Number theory (or arithmetic^{[note 1]}) is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well as the properties of objects made out of integers (such as rational numbers) or defined as generalizations of the integers (such as, for example, algebraic integers).

Integers can be considered either in themselves or as solutions to equations (diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter (diophantine approximation).

The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory".^{[note 2]} (The word "arithmetic" is used by the general public to mean "elementary calculations"; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence.^{[note 3]} In particular, arithmetical is preferred as an adjective to number-theoretic.

History[link]

Origins[link]

Dawn of arithmetic[link]

The first historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 (Larsa, Mesopotamia, ca. 1800 BCE) contains a list of "Pythagorean triples", i.e., integers Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle (a,b,c)

such that Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle a^2+b^2=c^2

. The triples are too many and too large to have been obtained by brute force. The heading over the first column reads: "The takiltum of the diagonal which has been substracted such that the width..."^[1]

The Plimpton 322 tablet

The table's layout suggests^[2] that it was constructed by means of what amounts, in modern language, to the identity

which is implicit in routine Old Babylonian exercises.^[3] If some other method was used,^[4] the triples were first constructed and then reordered by Failed to parse (Missing texvc executable; please see math/README to configure.): c/a , presumably for actual use as a "table", i.e., with a view to applications.

We do not know what these applications may have been, or whether there could have been any; Babylonian astronomy, for example, truly flowered only later. It has been suggested instead that the table was a source of numerical examples for school problems.^{[5][note 4]}

While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, Babylonian algebra (in the secondary-school sense of "algebra") was exceptionally well developed.^[6] Late Neoplatonic sources^[7] state that Pythagoras learned mathematics from the Babylonians. (Much earlier sources^[8] state that Thales and Pythagoras traveled and studied in Egypt.)

Euclid IX 21—34 is very probably Pythagorean;^[9] it is very simple material ("odd times even is even", "if an odd number measures [= divides] an even number, then it also measures [= divides] half of it"), but it is all that is needed to prove that Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \sqrt{2}

is irrational.^[10] Pythagoraean mystics gave great importance to the odd and the even.^[11] The discovery that Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \sqrt{2}

is irrational is credited to the early Pythagoreans (pre-Theodorus).[12] By revealing (in modern terms) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; its proof or its divulgation are sometimes credited to Hippasus, who was expelled or split from the Pythagorean sect.[13] It is only here that we can start to speak of a clear, conscious division between numbers (integers and the rationals—the subjects of arithmetic) and lengths (real numbers, whether rational or not).

The Pythagorean tradition spoke also of so-called polygonal or figured numbers.^[14] While square numbers, cubic numbers, etc., are seen now as more natural than triangular numbers, square numbers, pentagonal numbers, etc., the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period (17th to early 19th century).

We know of no clearly arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in both. The Chinese remainder theorem appears as an exercise ^[15] in Sun Zi's Suan Ching (also known as The Mathematical Classic of Sun Zi (3rd, 4th or 5th century CE.)^[16] (There is one important step glossed over in Sun Zi's solution:^{[note 5]} it is the problem that was later solved by Āryabhaṭa's kuṭṭaka – see below.)

There is also some numerical mysticism in Chinese mathematics,^{[note 6]} but, unlike that of the Pythagoreans, it seems to have led nowhere. Like the Pythagoreans' perfect numbers, magic squares have passed from superstition into recreation.

Classical Greece and the early Hellenistic period[link]

Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from the early Hellenistic period.^[17] In the case of number theory, this means, by and large, Plato and Euclid, respectively.

Plato had a keen interest in mathematics, and distinguished clearly between arithmetic and calculation. (By arithmetic he meant, in part, theorising on number, rather than what arithmetic or number theory have come to mean.) It is through one of Plato's dialogues—namely, Theaetetus – that we know that Theodorus had proven that Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \sqrt{3}, \sqrt{5}, \dots, \sqrt{17}

are irrational. Theaetetus was, like Plato, a disciple of Theodorus's; he worked on distinguishing different kinds of incommensurables, and was thus arguably a pioneer in the study of number systems. (Book X  of Euclid's Elements is described by Pappus as being largely based on Theaetetus's work.)

Euclid devoted part of his Elements to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic thereto (Books VII to IX of Euclid's Elements). In particular, he gave an algorithm for computing the greatest common divisor of two numbers (the Euclidean algorithm; Elements, Prop. VII.2) and the first known proof of the infinitude of primes (Elements, Prop. IX.20).

In 1773, Lessing published an epigram he had found in a manuscript during his work as a librarian; it claimed to be a letter sent by Archimedes to Eratosthenes.^[18][19] The epigram proposed what has become known as Archimedes' cattle problem; its solution (absent from the manuscript) requires solving an indeterminate quadratic equation (which reduces to what would later be misnamed Pell's equation). As far as we know, such equations were first successfully treated by the Indian school. It is not known whether Archimedes himself had a method of solution.

Diophantus[link]

Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.

Very little is known about Diophantus of Alexandria; he probably lived in the third century CE, that is, about five hundred years after Euclid. Six out of the thirteen books of Diophantus's Arithmetica survive in the original Greek; four more books survive in an Arabic translation. The Arithmetica is a collection of worked-out problems where the task is invariably to find rational solutions to a system of polynomial equations, usually of the form Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle f(x,y)=z^2

or Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle f(x,y,z)=w^2

. Thus, nowadays, we speak of Diophantine equations when we speak of polynomial equations to which rational or integer solutions must be found.

One may say that Diophantus was studying rational points — i.e., points whose coordinates are rational — on curves and algebraic varieties; however, unlike the Greeks of the Classical period, who did what we would now call basic algebra in geometrical terms, Diophantus did what we would now call basic algebraic geometry in purely algebraic terms. In modern language, what Diophantus does is to find rational parametrisations of many varieties; in other words, he shows how to obtain infinitely many rational numbers satisfying a system of equations by giving a procedure that can be made into an algebraic expression (say, Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle x=g_1(r,s) , Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle y=g_2(r,s) , Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle z=g_3(r,s) , where Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle g_1 , Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle g_2

and Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle g_3
are polynomials

or quotients of polynomials; this would be what is sought for if such Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle x, y, z

satisfied

a given equation Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle f(x,y)=z^2

(say) for all values of r and s).

Diophantus also studies the equations of some non-rational curves, for which no rational parametrisation is possible. He manages to find some rational points on these curves – elliptic curves, as it happens, in what seems to be their first known occurrence—by means of what amounts to a tangent construction: translated into coordinate geometry (which did not exist in Diophantus's time), his method would be visualised as drawing a tangent to a curve at a known rational point, and then finding the other point of intersection of the tangent with the curve; that other point is a new rational point. (Diophantus also resorts to what could be called a special case of a secant construction.)

While Diophantus is concerned largely with rational solutions, he assumes some results on integer numbers; in particular, he seems to assume that every integer is the sum of four squares, though he never states as much explicitly.

Indian school: Āryabhaṭa, Brahmagupta, Bhāskara[link]

While Greek astronomy—thanks to Alexander's conquests—probably influenced Indian learning, to the point of introducing trigonometry,^[20] it seems to be the case that Indian mathematics is otherwise an indigenous tradition;^[21] in particular, there is no evidence that Euclid's Elements reached India before the 18th century.^[22]

Āryabhaṭa (476–550 CE) showed that pairs of simultaneous congruences Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle n\equiv a_1 \pmod m_1 , Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle n\equiv a_2 \pmod m_2

could be solved by a method he called kuṭṭaka, or pulveriser;[23]  this is a procedure close to (a generalisation of) the Euclidean algorithm, which was probably discovered independently in India.[24] Āryabhaṭa seems to have had in mind applications to astronomical calculations.[25]

Brahmagupta (628 CE) started the systematic study of indefinite quadratic equations—in particular, the misnamed Pell equation, in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. A general procedure (the chakravala, or "cyclic method") for solving Pell's equation was finally found by Jayadeva (cited in the eleventh century; his work is otherwise lost); the earliest surviving exposition appears in Bhāskara II's Bīja-gaṇita (twelfth century).^[26]

Unfortunately, Indian mathematics remained largely unknown in the West until the late eighteenth century;^[27] Brahmagupta and Bhāskara's work was translated into English in 1817 by Henry Colebrooke.^[28]

Arithmetic in the Islamic golden age[link]

Al-Haytham seen by the West: frontispice of Selenographia, showing Alhasen [sic] representing knowledge through reason, and Galileo representing knowledge through the senses.

