept of aleph-null (the cardinal number of the infinite set of integers 1, 2, ...), the smallest cardinal transfinite number. The Jains also defined a whole system of transfinite cardinal numbers, of which is the smallest. In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ...

In the Jaina work on the theory of sets, two basic types of transfinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asmkhyata and ananata, between rigidly bounded and loosely bounded infinities. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... This article is about ontology in philosophy. ...

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. The first Greek mathematician to study these equations was Diophantus. 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 AD around the Eastern shores of the Mediterranean. ... This article is about the city in Egypt. ... In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. ... Title page of the 1621 edition of Diophantus Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac. ...

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. 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. Graph sample of linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. ... An indeterminate equation is an equation for which there is an infinite set of solutions – for example, 2x = y. ...

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. 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. 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. 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. In mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. ... For other uses, see Aryabhata (disambiguation). ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... An indeterminate equation is an equation for which there is an infinite set of solutions – for example, 2x = y. ... Graph sample of linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. ...

Brahmagupta in 628 handled more difficult Diophantine equations. He used the chakravala method to solve quadratic Diophantine equations, including forms of Pell's equation, such as 61x² + 1 = y². His Brahma Sphuta Siddhanta was translated into Arabic in 773 and was subsequently translated into Latin in 1126. The equation 61x² + 1 = y² was later posed as a problem in 1657 by the French mathematician Pierre de Fermat. 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'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. Narayana Pandit further improved on the chakravala method and found more general solutions to other indeterminate quadratic and higher-order polynomial equations. Brahmagupta (ब्रह्मगुप्त) ( ) (589–668) was an Indian mathematician and astronomer. ... The Chakravala method is a cyclic algorithm to solve quadratic integer equations. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ... The main work of Brahmagupta, Brahmasphutasiddhanta (The Opening of the Universe), written in 628, contains some remarkably advanced ideas, including a good understanding of the mathematical role of zero, rules for manipulating both positive and negative numbers, a method for computing square roots, methods of solving linear and some quadratic... Arabic can mean: From or related to Arabia From or related to the Arabs The Arabic language; see also Arabic grammar The Arabic alphabet, used for expressing the languages of Arabic, Persian, Malay ( Jawi), Kurdish, Panjabi, Pashto, Sindhi and Urdu, among others. ... For other uses, see Latin (disambiguation). ... Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ... Euler redirects here. ... Joseph-Louis, comte de Lagrange (January 25, 1736 Turin, Kingdom of Sardinia - April 10, 1813 Paris) was an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics as arguably the greatest mathematician of the 18th century. ... Bhāskara (1114-1185), also called Bhāskara II and Bhāskarācārya (Bhaskara the teacher) was an Indian mathematician. ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... In mathematics, a quartic equation is the result of setting a quartic function equal to zero. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... Narayana Pandit (नारायण पण्डित) (1340-1400) 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. In the 10th century, Al-Baghdadi looked at a slight variant of Thabit ibn Qurra's method. In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed by the Islamic civilization between 622 and 1600. ... Abul Hasan Thabit ibn Qurra ibn Marwan al-Sabi al-Harrani, (826 – February 18, 901) was an Arab astronomer and mathematician. ... Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself. ... Al-Baghdadi or just Baghdadi is an Arabic nesbat, meaning from Baghdad. It is usually added at the end of names as a specifier. ...

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. 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. 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. 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. Alhazen Abu Ali al-Hasan Ibn Al-Haitham (also: Ibn al Haitham) (965-1040) (Arabic: أبو علي الحسن بن الهيثم) was an Arab Muslim mathematician; he is sometimes called al-Basri (Arabic: البصري), after his birthplace Basra, Arab Islamic Caliphate (now Iraq). ... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... In mathematics, Wilsons theorem (also known as Al-Haythams theorem) states that p > 1 is a prime number if and only if (see factorial and modular arithmetic for the notation). ... Edward Waring (1736 - August 15, 1798) was British mathematician who was born in Old Heath (near Shrewsbury) Shropshire England and died in Pontesbury Shropshire England He was Lucasian professor of mathematics at Cambridge University from 1760 until his death. ... John Wilson (1741 – 1793) was an English mathematician who had a theorem, Wilsons Theorem, named after him for its discovery, not its proof. ...

