View source for Number theory

{{Short description|Branch of mathematics}}
{{for|the book by André Weil|Number Theory: An Approach Through History from Hammurapi to Legendre{{!}}''Number Theory: An Approach Through History from Hammurapi to Legendre''}}
{{distinguish|Numerology}}
[[File:Spirale Ulam 150.jpg|thumb|upright=1.2|The distribution of [[prime number]]s is a central point of study in number theory. This [[Ulam spiral]] serves to illustrate it, hinting, in particular, at the conditional [[Independence (probability theory)|independence]] between being prime and being a value of certain quadratic polynomials.]]
{{Math topics TOC}}
'''Number theory''' is a branch of [[pure mathematics]] devoted primarily to the study of the [[integer]]s and [[arithmetic function]]s. German mathematician [[Carl Friedrich Gauss]] (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."{{sfn|Long|1972|p=1}} Number theorists study [[prime number]]s as well as the properties of [[mathematical object]]s constructed from integers (for example, [[rational number]]s), or defined as generalizations of the integers (for example, [[algebraic integer]]s).

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 [[Complex analysis|analytical]] objects (for example, 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 number]]s in relation to rational numbers; for example, 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 By the early twentieth century, it had been superseded by ''number theory''.<ref group="note">Already in 1921, [[T. L. Heath]] had to explain: "By arithmetic, Plato meant, not arithmetic in our sense, but the science which considers numbers in themselves, in other words, what we mean by the Theory of Numbers." {{harv|Heath|1921|p=13}}</ref> (The word ''[[arithmetic]]'' is used by the general public to mean [[Elementary arithmetic|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 twentieth century, arguably in part due to French influence.<ref group="note">Take, for example, {{harvnb|Serre|1996}}. In 1952, [[Harold Davenport|Davenport]] still had to specify that he meant ''The Higher Arithmetic''. [[G. H. Hardy|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." {{harv|Hardy|Wright|2008}}</ref> In particular, ''arithmetical'' is commonly preferred as an adjective to ''number-theoretic''.

== History ==
=== Origins ===
==== Dawn of arithmetic ====
[[File:Plimpton 322.jpg|thumb|The Plimpton 322 tablet]]
The earliest historical find of an arithmetical nature is a fragment of a table: the broken clay tablet [[Plimpton 322]] ([[Larsa]], Mesopotamia, c.&nbsp;1800&nbsp;BC) contains a list of "[[Pythagorean triple]]s", that is, integers <math>(a,b,c)</math> such that <math>a^2+b^2=c^2</math>.
The triples are too many and too large to have been obtained by [[brute force method|brute force]]. The heading over the first column reads: "The ''takiltum'' of the diagonal which has been subtracted such that the width..."<ref>{{harvnb|Neugebauer|Sachs|1945|p=40}}. The term ''takiltum'' 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...".{{harvnb|Robson|2001|p=192}}</ref>

The table's layout suggests<ref>{{harvnb|Robson|2001|p=189}}. 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.{{harv|van der Waerden|1961|p=79}}</ref> that it was constructed by means of what amounts, in modern language, to the [[Identity (mathematics)|identity]]

:<math>\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 language|Old Babylonian]] exercises.{{sfn|van der Waerden|1961|p=184}} If some other method was used,<ref>Neugebauer {{harv|Neugebauer|1969|pp=36–40}} discusses the table in detail and mentions in passing Euclid's method in modern notation {{harv|Neugebauer|1969|p=39}}.</ref> 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.