In the early ninth century, the caliph Al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (the Sindhind, which may ^[29] or may not^[30] be Brahmagupta's Brāhmasphuţasiddhānta), thus giving rise to the rich tradition of Islamic mathematics. Diophantus's main work, the Arithmetica, was translated into Arabic by Qusta ibn Luqa (820–912). Part of the treatise al-Fakhri (by al-Karajī, 953 – ca. 1029) builds on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporary Ibn al-Haytham knew^[31] what would later be called Wilson's theorem.

Other than a treatise on squares in arithmetic progression by Fibonacci — who lived and studied in north Africa and Constantinople during his formative years, ca. 1175–1200 — no number theory to speak of was done in western Europe during the Middle Ages. Matters started to change in Europe in the late Renaissance, thanks to a renewed study of the works of Greek antiquity. A key catalyst was the textual emendation and translation into Latin of Diophantus's Arithmetica (Bachet, 1621, following a first attempt by Xylander, 1575). [[File:|upright|left|thumb|Al-Haytham seen by the West: frontispice of Selenographia, showing Alhasen [sic] representing knowledge through reason, and Galileo representing knowledge through the senses.]]

Early modern number theory[link]

Fermat[link]

Pierre de Fermat

Pierre de Fermat (1601–1665) never published his writings; in particular, his work on number theory is contained almost entirely in letters to mathematicians and in private marginal notes.^[32] He wrote down nearly no proofs in number theory; he had no models in the area.^[33] He did make repeated use of mathematical induction, introducing the method of infinite descent.

One of Fermat's first interests was perfect numbers (which appear in Euclid, Elements IX) and amicable numbers;^{[note 7]} this led him to work on integer divisors, which were from the beginning among the subjects of the correspondence (1636 onwards) that put him in touch with the mathematical community of the day.^[34] He had already studied Bachet's edition of Diophantus carefully;^[35] by 1643, his interests had shifted largely to diophantine problems and sums of squares^[36] (also treated by Diophantus).

Fermat's achievements in arithmetic include:

• Fermat's little theorem (1640),^[37] stating that, if a is not divisible by a prime p, then Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle a^{p-1} \equiv 1 \pmod p.

^{[note 8]}

• If a and b are coprime, then Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle a^2 + b^2

is not divisible by any prime congruent to −1 modulo 4;[38] and Every prime congruent to 1 modulo 4 can be written in the form Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle a^2 + b^2

.^[39] These two statements also date from 1640; in 1659, Fermat stated to Huygens that he had proven the latter statement by the method of descent.^[40] Fermat and Frenicle also did some work (some of it erroneous or non-rigorous)^[41] on other quadratic forms.

• Fermat posed the problem of solving Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle x^2 - N y^2 = 1

as a challenge to English mathematicians (1657). The problem was solved in a few months by Wallis and Brouncker.[42] Fermat considered their solution valid, but pointed out they had provided an algorithm without a proof (as had Jayadeva and Bhaskara, though Fermat would never know this.) He states that a proof can be found by descent.

• Fermat developed methods for (doing what in our terms amounts to) finding points on curves of genus 0 and 1. As in Diophantus, there are many special procedures and what amounts to a tangent construction, but no use of a secant construction.^[43]
• Fermat states and proves (by descent) in the appendix to Observations on Diophantus (Obs. XLV)^[44] that Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle x^{4} + y^{4} = z^{4}

has no non-trivial solutions in the integers. Fermat also mentioned to his correspondents that Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle x^3 + y^3 = z^3
has no non-trivial solutions, and that this could be proven by descent.[45] The first known proof is due to Euler (1753; indeed by descent).[46]

Fermat's claim ("Fermat's last theorem") to have shown there are no solutions to Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle x^n + y^n = z^n

for all Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle n\geq 3
(a fact completely beyond his methods) appears only on his annotations on the margin of his copy of Diophantus; he never claimed this to others[47] and thus had no need to retract it if he found a mistake in his alleged proof.

Euler[link]

Leonhard Euler

The interest of Leonhard Euler (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur^{[note 9]} Goldbach pointed him towards some of Fermat's work on the subject. ^[48][49] This has been called the "rebirth" of modern number theory,^[35] after Fermat's relative lack of success in getting his contemporaries' attention for the subject.^[50] Euler's work on number theory includes the following:^[51]

• Proofs for Fermat's statements. This includes Fermat's little theorem (generalised by Euler to non-prime moduli); the fact that Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle p = x^2 + y^2

if and only if Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle p\equiv 1\; mod\; 4

initial work towards a proof that every integer is the sum of four squares (the first complete proof is by Lagrange (1770), soon improved by Euler himself^[52]; the lack of non-zero integer solutions to Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle x^4 + y^4 = z^2

(implying the case n=4 of Fermat's last theorem, the case n=3 of which Euler also proved by a related method).

• Pell's equation, first misnamed by Euler.^[53] He wrote on the link between continued fractions and Pell's equation.^[54]
• First steps towards analytic number theory. In his work of sums of four squares, partitions, pentagonal numbers, and the distribution of prime numbers, Euler pioneered the use of what can be seen as analysis (in particular, infinite series) in number theory. Since he lived before the development of complex analysis, most of his work is restricted to the formal manipulation of power series. He did, however, do some very notable (though not fully rigorous) early work on what would later be called the Riemann zeta function.^[55]
• Quadratic forms. Following Fermat's lead, Euler did further research on the question of which primes are can be expressed in the form Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle x^2 + N y^2

, some of it prefiguring quadratic reciprocity.^{[56] [57][58]}

• Diophantine equations. Euler worked on some diophantine equations of genus 0 and 1.^[59][60] In particular, he studied Diophantus's work; he tried to systematise it, but the time was not yet ripe for such an endeavour – algebraic geometry was still in its infancy.^[61] He did notice there was a connection between diophantine problems and elliptic integrals,^[61] whose study he had himself initiated.

Lagrange, Legendre and Gauss[link]

Carl Friedrich Gauss's Disquisitiones Arithmeticae, first edition

Lagrange (1736–1813) was the first to give full proofs of some of Fermat's and Euler's work and observations - for instance, the four-square theorem and the basic theory of the misnamed "Pell's equation" (for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.) He also studied quadratic forms in full generality (as opposed to Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle m X^2 + n Y^2 ) — defining their equivalence relation, showing how to put them in reduced form, etc.

Legendre (1752–1833) was the first to state the law of quadratic reciprocity. He also conjectured what amounts to the prime number theorem and Dirichlet's theorem on arithmetic progressions. He gave a full treatment of the equation Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle a x^2 + b y^2 + c z^2 = 0 ^[62] and worked on quadratic forms along the lines later developed fully by Gauss.^[63] In his old age, he was the first to prove "Fermat's last theorem" for Failed to parse (Missing texvc executable; please see math/README to configure.): n=5

(completing work by Dirichlet, and crediting both him and Sophie Germain).[64]

Carl Friedrich Gauss

In his Disquisitiones Arithmeticae (1798), Gauss (1777–1855) proved the law of quadratic reciprocity and developed the theory of quadratic forms (in particular, defining their composition). He also introduced some basic notation (congruences) and devoted a section to computational matters, including primality tests.^[65] The last section of the Disquisitiones established a link between roots of unity and number theory:

The theory of the division of the circle...which is treated in sec. 7 does not belong by itself to arithmetic, but its principles can only be drawn from higher arithmetic.^[66]

In this way, Gauss arguably made a first foray towards both Galois's work and algebraic number theory.

