Number theory

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File:A 150x150 Ulam spiral of dots with varying widths (emphasis primes).svg

The distribution of prime numbers, a central point of study in number theory, illustrated by an Ulam spiral. It shows the conditional independence between being prime and being a value of certain quadratic polynomials.

Template:Math topics TOC Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers).

Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of analytical objects, such as 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, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation).

Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is that it deals with statements that are simple to understand but are very difficult to solve. Examples of this are Fermat's Last Theorem, which was proved 358 years after the original formulation, and Goldbach's conjecture, which remains unsolved since the 18th century. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."Template:Sfn It was regarded as the example of pure mathematics with no applications outside mathematics until the 1970s, when it became known that prime numbers would be used as the basis for the creation of public-key cryptography algorithms.

HistoryEdit

Number theory is the branch of mathematics that studies integers and their properties and relations.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The integers comprise a set that extends the set of natural numbers <math>\{1, 2, 3, \dots\}</math> to include number <math>0</math> and the negation of natural numbers <math>\{-1, -2, -3, \dots\}</math>. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers).<ref name=":5">Template:Cite book</ref><ref name=":1" />

Number theory is closely related to arithmetic and some authors use the terms as synonyms.<ref>Template:Multiref</ref> However, the word "arithmetic" is used today to mean the study of numerical operations and extends to the real numbers.<ref>Template:Multiref</ref> In a more specific sense, number theory is restricted to the study of integers and focuses on their properties and relationships.<ref>Template:Multiref</ref> Traditionally, it is known as higher arithmetic.<ref>Template:Multiref</ref> By the early twentieth century, the term number theory had been widely adopted.<ref group="note">The term 'arithmetic' may have regained some ground, arguably due to French influence. Take, for example, Template:Harvnb. In 1952, Davenport still had to specify that he meant The Higher Arithmetic. Hardy and Wright wrote in the introduction to An Introduction to the Theory of Numbers (1938): "We proposed at one time to change [the title] to An introduction to arithmetic, a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book." Template:Harv</ref> The term number means whole numbers, which refers to either the natural numbers or the integers.<ref name=":4">Template:Cite book</ref><ref name=":6">Template:Cite book</ref><ref>Template:Cite book</ref>

Elementary number theory studies aspects of integers that can be investigated using elementary methods such as elementary proofs.<ref name=":3">Template:Multiref</ref> Analytic number theory, by contrast, relies on complex numbers and techniques from analysis and calculus.<ref>Template:Multiref</ref> Algebraic number theory employs algebraic structures such as fields and rings to analyze the properties of and relations between numbers. Geometric number theory uses concepts from geometry to study numbers.<ref>Template:Multiref</ref> Further branches of number theory are probabilistic number theory,<ref>Template:Harvnb</ref> combinatorial number theory,<ref>Template:Harvnb</ref> computational number theory,<ref>Template:Harvnb</ref> and applied number theory, which examines the application of number theory to science and technology.<ref>Template:Multiref</ref>

OriginsEdit

Ancient MesopotamiaEdit

File:Plimpton 322.jpg

Plimpton 322 tablet

The earliest historical find of an arithmetical nature is a fragment of a table: Plimpton 322 (Larsa, Mesopotamia, c. 1800 BC), a broken clay tablet, contains a list of "Pythagorean triples", that is, integers <math>(a,b,c)</math> such that <math>a^2+b^2=c^2</math>. The triples are too numerous and too large to have been obtained by brute force. The heading over the first column reads: "The Template:Tlit of the diagonal which has been subtracted such that the width..."<ref>Template:Harvnb. The term Template:Tlit is problematic. Robson prefers the rendering "The holding-square of the diagonal from which 1 is torn out, so that the short side comes up...".Template:Harvnb</ref>

The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity<ref>Template:Harvnb. Other sources give the modern formula <math>(p^2-q^2,2pq,p^2+q^2)</math>. Van der Waerden gives both the modern formula and what amounts to the form preferred by Robson.Template:Harv</ref>

<math display="block">\left(\frac{1}{2} \left(x - \frac{1}{x}\right)\right)^2 + 1 = \left(\frac{1}{2} \left(x + \frac{1}{x} \right)\right)^2,</math>

which is implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and then reordered by <math>c/a</math>, presumably for actual use as a "table", for example, with a view to applications.<ref>Neugebauer Template:Harv discusses the table in detail and mentions in passing Euclid's method in modern notation Template:Harv.</ref>

It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy, for example, truly came into its own many centuries later. It has been suggested instead that the table was a source of numerical examples for school problems.Template:Sfn<ref group="note">Template:Harvnb. This is controversial. See Plimpton 322. Robson's article is written polemically Template:Harv with a view to "perhaps [...] knocking [Plimpton 322] off its pedestal" Template:Harv; at the same time, it settles to the conclusion that

[...] the question "how was the tablet calculated?" does not have to have the same answer as the question "what problems does the tablet set?" The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems Template:Harv.