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 only later. It has been suggested instead that the table was a source of numerical examples for school problems.{{sfn|Friberg|1981|p=302}}<ref group="note">{{harvnb|Robson|2001|p=201}}. This is controversial. See [[Plimpton 322]]. Robson's article is written polemically {{harv|Robson|2001|p=202}} with a view to "perhaps [...] knocking [Plimpton 322] off its pedestal" {{harv|Robson|2001|p=167}}; at the same time, it settles to the conclusion that <blockquote>[...] 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 {{harv|Robson|2001|p=202}}.</blockquote>
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".{{harv|Robson|2001|pp=199–200}}</ref>

While evidence of Babylonian number theory is only survived by the Plimpton 322 tablet, some authors assert that [[Babylonian mathematics#Algebra|Babylonian algebra]] was exceptionally well developed and included the foundations of modern [[elementary algebra]].{{sfn|van der Waerden|1961|p=43}} Late Neoplatonic sources<ref name="vanderW2">[[Iamblichus]], ''Life of Pythagoras'',(trans., for example, {{harvnb|Guthrie|1987}}) cited in {{harvnb|van der Waerden|1961|p=108}}. See also [[Porphyry (philosopher)|Porphyry]], ''Life of Pythagoras'', paragraph 6, in {{harvnb|Guthrie|1987|para=6}}
Van der Waerden {{harv|van der Waerden|1961|pp=87–90}} sustains the view that Thales knew Babylonian mathematics.</ref> state that [[Pythagoras]] learned mathematics from the Babylonians. Much earlier sources<ref name="stanencyc">Herodotus (II. 81) and Isocrates (''Busiris'' 28), cited in: {{harvnb|Huffman|2011}}. On Thales, see Eudemus ap. Proclus, 65.7, (for example, {{harvnb|Morrow|1992|p=52}}) cited in: {{harvnb|O'Grady|2004|p=1}}. Proclus was using a work by [[Eudemus of Rhodes]] (now lost), the ''Catalogue of Geometers''. See also introduction, {{harvnb|Morrow|1992|p=xxx}} on Proclus's reliability.</ref> state that [[Thales]] and Pythagoras traveled and studied in Egypt.

In book nine of [[Euclid's Elements]], propositions 21–34 are very probably influenced by [[Pythagoreanism|Pythagorean teachings]];<ref name="Becker">{{harvnb|Becker|1936|p=533}}, cited in: {{harvnb|van der Waerden|1961|p=108}}.</ref> 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 [[square root of 2|<math>\sqrt{2}</math>]]
is [[Irrational number|irrational]].{{sfn|Becker|1936}} [[Pythagoreanism|Pythagorean mystics]] gave great importance to the odd and the even.{{sfn|van der Waerden|1961|p=109}}
The discovery that <math>\sqrt{2}</math> is irrational is credited to the early Pythagoreans (pre-[[Theodorus of Cyrene|Theodorus]]).<ref name="Thea">Plato, ''Theaetetus'', p. 147 B, (for example, {{harvnb|Jowett|1871}}), cited
in {{harvnb|von Fritz|2004|p=212}}: "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> 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.{{sfn|von Fritz|2004}} This forced a distinction between ''[[number]]s'' (integers and the rationals—the subjects of arithmetic), on the one hand, and ''lengths'' and ''proportions'' (which may be identified with real numbers, whether rational or not), on the other hand.

The Pythagorean tradition spoke also of so-called [[polygonal number|polygonal]] or [[figurate numbers]].{{sfn|Heath|1921|p=76}} While [[square number]]s, [[cubic number]]s, etc., are seen now as more natural than [[triangular number]]s, [[pentagonal number]]s, etc., the study of the sums of triangular and pentagonal numbers would prove fruitful in the [[early modern period]] (seventeenth to early nineteenth centuries).

The [[Chinese remainder theorem]] appears as an exercise<ref>''Sunzi Suanjing'', Chapter 3, Problem 26. This can be found in {{harvnb|Lam|Ang|2004|pp=219–220}}, which contains a full translation of the ''Suan Ching'' (based on {{harvnb|Qian|1963}}). See also the discussion in {{harvnb|Lam|Ang|2004|pp=138–140}}.</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 CE (Yan Dunjie) or 280–473 CE (Wang Ling) through internal evidence (= taxation systems assumed in the text). See {{harvnb|Lam|Ang|2004|pp=27–28}}.</ref> (There is one important step glossed over in Sunzi's solution:<ref group="note">''Sunzi Suanjing'', Ch. 3, Problem 26,
in {{harvnb|Lam|Ang|2004|pp=219–220}}:<blockquote>
[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.<br />
''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.</blockquote></ref> it is the problem that was later solved by [[Āryabhaṭa]]'s [[Kuṭṭaka]] – see [[#Āryabhaṭa, Brahmagupta, Bhāskara|below]].) The result was later generalized with a complete solution called ''Da-yan-shu'' ({{lang|zh|大衍術}}) in [[Qin Jiushao]]'s 1247 ''[[Mathematical Treatise in Nine Sections]]'' <ref>{{harvnb|Dauben|2007|page=310}}</ref> which was translated into English in early nineteenth century by British missionary [[Alexander Wylie (missionary)|Alexander Wylie]].<ref>{{harvnb|Libbrecht|1973}}</ref>