Maturity and division into subfields[link]

Peter Gustav Lejeune Dirichlet

Starting early in the nineteenth century, the following developments gradually took place:

• The rise to self-consciousness of number theory (or higher arithmetic) as a field of study.^[67]
• The development of much of modern mathematics necessary for basic modern number theory: complex analysis, group theory, Galois theory—accompanied by greater rigor in analysis and abstraction in algebra.
• The rough subdivision of number theory into its modern subfields—in particular, analytic and algebraic number theory.

Algebraic number theory may be said to start with the study of reciprocity and cyclotomy, but truly came into its own with the development of abstract algebra and early ideal theory and valuation theory; see below. A conventional starting point for analytic number theory is Dirichlet's theorem on arithmetic progressions (1837),^{[68] [69]} whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable.^[70] The first use of analytic ideas in number theory actually goes back to Euler (1730s),^{[71] [72]} who used formal power series and non-rigorous (or implicit) limiting arguments. The use of complex analysis in number theory comes later: the work of Riemann (1859) on the zeta function is the canonical starting point;^[73] Jacobi's four-square theorem (1839), which predates it, belongs to an initially different strand that has by now taken a leading role in analytic number theory (modular forms).^[74]

The history of each subfield is briefly addressed in its own section below; see the main article of each subfield for fuller treatments. Many of the most interesting questions in each area remain open and are being actively worked on.

Main subdivisions[link]

Elementary tools[link]

The term elementary generally denotes a method that does not use complex analysis. For example, the prime number theorem was first proven in 1896, but an elementary proof was found only in 1949 by Erdős and Selberg.^[75] The term is somewhat ambiguous: for example, proofs based on complex Tauberian theorems (e.g. Wiener–Ikehara) are often seen as quite enlightening but not elementary, in spite of using Fourier analysis, rather than complex analysis as such. Here as elsewhere, an elementary proof may be longer and more difficult for most readers than a non-elementary one.

Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, the proofs of these results are not particularly accessible, in part because the range of tools they use is, if anything, unusually broad within mathematics.^[76]

Analytic number theory[link]

Riemann zeta function ζ(s) in the complex plane. The color of a point s gives the value of ζ(s): dark colors denote values close to zero and hue gives the value's argument.

The action of the modular group on the upper half plane. The region in grey is the standard fundamental domain.

Analytic number theory may be defined

• in terms of its tools, as the study of the integers by means of tools from real and complex analysis;^[68] or
• in terms of its concerns, as the study within number theory of estimates on size and density, as opposed to identities.^[77]

Some subjects generally considered to be part of analytic number theory, e.g., sieve theory,^{[note 10]} are better covered by the second rather than the first definition: some of sieve theory, for instance, uses little analysis,^{[note 11]} yet it is considered to be part of analytic number theory.

The following are examples of problems in analytic number theory: the prime number theorem, the Goldbach conjecture (or the twin prime conjecture, or the Hardy–Littlewood conjectures), the Waring problem and the Riemann Hypothesis. Some of the most important tools of analytic number theory are the circle method, sieve methods and L-functions (or, rather, the study of their properties). The theory of modular forms (and, more generally, automorphic forms) also occupies an increasingly central place in the toolbox of analytic number theory.

One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers living in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions, which are generalizations of the Riemann zeta function, an all-important analytic object that describes the distribution of prime numbers.

Algebraic number theory[link]

Algebraic number theory studies algebraic properties and algebraic objects of interest in number theory. (Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study.) A key topic is that of the algebraic numbers, which are generalizations of the rational numbers. Briefly, an algebraic number is any complex number that is a solution to some polynomial equation Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle f(x)=0

with rational coefficients; for example, every solution Failed to parse (Missing texvc executable; please see math/README to configure.): x
of Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle x^5 + (11/2) x^3 - 7 x^2 + 9 = 0 
(say) is an algebraic number. Fields of algebraic numbers are also called algebraic number fields, or shortly number fields.

It could be argued that the simplest kind of number fields (viz., quadratic fields) were already studied by Gauss, as the discussion of quadratic forms in Disquisitiones arithmeticae can be restated in terms of ideals and norms in quadratic fields. (A quadratic field consists of all numbers of the form Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle a + b \sqrt{d} , where Failed to parse (Missing texvc executable; please see math/README to configure.): a

and Failed to parse (Missing texvc executable; please see math/README to configure.): b
are rational numbers and Failed to parse (Missing texvc executable; please see math/README to configure.): d

is a fixed rational number whose square root is not rational.) For that matter, the 11th-century chakravala method amounts—in modern terms—to an algorithm for finding the units of a real quadratic number field. However, neither Bhāskara nor Gauss knew of number fields as such.

The grounds of the subject as we know it were set in the late nineteenth century, when ideal numbers, the theory of ideals and valuation theory were developed; these are three complementary ways of dealing with the lack of unique factorisation in algebraic number fields. (For example, in the field generated by the rationals and Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle \sqrt{-5} , the number Failed to parse (Missing texvc executable; please see math/README to configure.): 6

can be factorised both as Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle 6 = 2 \cdot 3
and

Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle 6 = (1 + \sqrt{-5}) ( 1 - \sqrt{-5})

all of Failed to parse (Missing texvc executable; please see math/README to configure.): 2

, Failed to parse (Missing texvc executable; please see math/README to configure.): 3 , Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle 1 + \sqrt{-5}

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle 1 - \sqrt{-5}

are irreducible, and thus, in a naïve sense, analogous to primes among the integers.) The initial impetus for the development of ideal numbers (by Kummer) seems to have come from the study of higher reciprocity laws,^[78]i.e., generalisations of quadratic reciprocity.

Number fields are often studied as extensions of smaller number fields: a field L is said to be an extension of a field K if L contains K. (For example, the complex numbers C are an extension of the reals R, and the reals R are an extension of the rationals Q.) Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions—that is, extensions L of K such that the Galois group^{[note 12]} Gal(L/K) of L over K is an abelian group—are relatively well understood. Their classification was the object of the programme of class field theory, which was initiated in the late 19th century (partly by Kronecker and Eisenstein) and carried out largely in 1900—1950.

An example of an active area of research in algebraic number theory is Iwasawa theory. The Langlands program, one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields.

Diophantine geometry[link]

The central problem of Diophantine geometry is to determine when a Diophantine equation has solutions, and if it does, how many. The approach taken is to think of the solutions of an equation as a geometric object.

For example, an equation in two variables defines a curve in the plane. More generally, an equation, or system of equations, in two or more variables defines a curve, a surface or some other such object in n-dimensional space. In Diophantine geometry, one asks whether there are any rational points (points all of whose coordinates are rationals) or integral points (points all of whose coordinates are integers) on the curve or surface. If there are any such points, the next step is to ask how many there are and how they are distributed. A basic question in this direction is: are there finitely or infinitely many rational points on a given curve (or surface)? What about integer points?

An example here may be helpful. Consider the Pythagorean equation Failed to parse (Missing texvc executable; please see math/README to configure.): x^2+y^2 = 1

we would like to study its rational solutions, i.e., its solutions Failed to parse (Missing texvc executable; please see math/README to configure.): (x,y)

such that

x and y are both rational. This is the same as asking for all integer solutions to Failed to parse (Missing texvc executable; please see math/README to configure.): a^2 + b^2 = c^2

any solution to the latter equation gives

us a solution Failed to parse (Missing texvc executable; please see math/README to configure.): x = a/c , Failed to parse (Missing texvc executable; please see math/README to configure.): y = b/c

to the former. It is also the

same as asking for all points with rational coordinates on the curve described by Failed to parse (Missing texvc executable; please see math/README to configure.): x^2 + y^2 = 1 . (This curve happens to be a circle of radius 1 around the origin.)

Two examples of an elliptic curve, i.e., a curve of genus 1 having at least one rational point. (Either graph can be seen as a slice of a torus in four-dimensional space.)