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. 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. This article is about the Persian people, an ethnic group found mainly in Iran. ... Kamal al-Din Abul Hasan Muhammad Al-Farisi (in Persian: كمال الدين ابوالحسن محمد الفارسي) (1260 - 1320) was a prominent Persian mathematician and physicist. ... 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. Fermat's last theorem was posed as a problem in 1637, a proof of which wasn't found until 1994. Fermat also posed the equation 61x² + 1 = y² as a problem in 1657. For other uses, see Europe (disambiguation). ... François Viète. ... Claude Gaspard Bachet de Méziriac (October 9, 1581 - February 26, 1638) was a French mathematician, born in Bourg_en_Bresse. ... Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parliament of Toulouse and a mathematician who is given credit for the development of modern calculus. ... In mathematics, a proof by infinite descent is a particular kind of proof by mathematical induction. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ...

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. 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. Analytic number theory is the branch of number theory that uses methods from mathematical analysis. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... The Chakravala method is a cyclic algorithm to solve quadratic integer equations. ...

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. Gauss's Disquisitiones Arithmeticae (1801) may be said to begin the modern theory of numbers. Adrien-Marie Legendre (September 18, 1752–January 10, 1833) was a French mathematician. ... Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... The Disquisitiones Arithmeticae is a textbook of number theory written by German mathematician Carl Friedrich Gauss and first published in 1801 when Gauss was 24. ...

The formulation of the theory of congruences starts with Gauss's Disquisitiones. He introduced the symbolism In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ...

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 (Russian: ) (May 16 [O.S. May 4] 1821 – December 8 [O.S. November 26] 1894)[1] was a Russian mathematician. ... Joseph Alfred Serret (August 30, 1819 - March 2, 1885) was a french mathematician who was born in Paris France and died in Versailles France. ...

Besides summarizing previous work, Legendre stated the law of quadratic reciprocity. This law, discovered by induction and enunciated by Euler, was first proved by Legendre in his Théorie des Nombres (1798) for special cases. 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. The theory extends to include cubic and quartic reciprocity, (Gauss, Jacobi who first proved the law of cubic reciprocity, and Kummer). Adrien-Marie Legendre (September 18, 1752–January 10, 1833) was a French mathematician. ... In mathematics, in number theory, the law of quadratic reciprocity, conjectured by Euler and Legendre and first satisfactorily proved by Gauss, connects the solvability of two related quadratic equations in modular arithmetic. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ... Vorlesungen über Zahlentheorie (Lectures on Number Theory) is a textbook of number theory written by German mathematicians P.G.L. Dirichlet and Richard Dedekind, and published in 1863. ... Karl Gustav Jacob Jacobi Carl Gustav Jacob Jacobi (December 10, 1804 - February 18, 1851), was not only a great German mathematician but also considered by many as the most inspiring teacher of his time (Bell, p. ... The Jacobi symbol generalises the Legendre symbol. ... Joseph Liouville (born March 24, 1809, died September 8, 1882) was a French mathematician. ... Ferdinand Gotthold Max Eisenstein (April 16, 1823 - October 11, 1852) was a German mathematician. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ... This page may meet Wikipedias criteria for speedy deletion. ...

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 or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...

Chebyshev (1850) gave useful bounds for the number of primes between two given limits. Riemann introduced complex analysis into the theory of the Riemann zeta function. 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. However, an elementary proof was given later by Paul Erdős and Atle Selberg in 1949. 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. Pafnuty Lvovich Chebyshev Pafnuty Lvovich Chebyshev (Пафнутий Львович Чебышёв) (May 4, 1821 - November 26, 1894) was a Russian mathematician. ... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... This page is a candidate for speedy deletion. ... Charles-Jean de la Vallée-Poussin (August 14, 1866 - March 2, 1962) was a Belgian mathematician. ... Paul ErdÅ‘s (Hungarian: ErdÅ‘s Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian-born mathematician. ... Atle Selberg (born June 17, 1917) is a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory. ... There is also the Riemann hypothesis for curves over finite fields. ...