Robson takes issue with the notion that the scribe who produced Plimpton 322 (who had to "work for a living", and would not have belonged to a "leisured middle class") could have been motivated by his own "idle curiosity" in the absence of a "market for new mathematics".Template:Harv</ref> Plimpton 322 tablet is the only surviving evidence of what today would be called number theory within Babylonian mathematics, though a kind of Babylonian algebra was much more developed.Template:Sfn

Ancient GreeceEdit

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Although other civilizations probably influenced Greek mathematics at the beginning,<ref>Template:Harvnb</ref> all evidence of such borrowings appear relatively late,<ref name="vanderW2">Iamblichus, Life of Pythagoras,(trans., for example, Template:Harvnb) cited in Template:Harvnb. See also Porphyry, Life of Pythagoras, paragraph 6, in Template:Harvnb</ref><ref name="stanencyc">Herodotus (II. 81) and Isocrates (Busiris 28), cited in: Template:Harvnb. On Thales, see Eudemus ap. Proclus, 65.7, (for example, Template:Harvnb) cited in: Template:Harvnb. Proclus was using a work by Eudemus of Rhodes (now lost), the Catalogue of Geometers. See also introduction, Template:Harvnb on Proclus's reliability.</ref> and it is likely that Greek Template:Tlit (the theoretical or philosophical study of numbers) is an indigenous tradition. Aside from a few fragments, most of what is known about Greek mathematics in the 6th to 4th centuries BC (the Archaic and Classical periods) comes through either the reports of contemporary non-mathematicians or references from mathematical works in the early Hellenistic period.Template:Sfn In the case of number theory, this means largely Plato, Aristotle, and Euclid.

Plato had a keen interest in mathematics, and distinguished clearly between Template:Tlit and calculation (Template:Tlit). Plato reports in his dialogue Theaetetus that Theodorus had proven that <math>\sqrt{3}, \sqrt{5}, \dots, \sqrt{17}</math> are irrational. Theaetetus, a disciple of Theodorus's, worked on distinguishing different kinds of incommensurables, and was thus arguably a pioneer in the study of number systems. Aristotle further claimed that the philosophy of Plato closely followed the teachings of the Pythagoreans,<ref>Metaphysics, 1.6.1 (987a)</ref> and Cicero repeats this claim: {{#invoke:Lang|lang}} ("They say Plato learned all things Pythagorean").<ref>Tusc. Disput. 1.17.39.</ref>

Euclid devoted part of his Elements (Books VII–IX) to topics that belong to elementary number theory, including prime numbers and divisibility.<ref>Template:Cite book</ref> He gave an algorithm, the Euclidean algorithm, for computing the greatest common divisor of two numbers (Prop. VII.2) and a proof implying the infinitude of primes (Prop. IX.20). There is also older material likely based on Pythagorean teachings (Prop. IX.21–34), such as "odd times even is even" and "if an odd number measures [= divides] an even number, then it also measures [= divides] half of it".<ref name="Becker">Template:Harvnb, cited in: Template:Harvnb.</ref> This is all that is needed to prove that <math>\sqrt{2}</math> is irrational.Template:Sfn Pythagoreans apparently gave great importance to the odd and the even.Template:Sfn The discovery that <math>\sqrt{2}</math> is irrational is credited to the early Pythagoreans, sometimes assigned to Hippasus, who was expelled or split from the Pythagorean community as a result.<ref name="Thea">Plato, Theaetetus, p. 147 B, (for example, Template:Harvnb), cited in Template:Harvnb: "Theodorus was writing out for us something about roots, such as the roots of three or five, showing that they are incommensurable by the unit;..." See also Spiral of Theodorus.</ref>Template:Sfn This forced a distinction between numbers (integers and the rationals—the subjects of arithmetic) and lengths and proportions (which may be identified with real numbers, whether rational or not).

The Pythagorean tradition also spoke of so-called polygonal or figurate numbers.Template:Sfn While square numbers, cubic numbers, etc., are seen now as more natural than triangular 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 centuries).

An epigram published by Lessing in 1773 appears to be a letter sent by Archimedes to Eratosthenes.Template:Sfn Template:Sfn The epigram proposed what has become known as Archimedes's 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 it is known, such equations were first successfully treated by Indian mathematicians. It is not known whether Archimedes himself had a method of solution.

Late AntiquityEdit

File:Diophantus-cover.png

lang}}, translated into Latin by Bachet (1621)

Aside from the elementary work of Neopythagoreans such as Nicomachus and Theon of Smyrna, the foremost authority in Template:Tlit in Late Antiquity was Diophantus of Alexandria, who probably lived in the 3rd century AD, approximately five hundred years after Euclid. Little is known about his life, but he wrote two works that are extant: On Polygonal Numbers, a short treatise written in the Euclidean manner on the subject, and the Arithmetica, a work on pre-modern algebra (namely, the use of algebra to solve numerical problems). Six out of the thirteen books of Diophantus's Arithmetica survive in the original Greek and four more survive in an Arabic translation. The {{#invoke:Lang|lang}} 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 <math>f(x,y)=z^2</math> or <math>f(x,y,z)=w^2</math>. In modern parlance, Diophantine equations are polynomial equations to which rational or integer solutions are sought.