The rephrasing of questions on equations in terms of points on curves turns out to be felicitous. The finiteness or not of the number of rational or integer points on an algebraic curve—that is, rational or integer solutions to an equation Failed to parse (Missing texvc executable; please see math/README to configure.): f(x,y)=0 , where Failed to parse (Missing texvc executable; please see math/README to configure.): f

is a polynomial in two variables—turns out to depend crucially on the genus of the curve. The genus can be defined as follows:[note 13] allow the variables in Failed to parse (Missing texvc executable; please see math/README to configure.): f(x,y)=0
to be complex numbers; then Failed to parse (Missing texvc executable; please see math/README to configure.): f(x,y)=0
defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables, i.e., four dimensions). Count

the number of (doughnut) holes in the surface; call this number the genus of Failed to parse (Missing texvc executable; please see math/README to configure.): f(x,y)=0 . Other geometrical notions turn out to be just as crucial.

There is also the closely linked area of diophantine approximations: given a number Failed to parse (Missing texvc executable; please see math/README to configure.): x , how well can it be approximated by rationals? (We are looking for approximations that are good relative to the amount of space that it takes to write the rational: call Failed to parse (Missing texvc executable; please see math/README to configure.): a/q

(with Failed to parse (Missing texvc executable; please see math/README to configure.): gcd(a,q)=1

) a good approximation to Failed to parse (Missing texvc executable; please see math/README to configure.): x

if Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle |x-a/q|<\frac{1}{q^c}

, where Failed to parse (Missing texvc executable; please see math/README to configure.): c

is large.) This question is of special interest if Failed to parse (Missing texvc executable; please see math/README to configure.): x
is an algebraic number. If Failed to parse (Missing texvc executable; please see math/README to configure.): x
cannot be well approximated, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of height) turn out to be crucial both in diophantine geometry and in the study of diophantine approximations.

Diophantine geometry should not be confused with the geometry of numbers, which is a collection of graphical methods for answering certain questions in algebraic number theory. Arithmetic geometry, on the other hand, is a contemporary term for much the same domain as that covered by the term diophantine geometry. The term arithmetic geometry is arguably used most often when one wishes to emphasise the connections to modern algebraic geometry (as in, for instance, Faltings' theorem) rather than to techniques in diophantine approximations.

Recent approaches and subfields[link]

The areas below date as such from no earlier than the mid-twentieth century, even if they are based on older material. For example, as is explained below, the matter of algorithms in number theory is very old, in some sense older than the concept of proof; at the same time, the modern study of computability dates only from the 1930s and 1940s, and computational complexity theory from the 1970s.

Probabilistic number theory[link]

Take a number at random between one and a million. How likely is it to be prime? This is just another way of asking how many primes there are between one and a million. Very well; ask further: how many prime divisors will it have, on average? How many divisors will it have altogether, and with what likelihood? What is the probability that it have many more or many fewer divisors or prime divisors than the average?

Much of probabilistic number theory can be seen as an important special case of the study of variables that are almost, but not quite, mutually independent. For example, the event that a random integer between one and a million be divisible by two and the event that it be divisible by three are almost independent, but not quite.

It is sometimes said that probabilistic combinatorics uses the fact that whatever happens with probability greater than Failed to parse (Missing texvc executable; please see math/README to configure.): 0

must happen sometimes; one may say with equal justice that many applications of probabilistic number theory hinge on the fact that whatever is unusual must be rare. If certain algebraic objects (say, rational or integer solutions to certain equations) can be shown to be in the tail of certain sensibly defined distributions, it follows that there must be few of them; this is a very concrete non-probabilistic statement following from a probabilistic one.

At times non-rigorous, probabilistic approach leads to a number of heuristic algorithms and open problems, notably Cramér's conjecture.

Arithmetic combinatorics[link]

Let Failed to parse (Missing texvc executable; please see math/README to configure.): A

be a set of integers. Consider the set Failed to parse (Missing texvc executable; please see math/README to configure.): A+A
consisting of all sums of two elements of Failed to parse (Missing texvc executable; please see math/README to configure.): A

. Is Failed to parse (Missing texvc executable; please see math/README to configure.): A+A

much larger than A? Barely larger? If Failed to parse (Missing texvc executable; please see math/README to configure.): A + A
is barely larger than Failed to parse (Missing texvc executable; please see math/README to configure.): A

, must Failed to parse (Missing texvc executable; please see math/README to configure.): A

have plenty of arithmetic structure \pmod e.g., does it look like an arithmetic progression?

If we begin from a fairly "thick" infinite set Failed to parse (Missing texvc executable; please see math/README to configure.): A , does it contain many elements in arithmetic progression: Failed to parse (Missing texvc executable; please see math/README to configure.): a , Failed to parse (Missing texvc executable; please see math/README to configure.): a+b , Failed to parse (Missing texvc executable; please see math/README to configure.): a+2 b , Failed to parse (Missing texvc executable; please see math/README to configure.): a+3 b , Failed to parse (Missing texvc executable; please see math/README to configure.): \ldots

, Failed to parse (Missing texvc executable; please see math/README to configure.): a+10b

, say? Should it be possible to write large integers as sums of elements of Failed to parse (Missing texvc executable; please see math/README to configure.): A ?

These questions are characteristic of arithmetic combinatorics. This is a presently coalescing field; it subsumes additive number theory (which concerns itself with certain very specific sets Failed to parse (Missing texvc executable; please see math/README to configure.): A

of arithmetic significance, such as the primes or the squares) and, arguably, some of the geometry of numbers,

together with some rapidly developing new material. Its focus on issues of growth and distribution accounts in part for its developing links with ergodic theory, finite group theory, model theory, and other fields. The term additive combinatorics is also used; however, the sets Failed to parse (Missing texvc executable; please see math/README to configure.): A

being studied need not be sets of integers, but rather subsets of non-commutative groups, for which the multiplication symbol, not the addition symbol, is traditionally used; they can also be subsets of rings, in which case the growth of Failed to parse (Missing texvc executable; please see math/README to configure.): A+A
and Failed to parse (Missing texvc executable; please see math/README to configure.): A

·Failed to parse (Missing texvc executable; please see math/README to configure.): A

may be

compared.

Computations in number theory[link]

While the word algorithm goes back only to certain readers of al-Khwārizmī, careful descriptions of methods of solution are older than proofs: such methods (that is, algorithms) are as old as any recognisable mathematics—ancient Egyptian, Babylonian, Vedic, Chinese—whereas proofs appeared only with the Greeks of the classical period. An interesting early case is that of what we now call the Euclidean algorithm. In its basic form (namely, as an algorithm for computing the greatest common divisor) it appears as Proposition 2 of Book VII in Elements, together with a proof of correctness. However, in the form that is often used in number theory (namely, as an algorithm for finding integer solutions to an equation Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle a x + b y = c , or, what is the same, for finding the quantities whose existence is assured by the Chinese remainder theorem) it first appears in the works of Āryabhaṭa (5th–6th century CE) as an algorithm called kuṭṭaka ("pulveriser"), without a proof of correctness.

There are two main questions: "can we compute this?" and "can we compute it rapidly?". Anybody can test whether a number is prime or, if it is not, split it into prime factors; doing so rapidly is another matter. We now know fast algorithms for testing primality, but, in spite of much work (both theoretical and practical), no truly fast algorithm for factoring.

The difficulty of a computation can be useful: modern protocols for encrypting messages (e.g., RSA) depend on functions that are known to all, but whose inverses (a) are known only to a chosen few, and (b) would take one too long a time to figure out on one's own. For example, these functions can be such that their inverses can be computed only if certain large integers are factorized. While many difficult computational problems outside number theory are known, most working encryption protocols nowadays are based on the difficulty of a few number-theoretical problems.

On a different note — some things may not be computable at all; in fact, this can be proven in some instances. For instance, in 1970, it was proven that there is no Turing machine which can solve all Diophantine equations (see Hilbert's 10th problem). There are thus some problems in number theory that will never be solved. We even know the shape of some of them, viz., Diophantine equations in nine variables; we simply do not know, and cannot know, which coefficients give us equations for which the following two statements are both true: there are no solutions, and we shall never know that there are no solutions.