Nineteenth-century developments

Cauchy, Poinsot (1845), Lebesgue(?) (1859, 1868), and notably Hermite have added to the subject. 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. Smith gave a complete classification of ternary quadratic forms, and extended Gauss's researches concerning real quadratic forms to complex forms. 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. Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Louis Poinsot (1777 - 1859) was a French mathematician and physicist. ... Henri Léon Lebesgue (June 28, 1875 - July 26, 1941) was a French mathematician, most famous for his theory of integration. ... Charles Hermite (pronounced air meet) (December 24, 1822 - January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ... Henry John Stephen Smith (November 2, 1826 - February 9, 1883) was an Irish mathematician, remembered for his work in number theory (elementary divisors, quadratic forms) and matrices. ...

Dirichlet was the first to lecture upon the subject in a German university. Among his contributions is the extension of Fermat's last theorem: Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ...

which Euler and Legendre had proven for n = 3,4 (and therefore by implication, all multiples of 3 and 4), Dirichlet showing that . Among the later French writers are Borel; Poincaré, whose memoirs are numerous and valuable; Tannery, and Stieltjes. Among the leading contributors in Germany were Kronecker, Kummer, Schering, Bachmann, and Dedekind. In Austria Stolz's Vorlesungen über allgemeine Arithmetik (1885-86), and in England Mathews' Theory of Numbers (Part I, 1892) were scholarly general works. Genocchi, Sylvester, and J. W. L. Glaisher have also added to the theory. Félix Édouard Justin Émile Borel (January 7, 1871 – February 3, 1956) was a French mathematician and politician. ... Jules Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Thomas Joannes Stieltjes (December 29, 1856 – December 31, 1894) was a Dutch mathematician. ... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Ernst Christian Friedrich Schering (1824-1889) Ernst Christian Friedrich Schering (May 31, 1824 – December 27, 1889) was a German apothecary and industrialist who created the Schering Corporation. ... Paul Gustav Heinrich Bachmann (June 22, 1837 - March 31, 1920) was a German mathematician. ... Julius Wilhelm Richard Dedekind (October 6, 1831 - February 12, 1916) was a German mathematician and Ernst Eduard Kummers closest follower in arithmetic. ... Otto Stolz (1842–1905) was an Austrian mathematician noted for his work on mathematical analysis and infinitesimals. ... Mathews is the name of several places in the United States of America: Mathews, Louisiana Mathews, Virginia Mathews County, Virginia Mathews is also a persons name: Gregory Mathews (1876 - 1949) was an Australian ornithologist in England. ... Angelo Genocchi was an Italian mathematician who specialized in number theory. ... James Joseph Sylvester 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) was a prolific British mathematician. ...

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 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 approximation, and combinatorics. ... | name = David Hilbert | image = Hilbert1912. ... Hermann Minkowski. ... Adolf Hurwitz Adolf Hurwitz (26 March 1859- 18 November 1919) was a German mathematician, and one of the most important figures in mathematics in the second half of the nineteenth century (according to Jean-Pierre Serre, always something good in Hurwitz). He was born in a Jewish family in Hildesheim... Wacław Franciszek Sierpiński, was born on March 14, 1882 in Warsaw and died on October 21, 1969 in Warsaw. ... Derrick Norman Lehmer (27 July 1867, Somerset, Indiana, USA — 8 September 1938 in Berkeley, California, USA) was an American mathematician and number theorist. ...