AsiaEdit

The Chinese remainder theorem appears as an exercise<ref>Sunzi Suanjing, Chapter 3, Problem 26. This can be found in Template:Harvnb, which contains a full translation of the Suan Ching (based on Template:Harvnb). See also the discussion in Template:Harvnb.</ref> in Sunzi Suanjing (between the third and fifth centuries).<ref name="YongSe">The date of the text has been narrowed down to 220–420 AD (Yan Dunjie) or 280–473 AD (Wang Ling) through internal evidence (= taxation systems assumed in the text). See Template:Harvnb.</ref> (There is one important step glossed over in Sunzi's solution:<ref group="note">Sunzi Suanjing, Ch. 3, Problem 26,

in Template:Harvnb:

[26] Now there are an unknown number of things. If we count by threes, there is a remainder 2; if we count by fives, there is a remainder 3; if we count by sevens, there is a remainder 2. Find the number of things. Answer: 23.

Method: If we count by threes and there is a remainder 2, put down 140. If we count by fives and there is a remainder 3, put down 63. If we count by sevens and there is a remainder 2, put down 30. Add them to obtain 233 and subtract 210 to get the answer. If we count by threes and there is a remainder 1, put down 70. If we count by fives and there is a remainder 1, put down 21. If we count by sevens and there is a remainder 1, put down 15. When [a number] exceeds 106, the result is obtained by subtracting 105.

</ref> it is the problem that was later solved by Āryabhaṭa's Kuṭṭaka – see below.) The result was later generalized with a complete solution called Da-yan-shu ({{#invoke:Lang|lang}}) in Qin Jiushao's 1247 Mathematical Treatise in Nine Sections<ref>Template:Harvnb</ref> which was translated into English in early nineteenth century by British missionary Alexander Wylie.<ref>Template:Harvnb</ref> There is also some numerical mysticism in Chinese mathematics,<ref group="note">See, for example, Sunzi Suanjing, Ch. 3, Problem 36, in Template:Harvnb:

[36] Now there is a pregnant woman whose age is 29. If the gestation period is 9 months, determine the sex of the unborn child. Answer: Male.

Method: Put down 49, add the gestation period and subtract the age. From the remainder take away 1 representing the heaven, 2 the earth, 3 the man, 4 the four seasons, 5 the five phases, 6 the six pitch-pipes, 7 the seven stars [of the Dipper], 8 the eight winds, and 9 the nine divisions [of China under Yu the Great]. If the remainder is odd, [the sex] is male and if the remainder is even, [the sex] is female.

This is the last problem in Sunzi's otherwise matter-of-fact treatise.</ref> but, unlike that of the Pythagoreans, it seems to have led nowhere.

While Greek astronomy probably influenced Indian learning, to the point of introducing trigonometry,Template:Sfn it seems to be the case that Indian mathematics is otherwise an autochthonous tradition;<ref name="Plofbab">Any early contact between Babylonian and Indian mathematics remains conjectural Template:Harv.</ref> in particular, there is no evidence that Euclid's Elements reached India before the eighteenth century.Template:Sfn Āryabhaṭa (476–550 AD) showed that pairs of simultaneous congruences <math>n\equiv a_1 \bmod m_1</math>, <math>n\equiv a_2 \bmod m_2</math> could be solved by a method he called kuṭṭaka, or pulveriser;<ref>Āryabhaṭa, Āryabhatīya, Chapter 2, verses 32–33, cited in: Template:Harvnb. See also Template:Harvnb. A slightly more explicit description of the kuṭṭaka was later given in Brahmagupta, Brāhmasphuṭasiddhānta, XVIII, 3–5 (in Template:Harvnb, cited in Template:Harvnb).</ref> this is a procedure close to (a generalization of) the Euclidean algorithm, which was probably discovered independently in India.Template:Sfn Āryabhaṭa seems to have had in mind applications to astronomical calculations.Template:Sfn

Brahmagupta (628 AD) 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).Template:Sfn

Indian mathematics remained largely unknown in Europe until the late eighteenth century;Template:Sfn Brahmagupta and Bhāskara's work was translated into English in 1817 by Henry Colebrooke.Template:Sfn

Arithmetic in the Islamic golden ageEdit

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File:Selenographia 1647 (122459248) (cropped).jpg

Al-Haytham as seen by the West: on the frontispiece of Selenographia AlhasenTemplate:Sic represents knowledge through reason and Galileo 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<ref>Template:Harvnb, cited in Template:Harvnb. See also the preface in Template:Harvnb cited in Template:Harvnb</ref> or may not<ref name="Plofnot">Template:Harvnb, and Template:Harvnb, cited in Template:Harvnb.</ref> be Brahmagupta's Brāhmasphuṭasiddhānta). 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 – c. 1029) builds on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporary Ibn al-Haytham knewTemplate:Sfn what would later be called Wilson's theorem.

Western Europe in the Middle AgesEdit

Other than a treatise on squares in arithmetic progression by Fibonacci—who traveled and studied in north Africa and Constantinople—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 catalyst was the textual emendation and translation into Latin of Diophantus' Arithmetica.<ref>Bachet, 1621, following a first attempt by Xylander, 1575</ref>

Early modern number theoryEdit

FermatEdit

File:Pierre de Fermat.png

Pierre de Fermat

Pierre de Fermat (1607–1665) never published his writings but communicated through correspondence instead. Accordingly, his work on number theory is contained almost entirely in letters to mathematicians and in private marginal notes.Template:Sfn Although he drew inspiration from classical sources, in his notes and letters Fermat scarcely wrote any proofs—he had no models in the area.<ref>Template:Harvnb. This was more so in number theory than in other areas (Template:Harvnb). Bachet's own proofs were "ludicrously clumsy" Template:Harv.</ref>