Applications[link]

The number-theorist Leonard Dickson (1874-1954) said "Thank God that number theory is unsullied by any application". Such a view is no longer applicable to number theory.^[79] In 1974 Donald Knuth said "...virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem of making computers do high-speed numerical calculations".^[80] Elementary number theory is taught in discrete mathematics courses for computer scientists, and on the other hand number theory also has applications to the continuous in numerical analysis.^[81] As well as the well-known applications to cryptography, there are also applications to many other areas of mathematics.^[82][83]

Literature[link]

Two of the most popular introductions to the subject are:

• G. H. Hardy; E. M. Wright (2008) [1938]. An introduction to the theory of numbers (rev. by D. R. Heath-Brown and J. H. Silverman, 6th ed.). Oxford University Press. ISBN 978-0-19-921986-5. http://books.google.com/books?id=rey9wfSaJ9EC&dq. 

• Vinogradov, I. M. (2003) [1954]. Elements of Number Theory (reprint of the 1954 ed.). Mineola, NY: Dover Publications. 

Hardy and Wright's book is a comprehensive classic, though its clarity sometimes suffers due to the authors' insistence on elementary methods.^[84] Vinogradov's main attraction consists in its set of problems, which quickly lead to Vinogradov's own research interests; the text itself is very basic and close to minimal.

Popular choices for a second textbook include:

• Borevich, A. I.; Shafarevich, Igor R. (1966). Number theory. Pure and Applied Mathematics. 20. Boston, MA: Academic Press. ISBN 978-0-12-117850-5. MR 0195803. http://books.google.com/books?id=njgVUjjO-EAC. 

• Serre, Jean-Pierre (1996) [1973]. A course in arithmetic. Graduate texts in mathematics. 7. Springer. ISBN 978-0-387-90040-7. 

See also[link]

Number theory portal

• Algebraic function field
• Finite field
• p-adic number

Notes[link]

References[link]

Sources[link]

This article incorporates material from the Citizendium article "Number theory", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.

External links[link]

• Quotations related to Number theory at Wikiquote
• Number Theory Web

Michael Jackson
Jackson at the White House in 1984
Background information
Birth name	Michael Joseph Jackson[1]
Also known as	King of Pop, MJ
Born	(1958-08-29)August 29, 1958 Gary, Indiana, U.S.
Died	June 25, 2009(2009-06-25) (aged 50) Los Angeles, California, U.S.
Genres	R&B, pop, rock, soul, dance, funk, disco, new jack swing
Occupations	Singer-songwriter, musician, composer, dancer, choreographer, record producer, actor, businessman, philanthropist
Instruments	Vocals
Years active	1964–2009
Labels	Motown, Epic, Legacy, MJJ Productions
Associated acts	The Jackson 5 Michael Jackson's Signature
Website	www.michaeljackson.com

Michael Joseph Jackson^[1] (August 29, 1958 – June 25, 2009) was an American recording artist, entertainer, and businessman. Often referred to as the King of Pop, or by his initials MJ,^[2] Jackson is recognized as the most successful entertainer of all time by Guinness World Records. His contribution to music, dance, and fashion, along with a much-publicized personal life, made him a global figure in popular culture for over four decades. The seventh child of the Jackson family, he debuted on the professional music scene along with his brothers as a member of The Jackson 5 in 1964, and began his solo career in 1971.

In the early 1980s, Jackson became a dominant figure in popular music. The music videos for his songs, including those of "Beat It", "Billie Jean", and "Thriller", were credited with transforming the medium into an art form and a promotional tool, and the popularity of these videos helped to bring the relatively new television channel MTV to fame. Videos such as "Black or White" and "Scream" made him a staple on MTV in the 1990s. Through stage performances and music videos, Jackson popularized a number of complicated dance techniques, such as the robot and the moonwalk, to which he gave the name. His distinctive musical sound and vocal style influenced numerous hip hop, post-disco, contemporary R&B, pop and rock artists.

Jackson's 1982 album Thriller is the best-selling album of all time. His other records, including Off the Wall (1979), Bad (1987), Dangerous (1991), and HIStory (1995), also rank among the world's best-selling. Jackson is one of the few artists to have been inducted into the Rock and Roll Hall of Fame twice. He was also inducted into the Dance Hall of Fame as the first (and currently only) dancer from the world of pop and rock 'n' roll. Some of his other achievements include multiple Guinness World Records; 13 Grammy Awards (as well as the Grammy Legend Award and the Grammy Lifetime Achievement Award); 26 American Music Awards (more than any other artist, including the "Artist of the Century"); 13 number-one singles in the United States in his solo career (more than any other male artist in the Hot 100 era); and the estimated sale of over 750 million records worldwide. Jackson won hundreds of awards, which made him the most-awarded recording artist in the history of popular music.^[3] He was also one of the world's most prominent humanitarians and philanthropists; personally, and through his Heal the World Foundation, he donated more than $300 million in charity, and held the Guinness World Record for having supported the most charities out of any pop star.^[4][5]

Aspects of Jackson's personal life, including his changing appearance, personal relationships, and behavior, generated controversy. In 1993, he was accused of child sexual abuse, but the case was settled out of court and no formal charges were brought. In 2005, he was tried and acquitted of further child sexual abuse allegations and several other charges after the jury found him not guilty on all counts. While preparing for his concert series titled This Is It, Jackson died of acute propofol and benzodiazepine intoxication on June 25, 2009, after suffering from cardiac arrest. The Los Angeles County Coroner ruled his death a homicide, and his personal physician was convicted of involuntary manslaughter. Jackson's death triggered a global outpouring of grief, and as many as one billion people around the world reportedly watched his public memorial service on live television. In March 2010, Sony Music Entertainment signed a $250 million deal with Jackson's estate to retain distribution rights to his recordings until 2017, and to release seven posthumous albums over the decade following his death.^[6]

Life and career[link]

Early life and The Jackson 5 (1958–1975)[link]

Jackson's childhood home in Gary, Indiana, showing floral tributes after his death

Jackson (center) as a member of The Jackson 5 in 1972

Michael Jackson was born on August 29, 1958, the eighth of ten children in an African American working-class family who lived in a small 3-room house in Gary, Indiana,^[7] an industrial city near Chicago. His mother, Katherine Esther Scruse, was a devout Jehovah's Witness, and his father, Joseph Walter "Joe" Jackson, was a steel mill worker who performed with an R&B band called The Falcons. Jackson had three sisters: Rebbie, La Toya, and Janet, and five brothers: Jackie, Tito, Jermaine, Marlon, and Randy.^[8] A sixth brother, Brandon, died shortly after birth.^[9]

Jackson had a troubled relationship with his father, Joe.^[10][11][12] Joseph acknowledged in 2003 that he regularly whipped Jackson as a boy.^[12] Jackson stated that he was physically and emotionally abused during incessant rehearsals, though he also credited his father's strict discipline with playing a large role in his success.^[10] Jackson first spoke openly about his childhood abuse in an interview with Oprah Winfrey, broadcast in February 1993. He admitted that he had often cried from loneliness and he would vomit on the sight of his father. Jackson's father was also said to have verbally abused Jackson, saying that he had a fat nose on numerous occasions.^[13] In fact, Michael Jackson's deep dissatisfaction with his appearance, his nightmares and chronic sleep problems, his tendency to remain hyper-compliant especially with his father, and to remain child-like throughout his adult life are in many ways consistent with the effects of this chronic maltreatment he endured as a young child.^[14]

In an interview with Martin Bashir, later included in the 2003 broadcast of Living with Michael Jackson, Jackson acknowledged that his father hurt him when he was a child, but was nonetheless a "genius", as he admitted his father's strict discipline played a huge role in his success. When Bashir dismissed the positive remark and continued asking about beatings, Jackson put his hand over his face and objected to the questions. He recalled that Joseph sat in a chair with a belt in his hand as he and his siblings rehearsed, and that "if you didn't do it the right way, he would tear you up, really get you".^[15][16]