There is also some numerical mysticism in Chinese mathematics,<ref group="note">See, for example, ''Sunzi Suanjing'', Ch. 3, Problem 36, in {{harvnb|Lam|Ang|2004|pp=223–224}}:<blockquote>
[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.<br />
''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.</blockquote>
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.

==== Classical Greece and the early Hellenistic period ====
{{Further|Ancient Greek mathematics}}

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]].{{sfn|Boyer|Merzbach|1991|p=82}} In the case of number theory, this means, by and large, ''[[Plato]]'' and ''Euclid'', respectively.

While Asian mathematics influenced Greek and Hellenistic learning, it seems to be the case that Greek mathematics is also an indigenous tradition.

[[Eusebius]], PE X, chapter 4 mentions of [[Pythagoras]]:

<blockquote>"In fact the said Pythagoras, while busily studying the wisdom of each nation, visited Babylon, and Egypt, and all Persia, being instructed by the Magi and the priests: and in addition to these he is related to have studied under the Brahmans (these are Indian philosophers); and from some he gathered astrology, from others geometry, and arithmetic and music from others, and different things from different nations, and only from the wise men of Greece did he get nothing, wedded as they were to a poverty and dearth of wisdom: so on the contrary he himself became the author of instruction to the Greeks in the learning which he had procured from abroad."<ref>{{Cite web |url=http://www.tertullian.org/fathers/eusebius_pe_10_book10.htm |title=Eusebius of Caesarea: Praeparatio Evangelica (Preparation for the Gospel). Tr. E. H. Gifford (1903) – Book 10 |access-date=2017-02-20 |archive-date=2016-12-11 |archive-url=https://web.archive.org/web/20161211194042/http://www.tertullian.org/fathers/eusebius_pe_10_book10.htm |url-status=live}}</ref></blockquote>

Aristotle 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: {{lang|la|Platonem ferunt didicisse Pythagorea omnia}} ("They say Plato learned all things Pythagorean").<ref>Tusc. Disput. 1.17.39.</ref>

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 (dialogue)|Theaetetus]]''—that it is known that [[Theodorus of Cyrene|Theodorus]] had proven that <math>\sqrt{3}, \sqrt{5}, \dots, \sqrt{17}</math> are irrational. [[Theaetetus of Athens|Theaetetus]] was, like Plato, a disciple of Theodorus's; he worked on distinguishing different kinds of [[Commensurability (mathematics)|incommensurables]], and was thus arguably a pioneer in the study of [[number systems]]. (Book X of [[Euclid's Elements]] is described by [[Pappus of Alexandria|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 to it (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, [[Gotthold Ephraim Lessing|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]].{{sfn|Vardi|1998|pp=305–319}}{{sfn|Weil|1984|pp=17–24}} 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 the [[#Āryabhaṭa, Brahmagupta, Bhāskara|Indian school]]. It is not known whether Archimedes himself had a method of solution.

==== Diophantus ====
[[File:Diophantus-cover.png|thumb|upright=0.8|Title page of the 1621 edition of [[Diophantus of Alexandria]]'s ''{{lang|la|[[Arithmetica]]}}'', translated into Latin by [[Claude Gaspard Bachet de Méziriac]]]]

Little is known about [[Diophantus of Alexandria]]; he probably lived in the third century&nbsp;AD, that is, about five hundred years after Euclid. Six out of the thirteen books of Diophantus's ''{{lang|la|[[Arithmetica]]}}'' survive in the original Greek and four more survive in an Arabic translation. The ''{{lang|la|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 <math>f(x,y)=z^2</math> or <math>f(x,y,z)=w^2</math>. In modern parlance, [[Diophantine equation]]s are [[polynomial equation]]s to which rational or integer solutions are sought.