Over his lifetime, Fermat made the following contributions to the field:

One of Fermat's first interests was perfect numbers (which appear in Euclid, Elements IX) and amicable numbers;<ref group="note">Perfect and especially amicable numbers are of little or no interest nowadays. The same was not true in earlier times—whether in the West or the Arab-speaking world—due in part to the importance given to them by the Neopythagorean Nicomachus (c. 100 AD), who wrote a very elementary but influential book entitled Introduction to Arithmetic. See Template:Harvnb.</ref> these topics 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.<ref>Template:Harvnb. The initial subjects of Fermat's correspondence included divisors ("aliquot parts") and many subjects outside number theory; see the list in the letter from Fermat to Roberval, 22.IX.1636, Template:Harvnb, cited in Template:Harvnb.</ref>
In 1638, Fermat claimed, without proof, that all whole numbers can be expressed as the sum of four squares or fewer.<ref>Template:Cite encyclopedia</ref>
Fermat's little theorem (1640):<ref>Template:Harvnb, Letter XLVI from Fermat to Frenicle, 1640,

cited in Template:Harvnb</ref> if a is not divisible by a prime p, then <math>a^{p-1} \equiv 1 \bmod p.</math><ref group="note">Here, as usual, given two integers a and b and a non-zero integer m, we write <math>a \equiv b \bmod m</math> (read "a is congruent to b modulo m") to mean that m divides a − b, or, what is the same, a and b leave the same residue when divided by m. This notation is actually much later than Fermat's; it first appears in section 1 of Gauss's {{#invoke:Lang|lang}}. Fermat's little theorem is a consequence of the fact that the order of an element of a group divides the order of the group. The modern proof would have been within Fermat's means (and was indeed given later by Euler), even though the modern concept of a group came long after Fermat or Euler. (It helps to know that inverses exist modulo p, that is, given a not divisible by a prime p, there is an integer x such that <math> x a \equiv 1 \bmod p</math>); this fact (which, in modern language, makes the residues mod p into a group, and which was already known to Āryabhaṭa; see above) was familiar to Fermat thanks to its rediscovery by Bachet Template:Harv. Weil goes on to say that Fermat would have recognised that Bachet's argument is essentially Euclid's algorithm.</ref>

If a and b are coprime, then <math>a^2 + b^2</math> is not divisible by any prime congruent to −1 modulo 4;<ref>Template:Harvnb, cited in Template:Harvnb. All of the following citations from Fermat's Varia Opera are taken from Template:Harvnb. The standard Tannery & Henry work includes a revision of Fermat's posthumous Varia Opera Mathematica originally prepared by his son Template:Harv.</ref> and every prime congruent to 1 modulo 4 can be written in the form <math>a^2 + b^2</math>.Template:Sfn These two statements also date from 1640; in 1659, Fermat stated to Huygens that he had proven the latter statement by the method of infinite descent.Template:Sfn
In 1657, Fermat posed the problem of solving <math>x^2 - N y^2 = 1</math> as a challenge to English mathematicians. The problem was solved in a few months by Wallis and Brouncker.Template:Sfn Fermat considered their solution valid, but pointed out they had provided an algorithm without a proof (as had Jayadeva and Bhaskara, though Fermat was not aware of this). He stated that a proof could be found by infinite descent.
Fermat stated and proved (by infinite descent) in the appendix to Observations on Diophantus (Obs. XLV)Template:Sfn that <math>x^{4} + y^{4} = z^{4}</math> has no non-trivial solutions in the integers. Fermat also mentioned to his correspondents that <math>x^3 + y^3 = z^3</math> has no non-trivial solutions, and that this could also be proven by infinite descent.Template:Sfn The first known proof is due to Euler (1753; indeed by infinite descent).Template:Sfn
Fermat claimed (Fermat's Last Theorem) to have shown there are no solutions to <math>x^n + y^n = z^n</math> for all <math>n\geq 3</math>; this claim appears in his annotations in the margins of his copy of Diophantus.

EulerEdit

File:Leonhard Euler.jpg

Leonhard Euler

The interest of Leonhard Euler (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur<ref group="note">Up to the second half of the seventeenth century, academic positions were very rare, and most mathematicians and scientists earned their living in some other way Template:Harv. (There were already some recognisable features of professional practice, viz., seeking correspondents, visiting foreign colleagues, building private libraries Template:Harv. Matters started to shift in the late seventeenth century Template:Harv; scientific academies were founded in England (the Royal Society, 1662) and France (the Académie des sciences, 1666) and Russia (1724). Euler was offered a position at this last one in 1726; he accepted, arriving in St. Petersburg in 1727 (Template:Harvnb and Template:Harvnb). In this context, the term amateur usually applied to Goldbach is well-defined and makes some sense: he has been described as a man of letters who earned a living as a spy Template:Harv; cited in Template:Harvnb). Notice, however, that Goldbach published some works on mathematics and sometimes held academic positions.</ref> Goldbach, pointed him towards some of Fermat's work on the subject.Template:Sfn Template:Sfn This has been called the "rebirth" of modern number theory,Template:Sfn after Fermat's relative lack of success in getting his contemporaries' attention for the subject.<ref>Template:Harvnb and Template:Harvnb</ref> Euler's work on number theory includes the following:<ref>Template:Harvnb and Template:Harvnb</ref>