In 1964, Michael and Marlon joined the Jackson Brothers—a band formed by brothers Jackie, Tito, and Jermaine—as backup musicians playing congas and tambourine. Jackson later began performing backup vocals and dancing. When he was eight, Jackson began sharing the lead vocals with his older brother Jermaine, and the group's name was changed to The Jackson 5.^[8] The band toured the Midwest extensively from 1966 to 1968, frequently performing at a string of black clubs known as the "chitlin' circuit", where they often opened stripteases and other adult acts. In 1966, they won a major local talent show with renditions of Motown hits and James Brown's "I Got You (I Feel Good)", led by Michael.^[17]

The Jackson 5 recorded several songs, including "Big Boy", for the local record label Steeltown in 1967, before signing with Motown Records in 1968.^[8] Rolling Stone magazine later described the young Michael as "a prodigy" with "overwhelming musical gifts," writing that he "quickly emerged as the main draw and lead singer."^[18] The group set a chart record when its first four singles ("I Want You Back", "ABC", "The Love You Save", and "I'll Be There") peaked at number one on the Billboard Hot 100.^[8] Between 1972 and 1975, Michael released four solo studio albums with Motown, among them Got to Be There and Ben, released as part of the Jackson 5 franchise, and producing successful singles such as "Got to Be There", "Ben", and a remake of Bobby Day's "Rockin' Robin".

The Jackson 5 "became a cutting-edge example of black crossover artists... five working-class black boys with afros and bell bottoms, and they really didn't have to trade any of that stuff in order to become mainstream stars."^[19]

The group's sales began declining in 1973, and the band members chafed under Motown's strict refusal to allow them creative control or input. Although they scored several top 40 hits, including the top 5 disco single "Dancing Machine" and the top 20 hit "I Am Love", the Jackson 5 left Motown in 1975.^[20]

[edit] Move to Epic and Off the Wall (1975–81)

In June 1975, the Jackson 5 signed with Epic Records, a subsidiary of CBS Records^[20] and renamed themselves the Jacksons. Younger brother Randy formally joined the band around this time, while Jermaine left to pursue a solo career.^[21] They continued to tour internationally, releasing six more albums between 1976 and 1984, during which Michael was the lead songwriter, writing hits such as "Shake Your Body (Down to the Ground)", "This Place Hotel", and "Can You Feel It".^[17] In 1978, he starred as the scarecrow in the musical, The Wiz, a box-office disaster. It was here that he teamed up with Quincy Jones, who was arranging the film's musical score. Jones agreed to produce Jackson's next solo album, Off the Wall.^[22] In 1979, Jackson broke his nose during a complex dance routine. His subsequent rhinoplasty was not a complete success; he complained of breathing difficulties that would affect his career. He was referred to Dr. Steven Hoefflin, who performed Jackson's second rhinoplasty and subsequent operations.^[23]

Jones and Jackson produced the Off the Wall album together. Songwriters for the album included Jackson, Rod Temperton, Stevie Wonder, and Paul McCartney. Released in 1979, it was the first solo album to generate four U.S. top 10 hits, including the chart-topping singles "Don't Stop 'til You Get Enough" and "Rock with You".^[24][25] It reached number three on the Billboard 200 and eventually sold over 20 million copies worldwide.^[26] In 1980, Jackson won three awards at the American Music Awards for his solo efforts: Favorite Soul/R&B Album, Favorite Soul/R&B Male Artist, and Favorite Soul/R&B Single for "Don't Stop 'Til You Get Enough".^[27][28] That year, he also won Billboard Year-End for Top Black Artist and Top Black Album and a Grammy Award for Best Male R&B Vocal Performance, also for "Don't Stop 'Til You Get Enough".^[24] Jackson again won at the American Music Awards in 1981 for Favorite Soul/R&B Album and Favorite Soul/R&B Male Artist.^[29] Despite its commercial success, Jackson felt Off the Wall should have made a much bigger impact, and was determined to exceed expectations with his next release.^[30] In 1980, he secured the highest royalty rate in the music industry: 37 percent of wholesale album profit.^[31]

[edit] Thriller and Motown 25: Yesterday, Today, Forever (1982–83)

In 1982, Jackson contributed the song "Someone In the Dark" to the storybook for the film E.T. the Extra-Terrestrial; the record won a Grammy for Best Recording for Children in 1984. In the same year he won another seven Grammys and eight American Music Awards (including the Award of Merit, the youngest artist to win it), making him the most awarded in one night for both award shows.^[32][33] These awards were thanks to the Thriller album, released in late 1982, which was 1983's best-selling album worldwide^[34][35] and became the best-selling album of all time in the United States,^[36] as well as the best-selling album of all time worldwide, selling an estimated 110 million copies so far.^[37] The album topped the Billboard 200 chart for 37 weeks and was in the top 10 of the 200 for 80 consecutive weeks. It was the first album to have seven Billboard Hot 100 top 10 singles, including "Billie Jean", "Beat It", and "Wanna Be Startin' Somethin'".^[38] Thriller was certified for 29 million shipments by the RIAA, giving it Double Diamond status in the United States. The album won also another Grammy for Best Engineered Recording – Non Classical in 1984, awarding Bruce Swedien for his work.^[39] Jackson's attorney John Branca noted that Jackson had the highest royalty rate in the music industry at that point: approximately $2 for every album sold. He was also making record-breaking profits from sales of his recordings. The videocassette of the documentary The Making of Michael Jackson's Thriller sold over 350,000 copies in a few months. The era saw the arrival of novelties like dolls modeled after Michael Jackson, which appeared in stores in May 1984 at a price of $12.^[40] Biographer J. Randy Taraborrelli writes that, "Thriller stopped selling like a leisure item—like a magazine, a toy, tickets to a hit movie—and started selling like a household staple."^[41] In 1985, The Making of Michael Jackson's Thriller won a Grammy for Best Music Video, Longform.^[32] In December 2009, the music video for "Thriller" was selected for the National Film Registry by the Library of Congress, "Thriller" is the first music video ever to be inducted.^[42][43][44]

Time described Jackson's influence at that point as "Star of records, radio, rock video. A one-man rescue team for the music business. A songwriter who sets the beat for a decade. A dancer with the fanciest feet on the street. A singer who cuts across all boundaries of taste and style and color too".^[40] The New York Times wrote that, "in the world of pop music, there is Michael Jackson and there is everybody else".^[45]

On March 25th 1983, Michael Jackson reunited with his brothers for a legendary live performance which was taped for a Motown 25: Yesterday, Today, Forever television special. The show aired on May 16, 1983, to an audience of 47 million viewers, and featured the Jacksons and a number of other Motown stars. It is best remembered for Jackson's solo performance of "Billie Jean". Wearing a distinctive black sequin jacket and golf glove decorated with rhinestones, he debuted his signature dance move, the moonwalk, which former Soul Train dancer and Shalamar member Jeffrey Daniel had taught him three years before. The Jacksons' performance drew comparisons to Elvis Presley's and The Beatles' appearances on The Ed Sullivan Show.^[46] Anna Kisselgoff of The New York Times later wrote, "The moonwalk that he made famous is an apt metaphor for his dance style. How does he do it? As a technician, he is a great illusionist, a genuine mime. His ability to keep one leg straight as he glides while the other bends and seems to walk requires perfect timing."^[47]

Pepsi, "We Are the World" and business career (1984–85)[link]

Jackson with a Knott's Berry Farm Snoopy mascot in April 1984

On January 27, 1984, Michael and other members of the Jacksons filmed a Pepsi Cola commercial, overseen by executive Phil Dusenberry,^[48] from ad agency BBDO and Pepsi's Worldwide Creative Director, Alan Pottasch at the Shrine Auditorium in Los Angeles. In front of a full house of fans during a simulated concert, pyrotechnics accidentally set Jackson's hair on fire. He suffered second-degree burns to his scalp. Jackson underwent treatment to hide the scars on his scalp, and he also had his third rhinoplasty shortly thereafter.^[23] Jackson never recovered from this injury. Pepsi settled out of court, and Jackson donated his $1.5 million settlement to the Brotman Medical Center in Culver City, California, which now has a "Michael Jackson Burn Center" in honor of his donation.^[49] Dusenberry later recounted the episode in his memoir, Then We Set His Hair on Fire: Insights and Accidents from a Hall of Fame Career in Advertising.