==== Āryabhaṭa, Brahmagupta, Bhāskara ====
While Greek astronomy probably influenced Indian learning, to the point of introducing [[trigonometry]],{{sfn|Plofker|2008|p=119}} it seems to be the case that Indian mathematics is otherwise an indigenous tradition;<ref name="Plofbab">Any early contact between Babylonian and Indian mathematics remains conjectural {{harv|Plofker|2008|p=42}}.</ref> in particular, there is no evidence that Euclid's Elements reached India before the eighteenth century.{{sfn|Mumford|2010|p=387}}

Āryabhaṭa (476–550&nbsp;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: {{harvnb|Plofker|2008|pp=134–140}}. See also {{harvnb|Clark|1930|pp=42–50}}. A slightly more explicit description of the kuṭṭaka was later given in [[Brahmagupta]], ''Brāhmasphuṭasiddhānta'', XVIII, 3–5 (in {{harvnb|Colebrooke|1817|p=325}}, cited in {{harvnb|Clark|1930|p=42}}).</ref> this is a procedure close to (a generalisation of) the Euclidean algorithm, which was probably discovered independently in India.{{sfn|Mumford|2010|p=388}} Āryabhaṭa seems to have had in mind applications to astronomical calculations.{{sfn|Plofker|2008|p=119}}

Brahmagupta (628&nbsp;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 (mathematician)|Jayadeva]] (cited in the eleventh century; his work is otherwise lost); the earliest surviving exposition appears in [[Bhāskara II]]'s Bīja-gaṇita (twelfth century).{{sfn|Plofker|2008|p=194}}

Indian mathematics remained largely unknown in Europe until the late eighteenth century;{{sfn|Plofker|2008|p=283}} Brahmagupta and Bhāskara's work was translated into English in 1817 by [[Henry Colebrooke]].{{sfn|Colebrooke|1817}}

==== Arithmetic in the Islamic golden age ====
{{Further|Mathematics in medieval Islam|Islamic Golden Age}}

[[File:Hevelius Selenographia frontispiece.png|upright=0.8|thumb|[[Al-Haytham]] as seen by the West: on the frontispiece of ''[[Selenographia]]'' Alhasen{{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>{{harvnb|Colebrooke|1817|p=lxv}}, cited in {{harvnb|Hopkins|1990|p=302}}. See also the preface in
{{harvnb|Sachau|Bīrūni|1888}} cited in {{harvnb|Smith|1958|pp=168}}</ref> or may not<ref name="Plofnot">{{harvnb|Pingree|1968|pp=97–125}}, and {{harvnb|Pingree|1970|pp=103–123}}, cited in {{harvnb|Plofker|2008|p=256}}.</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&nbsp;– c.&nbsp;1029) builds on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporary [[Ibn al-Haytham]] knew{{sfn|Rashed|1980|pp=305–321}} what would later be called [[Wilson's theorem]].

==== Western Europe in the Middle Ages ====
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 [[Guilielmus Xylander|Xylander]], 1575</ref>
<!--Fibonaaci sequence, unknown author of 1486 ms, Luca Pacioli.. -->

=== Early modern number theory ===
==== Fermat ====
[[File:Pierre de Fermat.png|thumb|upright=0.8|[[Pierre de Fermat]]]]

[[Pierre de Fermat]] (1607–1665) never published his writings; in particular, his work on number theory is contained almost entirely in letters to mathematicians and in private marginal notes.{{sfn|Weil|1984|pp=45–46}} In his notes and letters, he scarcely wrote any proofs—he had no models in the area.<ref>{{harvnb|Weil|1984|p=118}}. This was more so in number theory than in other areas (remark in {{harvnb|Mahoney|1994|p=284}}). Bachet's own proofs were "ludicrously clumsy" {{harv|Weil|1984|p=33}}.</ref>