Proofs for Fermat's statements. This includes Fermat's little theorem (generalised by Euler to non-prime moduli); the fact that <math>p = x^2 + y^2</math> if and only if <math>p\equiv 1 \bmod 4</math>; initial work towards a proof that every integer is the sum of four squares (the first complete proof is by Joseph-Louis Lagrange (1770), soon improved by Euler himselfTemplate:Sfn); the lack of non-zero integer solutions to <math>x^4 + y^4 = z^2</math> (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.<ref name="Eulpell">Template:Harvnb. Euler was generous in giving credit to others Template:Harv, not always correctly.</ref> He wrote on the link between continued fractions and Pell's equation.Template:Sfn
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.<ref>Template:Harvnb; see also chapter III.</ref>
Quadratic forms. Following Fermat's lead, Euler did further research on the question of which primes can be expressed in the form <math>x^2 + N y^2</math>, some of it prefiguring quadratic reciprocity.Template:Sfn Template:Sfn Template:Sfn
Diophantine equations. Euler worked on some Diophantine equations of genus 0 and 1.Template:Sfn Template:Sfn 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.Template:Sfn He did notice there was a connection between Diophantine problems and elliptic integrals,Template:Sfn whose study he had himself initiated.

Lagrange, Legendre, and GaussEdit

File:Carl Friedrich Gauss 1840 by Jensen.jpg

Carl Friedrich Gauss

Joseph-Louis 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 <math>m X^2 + n Y^2</math>), including defining their equivalence relation, showing how to put them in reduced form, etc.

Adrien-Marie 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 <math>a x^2 + b y^2 + c z^2 = 0</math>Template:Sfn and worked on quadratic forms along the lines later developed fully by Gauss.Template:Sfn In his old age, he was the first to prove Fermat's Last Theorem for <math>n=5</math> (completing work by Peter Gustav Lejeune Dirichlet, and crediting both him and Sophie Germain).Template:Sfn

Carl Friedrich Gauss (1777–1855) worked in a wide variety of fields in both mathematics and physics including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. The Disquisitiones Arithmeticae (1801), which he wrote three years earlier when he was 21, had an immense influence in the area of number theory and set its agenda for much of the 19th century. Gauss proved in this work 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.Template:Sfn 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.<ref>From the preface of Disquisitiones Arithmeticae; the translation is taken from Template:Harvnb</ref>

In this way, Gauss arguably made forays towards Évariste Galois's work and the area algebraic number theory.

Maturity and division into subfieldsEdit

File:Peter Gustav Lejeune Dirichlet.jpg

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.<ref>See the discussion in section 5 of Template:Harvnb. Early signs of self-consciousness are present already in letters by Fermat: thus his remarks on what number theory is, and how "Diophantus's work [...] does not really belong to [it]" (quoted in Template:Harvnb).</ref>
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),Template:Sfn Template:Sfn whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable.<ref>See the proof in Template:Harvnb</ref> The first use of analytic ideas in number theory actually goes back to Euler (1730s),Template:Sfn Template:Sfn 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 Bernhard Riemann (1859) on the zeta function is the canonical starting point;Template:Sfn 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).<ref>See the comment on the importance of modularity in Template:Harvnb</ref>

The American Mathematical Society awards the Cole Prize in Number Theory. Moreover, number theory is one of the three mathematical subdisciplines rewarded by the Fermat Prize.

Main subdivisionsEdit

Elementary number theoryEdit

File:Paul Erdos with Terence Tao.jpg

Number theorists Paul Erdős and Terence Tao in 1985, when Erdős was 72 and Tao was 10

Elementary number theory deals with the topics in number theory by means of basic methods in arithmetic.<ref name=":1">Template:Cite book</ref> Its primary subjects of study are divisibility, factorization, and primality, as well as congruences in modular arithmetic.<ref>Template:Cite book</ref><ref name=":3" /> Other topics in elementary number theory include Diophantine equations, continued fractions, integer partitions, and Diophantine approximations.<ref name=":2">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations of addition, subtraction, multiplication, division, exponentiation, extraction of roots, and logarithms. Multiplication, for instance, is an operation that combines two numbers, referred to as factors, to form a single number, termed the product, such as <math>2 \times 3 = 6</math>.<ref>Template:Multiref</ref>

Divisibility is a property between two nonzero integers related to division. An integer <math>a</math> is said to be divisible by a nonzero integer <math>b</math> if <math>a</math> is a multiple of <math>b</math>; that is, if there exists an integer <math>q</math> such that <math>a = bq</math>. An equivalent formulation is that <math>b</math> divides <math>a</math> and is denoted by a vertical bar, which in this case is <math>b | a</math>. Conversely, if this were not the case, then <math>a</math> would not be divided evenly by <math>b</math>, resulting in a remainder. Euclid's division lemma asserts that <math>a</math> and <math>b</math> can generally be written as <math>a = bq + r</math>, where the remainder <math>r < b</math> accounts for the leftover quantity. Elementary number theory studies divisibility rules in order to quickly identify if a given integer is divisible by a fixed divisor. For instance, it is known that any integer is divisible by 3 if its decimal digit sum is divisible by 3.<ref name="Richmond-Richmond-2009">Richmond & Richmond (2009), [[[:Template:Google books]] Section 3.4 (Divisibility Tests), p. 102–108]</ref><ref name=":4" />