On May 14, 1984, Jackson was invited to the White House to receive an award from President Ronald Reagan for his support of charities that helped people overcome alcohol and drug abuse.^[50] Jackson won eight awards during the Grammys that year. Unlike later albums, Thriller did not have an official tour to promote it, but the 1984 Victory Tour, headlined by The Jacksons, showcased much of Jackson's new solo material to more than two million Americans. He donated all the funds (around $8 million) raised from the Victory Tour to charity.^[51] He also co-wrote the charity single "We Are the World" in 1985 with Lionel Richie, which was released worldwide to aid the poor in the U.S. and Africa. It became one of the best-selling singles of all time, with nearly 30 million copies sold and millions of dollars donated to famine relief. In 1986, "We Are the World" won four Grammys (one for Jackson for Song of the Year). American Music Award directors removed the charity song from the competition because they felt it would be inappropriate, but recognized it with two special honors (one for the creation of the song and one for the USA for Africa idea). They are the only AMAs that Jackson won as non-solo artist.^{[52][53][54][55]}

Jackson at the White House South Portico with President Ronald Reagan and first lady Nancy Reagan, 1984

In 1984, ATV Music Publishing, which had the copyrights to nearly 4000 songs, including the Northern Songs catalog that contained the majority of the Lennon–McCartney compositions recorded by The Beatles, was put up for sale by Robert Holmes à Court.^[56] Jackson had become interested in owning music catalogs after working with Paul McCartney in the early 1980s: Jackson had learned McCartney made approximately $40 million a year from other people's songs.^[57] In 1981,^[58] McCartney was offered the ATV music catalog for £20 million ($40 million).^[57][59] According to McCartney, he contacted Yoko Ono about making a joint purchase by splitting the cost equally at £10 million each, but Ono thought they could buy it for £5 million each.^[57][59] When they were unable to make the joint purchase, McCartney let the offer fall through, not wanting to be the sole owner of the Beatles' songs.^[58][59]

According to a negotiator for Holmes à Court in the 1984 sale, "We had given Paul McCartney first right of refusal but Paul didn't want it at that time."^[60] Also, an attorney for McCartney assured Jackson's attorney, John Branca, that McCartney was not interested in bidding: McCartney reportedly said "It's too pricey".^[57][58] But there were several other companies and investors bidding. In September 1984, Jackson was first informed about the sale by Branca and sent a bid of $46 million on November 20, 1984.^[56] Jackson's agents thought they had a deal several times, but encountered new bidders or new areas of debate.^[56] In May 1985, Jackson's team walked away from talks after having spent over $1 million on four months of due diligence and on the negotiations.^[56]

In June 1985, Jackson and Branca learned that Charles Koppelman's and Marty Bandier's The Entertainment Co. had made a tentative agreement with Holmes à Court to buy ATV Music for $50 million.^[56] But in early August, Holmes à Court's team contacted Jackson and talks resumed. Jackson raised his bid to $47.5 million and it was accepted because he could close the deal more quickly, having already completed due diligence of ATV Music.^[56] He also agreed to visit Holmes à Court in Australia, where he would appear on the Channel Seven Perth Telethon.^[56][60] Jackson's purchase of ATV Music was finalized August 10, 1985.^[56]

[edit] Appearance, tabloids, Bad, films, autobiography and Neverland (1986–90)

Jackson's skin had been a medium-brown color for the entire duration of his youth, but starting in the mid 1980s, it gradually grew paler. The change gained widespread media coverage, including rumors that he might be bleaching his skin.^[61] According to J. Randy Taraborrelli's biography, in 1986, Jackson was diagnosed with vitiligo and lupus; the vitiligo partially lightened his skin, and the lupus was in remission; both illnesses made him sensitive to sunlight. The treatments he used for his condition further lightened his skin tone, and, with the application of pancake makeup to even out blotches, he could appear very pale.^[62] Jackson was also diagnosed with vitiligo in his autopsy.^[63] By the mid 1990s several surgeons speculated that he had undergone various nasal surgeries, a forehead lift, thinned lips, and cheekbone surgery—although Jackson denied this and insisted that he only had surgery on his nose.^[64] Jackson claimed that he had only two rhinoplasties and no other surgery on his face, although at one point he mentioned having a dimple created in his chin.^[65] Jackson lost weight in the early 1980s because of a change in diet and a desire for "a dancer's body".^[65] Witnesses reported that he was often dizzy and speculated that he was suffering from anorexia nervosa; periods of weight loss would become a recurring problem later in life.^[66]

During the course of his treatment, Jackson made two close friends: his dermatologist, Dr. Arnold Klein, and Klein's nurse Debbie Rowe. Rowe eventually became Jackson's second wife and the mother of his two eldest children. Long before becoming romantically involved with her, Jackson relied heavily on Rowe for emotional support. He also relied heavily on Klein, for medical and business advice.^[67]

Jackson became the subject of increasingly sensational reports. In 1986, the tabloids ran a story claiming that Jackson slept in a hyperbaric oxygen chamber to slow the aging process; he was pictured lying down in a glass box. Although the claim was untrue, according to tabloid reports that are widely cited, Jackson had disseminated the fabricated story himself.^[68][69] When Jackson bought a chimpanzee called Bubbles from a laboratory, he was reported to be increasingly detached from reality.^[70] It was reported that Jackson had offered to buy the bones of Joseph Merrick (the "elephant man") and although untrue, Jackson did not deny the story.^[68][69] Although initially he saw these stories as opportunities for publicity, he stopped leaking untruths to the press as they became more sensational. Consequently the media began making up their own stories.^[69][71][72] These reports became embedded in the public consciousness, inspiring the nickname "Wacko Jacko", which Jackson came to despise.^[73] Responding to the gossip, Jackson remarked to Taraborrelli:

Why not just tell people I'm an alien from Mars? Tell them I eat live chickens and do a voodoo dance at midnight. They'll believe anything you say, because you're a reporter. But if I, Michael Jackson, were to say, "I'm an alien from Mars and I eat live chickens and do a voodoo dance at midnight," people would say, "Oh, man, that Michael Jackson is nuts. He's cracked up. You can't believe a single word that comes out of his mouth."^[74]

Jackson wore a gold-plated military style jacket with belt in the Bad era

Jackson collaborated with George Lucas and Francis Ford Coppola on the 17-minute 3-D film Captain EO, which debuted in September 1986 at both the original Disneyland and at EPCOT in Florida, and in March 1987 at Tokyo Disneyland. The $30 million movie was a popular attraction at all three parks. A Captain EO attraction was later featured at Euro Disneyland after that park opened in 1992. All four parks' Captain EO installations stayed open well into the 1990s: Paris' installation was the last one to close, in 1998.^[75] The attraction would later return to Disneyland in 2010 after Jackson's death.^[76]

In 1987, Jackson disassociated himself from the Jehovah's Witnesses, in response to their disapproval of the Thriller video.^[77] With the industry expecting another major hit, Jackson's first album in five years, Bad (1987), was highly anticipated.^[78] It did not top Thriller as a commercial or artistic triumph, but Bad was still a substantial success in its own right.

