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 By the early twentieth century, it had been superseded by "number theory".^{[note 2]} (The word "arithmetic" is used by the general public to mean "elementary calculations"; it has also acquired other meanings in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