A common divisor of several nonzero integers is an integer that divides all of them. The greatest common divisor (gcd) is the largest of such divisors. Two integers are said to be coprime or relatively prime to one another if their greatest common divisor, and simultaneously their only divisor, is 1. The Euclidean algorithm computes the greatest common divisor of two integers <math>a,b</math> by means of repeatedly applying the division lemma and shifting the divisor and remainder after every step. The algorithm can be extended to solve a special case of linear Diophantine equations <math>ax + by = 1</math>. A Diophantine equation is an equation with several unknowns and integer coefficients. Another kind of Diophantine equation is described in the Pythagorean theorem, <math>x^2 + y^2 = z^2</math>, whose solutions are called Pythagorean triples if they are all integers.<ref name=":4" /><ref name=":6" />

Elementary number theory studies the divisibility properties of integers such as parity (even and odd numbers), prime numbers, and perfect numbers. Important number-theoric functions include the divisor-counting function, the divisor summatory function and its modifications, and Euler's totient function. A prime number is an integer greater than 1 whose only positive divisors are 1 and the prime itself. A positive integer greater than 1 that is not prime is called a composite number. Euclid's theorem demonstrates that there are infinitely many prime numbers that comprise the set {2, 3, 5, 7, 11, ...}. The sieve of Eratosthenes was devised as an efficient algorithm for identifying all primes up to a given natural number by eliminating all composite numbers.<ref>Template:Cite book</ref>

Factorization is a method of expressing a number as a product. Specifically in number theory, integer factorization is the decomposition of an integer into a product of integers. The process of repeatedly applying this procedure until all factors are prime is known as prime factorization. A fundamental property of primes is shown in Euclid's lemma. It is a consequence of the lemma that if a prime divides a product of integers, then that prime divides at least one of the factors in the product. The unique factorization theorem is the fundamental theorem of arithmetic that relates to prime factorization. The theorem states that every integer greater than 1 can be factorised into a product of prime numbers and that this factorisation is unique up to the order of the factors. For example, <math>120</math> is expressed uniquely as <math>2 \times 2 \times 2 \times 3 \times 5</math> or simply <math>2^3 \times 3 \times 5</math>.<ref>Template:Cite book</ref><ref name=":4" />

Modular arithmetic works with finite sets of integers and introduces the concepts of congruence and residue classes. A congruence of two integers <math>a, b</math> modulo <math>n</math> (a positive integer called the modulus) is an equivalence relation whereby <math>n | (a - b)</math> is true. Performing Euclidean division on both <math>a</math> and <math>n</math>, and on <math>b</math> and <math>n</math>, yields the same remainder. This written as <math display="inline">a \equiv b \pmod{n}</math>. In a manner analogous to the 12-hour clock, the sum of 4 and 9 is equal to 13, yet congruent to 1. A residue class modulo <math>n</math> is a set that contains all integers congruent to a specified <math>r</math> modulo <math>n</math>. For example, <math>6\Z + 1</math> contains all multiples of 6 incremented by 1. Modular arithmetic provides a range of formulas for rapidly solving congruences of very large powers. An influential theorem is Fermat's little theorem, which states that if a prime <math>p</math> is coprime to some integer <math>a</math>, then <math display="inline">a^{p - 1} \equiv 1 \pmod{p}</math> is true. Euler's theorem extends this to assert that every integer <math>n</math> satisfies the congruence<math display="block">a^{\varphi(n)} \equiv 1 \pmod{n},</math>where Euler's totient function <math>\varphi</math> counts all positive integers up to <math>n</math> that are coprime to <math>n</math>. Modular arithmetic also provides formulas that are used to solve congruences with unknowns in a similar vein to equation solving in algebra, such as the Chinese remainder theorem.<ref>Template:Cite book</ref>

Analytic number theoryEdit

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File:Complex zeta.jpg

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.

File:ModularGroup-FundamentalDomain.svg

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

Analytic number theory, in contrast to elementary number theory, relies on complex numbers and techniques from analysis and calculus. 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;Template:Sfn or
in terms of its concerns, as the study within number theory of estimates on the size and density of certain numbers (e.g., primes), as opposed to identities.<ref>Template:Harvnb: "The main difference is that in algebraic number theory [...] one typically considers questions with answers that are given by exact formulas, whereas in analytic number theory [...] one looks for good approximations."</ref>

It studies the distribution of primes, behavior of number-theoric functions, and irrational numbers.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, many of 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.<ref>See, for example, the initial comment in Template:Harvnb.</ref> The following are examples of problems in analytic number theory: the prime number theorem, the Goldbach conjecture, the twin prime conjecture, 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.<ref>See the remarks in the introduction to Template:Harvnb: "However much stronger...".</ref>

Analysis is the branch of mathematics that studies the limit, defined as the value to which a sequence or function tends as the argument (or index) approaches a specific value. For example, the limit of the sequence 0.9, 0.99, 0.999, ... is 1. In the context of functions, the limit of <math display="inline">\frac1x</math> as <math>x</math> approaches infinity is 0.<ref>Template:Cite book</ref> The complex numbers extend the real numbers with the imaginary unit <math>i</math> defined as the solution to <math>i^2 = -1</math>. Every complex number can be expressed as <math>x + iy</math>, where <math>x</math> is called the real part and <math>y</math> is called the imaginary part.<ref>Template:Cite book</ref>