The Bad album spawned seven hit singles in the U.S., five of which ("I Just Can't Stop Loving You", "Bad", "The Way You Make Me Feel", "Man in the Mirror" and "Dirty Diana") reached number one on the Billboard Hot 100 charts. This was a record for most number one Hot 100 singles from any one album, including Thriller.^[79] Although the title track's video was arguably derivative of the video for the earlier single "Beat It", the "Bad" video still proved to be one of Jackson's iconic moments. It was a gritty but colorful epic set against the backdrop of the New York City Subway system, with costuming and choreography inspired by West Side Story. As of 2012, the album sold between 30 and 45 million copies worldwide.^{[80][81][82][83][84][85][86]} Thanks to the Bad album, Bruce Swedien and Humberto Gatica won one Grammy in 1988 for Best Engineered Recording – Non Classical and Michael Jackson won one Grammy for Best Music Video, Short Form for "Leave Me Alone" in 1989.^[32][39] In the same year, Jackson won an Award of Achievement at the American Music Awards because Bad is the first album ever to generate five number one singles in the US, the first album to top in 25 countries and the best-selling album worldwide in 1987 and in 1988.^{[87][88][89][90]} In 1988, "Bad" won an American Music Award for Favorite Soul/R&B Single.^[91]

Front view: Jackson wearing the costume on a June 2, 1988 performance in Vienna, Austria

The Bad World Tour began on September 12 that year, finishing on January 14, 1989.^[92] In Japan alone, the tour had 14 sellouts and drew 570,000 people, nearly tripling the previous record of 200,000 in a single tour.^[93] Jackson broke a Guinness World Record   when 504,000 people attended seven sold-out shows at Wembley Stadium. He performed a total of 123 concerts to an audience of 4.4 million people. The Bad Tour turned out to be the last of Jackson's concert tours to include shows in the continental United States, although later tours did make it to Hawaii.

In 1988, Jackson released his first and only autobiography, Moonwalk, which took four years to complete and sold 200,000 copies.^[94] Jackson wrote about his childhood, The Jackson 5, and the abuse he had suffered.^[95] He also wrote about his facial appearance, saying he had had two rhinoplastic surgeries and a dimple created in his chin.^[65] He attributed much of the change in the structure of his face to puberty, weight loss, a strict vegetarian diet, a change in hair style, and stage lighting.^[65] Moonwalk reached the top position on The New York Times best sellers' list.^[96] The musician then released a film called Moonwalker, which featured live footage and short films that starred Jackson and Joe Pesci. The film was originally intended to be released to theaters but due to financial issues, the film was released direct-to-video. It saw a theatrical release in Germany, though. It debuted atop the Billboard Top Music Video Cassette chart, staying there for 22 weeks. It was eventually knocked off the top spot by Michael Jackson: The Legend Continues.^[97]

In March 1988, Jackson purchased land near Santa Ynez, California, to build Neverland Ranch at a cost of $17 million. He installed Ferris wheels, a menagerie, and a movie theater on the 2,700-acre (11 km²) property. A security staff of 40 patrolled the grounds. In 2003, it was valued at approximately $100 million.^[18][98] In 1989, his annual earnings from album sales, endorsements, and concerts was estimated at $125 million for that year alone.^[99] Shortly afterwards, he became the first Westerner to appear in a television ad in the Soviet Union.^[97]

His success resulted in his being dubbed the "King of Pop".^{[100][101][102][103]} The nickname was popularized by Elizabeth Taylor when she presented him with the Soul Train Heritage Award in 1989, proclaiming him "the true king of pop, rock and soul."^[104] President George H. W. Bush designated him the White House's "Artist of the Decade".^[105] From 1985 to 1990, he donated $500,000 to the United Negro College Fund, and all of the profits from his single "Man in the Mirror" went to charity.^[106][107] Jackson's live rendition of "You Were There" at Sammy Davis Jr.'s 60th birthday celebration received an Emmy nomination.^[97]

[edit] Dangerous, Heal the World Foundation and Super Bowl XXVII (1991–93)

In March 1991, Jackson renewed his contract with Sony for $65 million, a record-breaking deal at the time,^[108] displacing Neil Diamond's renewal contract with Columbia Records.^[109] He released his eighth album Dangerous in 1991. As of 2008, Dangerous had shipped seven million copies in the U.S. and had sold 32 million copies worldwide. The Dangerous album was co-produced by Teddy Riley, one of the pioneers of "new jack swing" which convinced Michael to feature a rapper on his album for the first time, the act worked and it turned out to be the best-selling album associated with that movement.^{[110][111][112]} In the United States, the album's first single "Black or White" was its biggest hit, reaching number one on the Billboard Hot 100 and remaining there for seven weeks, with similar chart performances worldwide.^[113] The album's second single "Remember the Time" spent eight weeks in the top five in the United States, peaking at number three on the Billboard Hot 100 singles chart.^[114] At the end of 1992, Dangerous was awarded 1992's best-selling album worldwide and "Black or White" was awarded 1992's best-selling single worldwide at the Billboard Music Awards. Additionally, he won an award as best-selling artist of the '80s.^[115] In 1993, Jackson performed the song at the Soul Train Music Awards in a chair, saying he had suffered an injury in rehearsals.^[116] In the UK and other parts of Europe, "Heal the World" was the biggest hit from the album; it sold 450,000 copies in the UK and spent five weeks at number two in 1992.^[114]

Jackson founded the Heal the World Foundation in 1992. The charity organization brought underprivileged children to Jackson's ranch to enjoy theme park rides that Jackson had built on the property. The foundation also sent millions of dollars around the globe to help children threatened by war, poverty, and disease. In the same year Jackson published his second book, the bestselling collection of poetry, Dancing the Dream. While it was a commercial success and revealed a more intimate side to Jackson's nature, the collection was mostly critically unacclaimed at the time of release. In 2009, the book was republished by Doubleday and was more positively received by some critics in the wake of Jackson's untimely death. The Dangerous World Tour grossed $100 million. The tour began on June 27, 1992, and finished on November 11, 1993. Jackson performed to 3.5 million people in 67 concerts.^[114][117] He sold the broadcast rights to his Dangerous world tour to HBO for $20 million, a record-breaking deal that still stands.^[118]

Following the illness and death of Ryan White, Jackson helped draw public attention to HIV/AIDS, something that was still controversial at the time. He publicly pleaded with the Clinton Administration at Bill Clinton's Inaugural Gala to give more money to HIV/AIDS charities and research.^[119][120] In a high-profile visit to Africa, Jackson visited several countries, among them Gabon and Egypt.^[121] His first stop to Gabon was greeted with a sizable and enthusiastic reception of more than 100,000 people, some of them carrying signs that read, "Welcome Home Michael."^[121] In his trip to Côte d'Ivoire, Jackson was crowned "King Sani" by a tribal chief.^[121] He then thanked the dignitaries in French and English, signed official documents formalizing his kingship and sat on a golden throne while presiding over ceremonial dances.^[121]

In January 1993, Jackson made a memorable appearance at the halftime show at Super Bowl XXVII. The performance began with Jackson catapulting onto the stage as fireworks went off behind him. As he landed on the canvas, he maintained a motionless "clenched fist, standing statue stance", dressed in a gold and black military outfit and sunglasses; he remained completely motionless for a minute and a half while the crowd cheered. He then slowly removed his sunglasses, threw them away and sang four songs: "Jam", "Billie Jean", "Black or White" and "Heal the World". It was the first Super Bowl where the audience figures increased during the half-time show, and was viewed by 135 million Americans alone; Jackson's Dangerous album rose 90 places up the album chart.^[61] Jackson was given the "Living Legend Award" at the 35th Grammy Awards in Los Angeles. "Black or White" was Grammy-nominated for best vocal performance. "Jam" gained two nominations: Best R&B Vocal Performance and Best R&B Song.^[114] The Dangerous album won a Grammy for Best Engineered – Non Classical, awarding the work of Bruce Swedien and Teddy Riley. In the same year, Michael Jackson won three American Music Awards for Favorite Pop/Rock Album (Dangerous), Favorite Soul/R&B Single ("Remember the Time") and was the first to win the International Artist Award, for his global performances and humanitarian concerns. This award will bear his name in the future.^{[32][39][122]}