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 (November 9, 1885 – December 9, 1955) was a German mathematician. ... N. G. Chebotaryov. ... Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ... Erich Hecke (September 20, 1887 – February 13, 1947) was a German mathematician. ... Helmut Hasse (pronounced HAHS uh) (25 August 1898- 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry (Hasse principle), and to local zeta functions. ... Alexander Gelfond Alexander Osipovich Gelfond (Russian: ; October 24, 1906 - November 7, 1968) was a Russian mathematician, author of the Gelfonds theorem. ... Yuri V. Linnik. ... Paul ErdÅ‘s (Hungarian: ErdÅ‘s Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian-born mathematician. ... Gerd Faltings, June 2006 Gerd Faltings (born July 28, 1954 in Gelsenkirchen-Buer) is a German Lutheran mathematician known for his work in arithmetic algebraic geometry. ... G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent English mathematician, known for his achievements in number theory and mathematical analysis. ... Edmund Georg Hermann (Yehezkel) Landau (February 14, 1877 – February 19, 1938) was a German Jew mathematician and author of over 250 papers on number theory. ... Louis Joel Mordell (28 January 1888 - 12 March 1972) was a British mathematician, known for pioneering research in number theory. ... John Edensor Littlewood (June 9, 1885 – September 6, 1977) was a British mathematician. ... Ramanujan redirects here. ... André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... Ivan Matveevich Vinogradov (September 14, 1891–March 20, 1983) was a Russian mathematician, who was one of the creators of modern analytic number theory, and also the dominant figure in mathematics in the USSR. He was born in the Velikiye Luki district, Pskov Oblast. ... Atle Selberg (born June 17, 1917) is a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory. ... Carl Ludwig Siegel (December 31, 1896 – April 4, 1981) was a German mathematician specialising in number theory. ... Igor R. Shafarevich. ... You may be looking for John Tate (boxer) John Torrence Tate, born March 13, 1925 in Minneapolis, Minnesota, is an American mathematician, distinguished for many fundamental contributions in algebraic number theory and related areas in algebraic geometry. ... Robert Langlands (born 1936 in Canada) is one of the most significant mathematicians of the 20th century, with profound insights in number theory and representation theory. ... Goro Shimura (志村 五郎, 1930 -) is a Japanese-American mathematician, and currently a professor of mathematics at Princeton University. ... Kenkichi Iwasawa (岩澤 健吉 Iwasawa Kenkichi, September 11, 1917 - October 26, 1998) was a Japanese mathematician who is known for his influence on algebraic number theory. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... Pierre Deligne, March 2005 Pierre Deligne (born 3 October 1944) is a Belgian mathematician. ... Enrico Bombieri (born November 26, 1940) is a Italian mathematician, born in Milan. ... Alan Baker (born on August 19, 1939) is an English mathematician. ... Sir Henry Peter Francis Swinnerton-Dyer (known as Peter Swinnerton-Dyer) is a English mathematician specialising in number theory at Cambridge University. ... Bryan John Birch is a British mathematician. ... Vladimir Gershonovich Drinfeld (Владимир Гершонович Дринфельд) is a mathematician born February 14, 1954 in Ukraine. ... Laurent Lafforgue (born November 6, 1966, in Antony, France) is a French mathematician. ... For the French mathematician with work in the area of elliptic curves, see André Weil. ... Richard Taylor (born 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. Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... For the French mathematician with work in the area of elliptic curves, see André Weil. ... In mathematics, the modularity theorem establishes an important connection between elliptic curves over the field of rational numbers and modular forms, certain analytic functions introduced in 19th century mathematics. ...

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. —Kronecker^[2]

Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ...

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. 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.
• O'Connor, John J. and Robertson, Edmund F. (2004). 'Index of Ancient Indian mathematics', MacTutor History of Mathematics archive.
• O'Connor, John J. and Robertson, Edmund F. (2004). 'Numbers and Number Theory Index', MacTutor History of Mathematics archive.
• Important publications in number theory

Wikibooks Discrete mathematics has a page on the topic of

Number theory

The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... This is a list of important publications in mathematics, organized by field. ... Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ...

External links

• Number Theory Web