The distribution of primes, described by the function <math>\pi</math> that counts all primes up to a given real number, is unpredictable and is a major subject of study in number theory. Elementary formulas for a partial sequence of primes, including Euler's prime-generating polynomials have been developed. However, these cease to function as the primes become too large. The prime number theorem in analytic number theory provides a formalisation of the notion that prime numbers appear less commonly as their numerical value increases. One distribution states, informally, that the function <math>\frac{x}{\log(x)}</math> approximates <math>\pi(x)</math>. Another distribution involves an offset logarithmic integral which converges to <math>\pi(x)</math> more quickly.<ref name=":5" />

File:Riemann Explicit Formula.gif

Corrections to an estimate of the prime-counting function using zeros of the zeta function.

The zeta function has been demonstrated to be connected to the distribution of primes. It is defined as the series<math display="block"> \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots</math>that converges if <math> s</math> is greater than 1. Euler demonstrated a link involving the infinite product over all prime numbers, expressed as the identity <math display="block">\zeta(s) = \prod_{p \text{ prime}} \left(1 - \frac{1}{p^{s}}\right)^{-1}.</math>Riemann extended the definition to a complex variable and conjectured that all nontrivial cases (<math>0 < \Re(s) < 1</math>) where the function returns a zero are those in which the real part of <math>s</math> is equal to <math display="inline">\frac12</math>. He established a connection between the nontrivial zeroes and the prime-counting function. In what is now recognised as the unsolved Riemann hypothesis, a solution to it would imply direct consequences for understanding the distribution of primes.<ref>Template:Cite book</ref>

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 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, a key analytic object at the roots of the subject.<ref>Template:Harvnb: "[Riemann] defined what we now call the Riemann zeta function [...] Riemann's deep work gave birth to our subject [...]"</ref> This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complex-valued function.<ref name=":0">See, for example, Template:Harvnb, p. 1.</ref>

Elementary number theory works with elementary proofs, a term that excludes the use of complex numbers but may include basic analysis.<ref name=":2" /> For example, the prime number theorem was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by Erdős and Selberg.Template:Sfn The term is somewhat ambiguous. For example, proofs based on complex Tauberian theorems, such as Wiener–Ikehara, are often seen as quite enlightening but not elementary despite using Fourier analysis, not complex analysis. Here as elsewhere, an elementary proof may be longer and more difficult for most readers than a more advanced proof.

Some subjects generally considered to be part of analytic number theory (e.g., sieve theory) are better covered by the second rather than the first definition.<ref group="note">Sieve theory figures as one of the main subareas of analytic number theory in many standard treatments; see, for instance, Template:Harvnb or Template:Harvnb</ref> Small sieves, for instance, use little analysis and yet still belong to analytic number theory.<ref group="note">This is the case for some combinatorial sieves such as the Brun sieve, rather than for large sieves. The study of the latter now includes ideas from harmonic and functional analysis.</ref>

Algebraic number theoryEdit

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An algebraic number is any complex number that is a solution to some polynomial equation <math>f(x)=0</math> with rational coefficients; for example, every solution <math>x</math> of <math>x^5 + (11/2) x^3 - 7 x^2 + 9 = 0 </math> is an algebraic number. Fields of algebraic numbers are also called algebraic number fields, or shortly number fields. Algebraic number theory studies algebraic number fields.Template:Sfn

It could be argued that the simplest kind of number fields, namely 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 <math> a + b \sqrt{d}</math>, where <math>a</math> and <math>b</math> are rational numbers and <math>d</math> is a fixed rational number whose square root is not rational.) For that matter, the eleventh-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 were set in the late nineteenth century, when ideal numbers, the theory of ideals and valuation theory were introduced; these are three complementary ways of dealing with the lack of unique factorization in algebraic number fields. (For example, in the field generated by the rationals and <math> \sqrt{-5}</math>, the number <math>6</math> can be factorised both as <math> 6 = 2 \cdot 3</math> and <math> 6 = (1 + \sqrt{-5}) ( 1 - \sqrt{-5})</math>; all of <math>2</math>, <math>3</math>, <math>1 + \sqrt{-5}</math> and <math> 1 - \sqrt{-5}</math> 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,Template:Sfn that is, generalizations 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<ref group="note">The Galois group of an extension L/K consists of the operations (isomorphisms) that send elements of L to other elements of L while leaving all elements of K fixed. Thus, for instance, Gal(C/R) consists of two elements: the identity element (taking every element x + iy of C to itself) and complex conjugation (the map taking each element x + iy to x − iy). The Galois group of an extension tells us many of its crucial properties. The study of Galois groups started with Évariste Galois; in modern language, the main outcome of his work is that an equation f(x) = 0 can be solved by radicals (that is, x can be expressed in terms of the four basic operations together with square roots, cubic roots, etc.) if and only if the extension of the rationals by the roots of the equation f(x) = 0 has a Galois group that is solvable in the sense of group theory. ("Solvable", in the sense of group theory, is a simple property that can be checked easily for finite groups.)</ref> 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 nineteenth 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 geometryEdit

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The central problem of Diophantine geometry is to determine when a Diophantine equation has integer or rational 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 Template:Math-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 whether there are finitely or infinitely many rational points on a given curve or surface.

Consider, for instance, the Pythagorean equation <math>x^2+y^2 = 1</math>. One would like to know its rational solutions, namely <math>(x,y)</math> such that x and y are both rational. This is the same as asking for all integer solutions to <math>a^2 + b^2 = c^2</math>; any solution to the latter equation gives us a solution <math>x = a/c</math>, <math>y = b/c</math> to the former. It is also the same as asking for all points with rational coordinates on the curve described by <math>x^2 + y^2 = 1</math> (a circle of radius 1 centered on the origin).

File:ECClines-3.svg

Two examples of elliptic curves, that is, curves of genus 1 having at least one rational point.

The rephrasing of questions on equations in terms of points on curves is 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 <math>f(x,y)=0</math>, where <math>f</math> is a polynomial in two variables) depends crucially on the genus of the curve.<ref group="note">The genus can be defined as follows: allow the variables in <math>f(x,y)=0</math> to be complex numbers; then <math>f(x,y)=0</math> defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables; that is, four dimensions). The number of doughnut-like holes in the surface is called the genus of the curve of equation <math>f(x,y)=0</math>.</ref> A major achievement of this approach is Wiles's proof of Fermat's Last Theorem, for which other geometrical notions are just as crucial.

There is also the closely linked area of Diophantine approximations: given a number <math>x</math>, determine how well it can be approximated by rational numbers. One seeks approximations that are good relative to the amount of space required to write the rational number: call <math>a/q</math> (with <math>\gcd(a,q)=1</math>) a good approximation to <math>x</math> if <math>|x-a/q|<\frac{1}{q^c}</math>, where <math>c</math> is large. This question is of special interest if <math>x</math> is an algebraic number. If <math>x</math> cannot be approximated well, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of height) are critical both in Diophantine geometry and in the study of Diophantine approximations. This question is also of special interest in transcendental number theory: if a number can be approximated better than any algebraic number, then it is a transcendental number. It is by this argument that [[Pi|Template:Pi]] and e have been shown to be transcendental.

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 is a contemporary term for the same domain covered by Diophantine geometry, particularly when one wishes to emphasize the connections to modern algebraic geometry (for example, in Faltings's theorem) rather than to techniques in Diophantine approximations.

Other subfieldsEdit

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Probabilistic number theory starts with questions such as the following: Take an integer Template:Mvar 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). How many prime divisors will Template:Mvar have on average? What is the probability that it will have many more or many fewer divisors or prime divisors than the average?{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

Combinatorics in number theory starts with questions like the following: Does a fairly "thick" infinite set <math>A</math> contain many elements in arithmetic progression: <math>a</math>,

<math>a+b, a+2 b, a+3 b, \ldots, a+10b</math>? Should it be possible to write large integers as sums of elements of <math>A</math>?{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

File:Computer History Museum (4145886786).jpg

A Lehmer sieve, a primitive digital computer used to find primes and solve simple Diophantine equations

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

ApplicationsEdit

For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of mathematics other than the use of prime numbered gear teeth to distribute wear evenly.<ref>Template:Cite book</ref> In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance.<ref>Template:Cite book</ref> 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.<ref>The Unreasonable Effectiveness of Number Theory, Stefan Andrus Burr, George E. Andrews, American Mathematical Soc., 1992, Template:Isbn</ref>

This vision of the purity of number theory was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public-key cryptography algorithms.<ref>Template:Cite book</ref> Schemes such as RSA are based on the difficulty of factoring large composite numbers into their prime factors.<ref>Template:Cite book</ref> These applications have led to significant study of algorithms for computing with prime numbers, and in particular of primality testing, methods for determining whether a given number is prime. Prime numbers are also used in computing for checksums, hash tables, and pseudorandom number generators.

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".<ref>Computer science and its relation to mathematics" DE Knuth – The American Mathematical Monthly, 1974</ref> Elementary number theory is taught in discrete mathematics courses for computer scientists. It also has applications to the continuous in numerical analysis.<ref>"Applications of number theory to numerical analysis", Lo-keng Hua, Luogeng Hua, Yuan Wang, Springer-Verlag, 1981, Template:Isbn</ref>

Number theory has now several modern applications spanning diverse areas such as:

Computer science: The fast Fourier transform (FFT) algorithm, which is used to efficiently compute the discrete Fourier transform, has important applications in signal processing and data analysis.<ref>Template:Cite book</ref>
Physics: The Riemann hypothesis has connections to the distribution of prime numbers and has been studied for its potential implications in physics.<ref>Template:Cite journal</ref>
Error correction codes: The theory of finite fields and algebraic geometry have been used to construct efficient error-correcting codes.<ref>Template:Cite book</ref>
Communications: The design of cellular telephone networks requires knowledge of the theory of modular forms, which is a part of analytic number theory.<ref>Template:Citation</ref>
Study of musical scales: the concept of "equal temperament", which is the basis for most modern Western music, involves dividing the octave into 12 equal parts.<ref>Template:Cite journal</ref> This has been studied using number theory and in particular the properties of the 12th root of 2.