Over his lifetime, Fermat made the following contributions to the field:
* One of Fermat's first interests was [[perfect number]]s (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 medieval times—whether in the West or the Arab-speaking world—due in part to the importance given to them by the Neopythagorean (and hence mystical) [[Nicomachus]] (c.&nbsp;100&nbsp;AD), who wrote a primitive but influential "[[Introduction to Arithmetic]]". See {{harvnb|van der Waerden|1961|loc=Ch. IV}}.</ref> these topics led him to work on integer [[divisor]]s, 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>{{harvnb|Mahoney|1994|pp=48, 53–54}}. 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, {{harvnb|Tannery|Henry|1891|loc=Vol. II, pp. 72, 74}}, cited in {{harvnb|Mahoney|1994|p=54}}.</ref>
* In 1638, Fermat claimed, without proof, that all whole numbers can be expressed as the sum of four squares or fewer.<ref>{{Cite encyclopedia |url=https://books.google.com/books?id=5tFFDwAAQBAJ |title=Numbers and Measurements |last1=Faulkner |first1=Nicholas |last2=Hosch |first2=William L. |date=2017 |encyclopedia=Encyclopaedia Britannica |isbn=978-1-5383-0042-8 |access-date=2019-08-06}}</ref>
* [[Fermat's little theorem]] (1640):<ref>{{harvnb|Tannery|Henry|1891|loc=Vol. II, p. 209}}, Letter XLVI from Fermat to Frenicle, 1640,
cited in {{harvnb|Weil|1984|p=56}}</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''&nbsp;−&nbsp;''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 [[Disquisitiones Arithmeticae]]. Fermat's little theorem is a consequence of the [[Lagrange's theorem (group theory)|fact]] that the [[Order (group theory)|order]] of an element of a group divides the [[Order (group theory)|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 [[#Indian school: Āryabhaṭa, Brahmagupta, Bhāskara|above]]) was familiar to Fermat thanks to its rediscovery by [[Bachet]] {{harv|Weil|1984|p=7}}. 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>{{harvnb|Tannery|Henry|1891|loc=Vol. II, p. 204}}, cited in {{harvnb|Weil|1984|p=63}}. All of the following citations from Fermat's ''Varia Opera'' are taken from {{harvnb|Weil|1984|loc=Chap. II}}. The standard Tannery & Henry work includes a revision of Fermat's posthumous ''Varia Opera Mathematica'' originally prepared by his son {{harv|Fermat|1679}}.</ref> and every prime congruent to 1 modulo 4 can be written in the form <math>a^2 + b^2</math>.{{sfn|Tannery|Henry|1891|loc=Vol. II, p. 213}} 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]].{{sfn|Tannery|Henry|1891|loc=Vol. II, p. 423}}
* 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.{{sfn|Weil|1984|p=92}} 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){{sfn |Tannery|Henry|1891|loc=Vol. I, pp. 340–341}} 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.{{sfn|Weil|1984|p=115}} The first known proof is due to Euler (1753; indeed by infinite descent).{{sfn|Weil|1984|pp=115–116}}
* 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.

==== Euler ====
[[File:Leonhard Euler.jpg|thumb|upright=0.8|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 {{harv|Weil|1984|pp=159, 161}}. (There were already some recognisable features of professional ''practice'', viz., seeking correspondents, visiting foreign colleagues, building private libraries {{harv|Weil|1984|pp=160–161}}. Matters started to shift in the late seventeenth century {{harv|Weil|1984|p=161}}; scientific academies were founded in England (the [[Royal Society]], 1662) and France (the [[Académie des sciences]], 1666) and [[Russian Academy of Sciences|Russia]] (1724). Euler was offered a position at this last one in 1726; he accepted, arriving in St. Petersburg in 1727 ({{harvnb|Weil|1984|p=163}} and
{{harvnb|Varadarajan|2006|p=7}}).
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 {{harv|Truesdell|1984|p=xv}}; cited in {{harvnb|Varadarajan|2006|p=9}}). Notice, however, that Goldbach published some works on mathematics and sometimes held academic positions.</ref> [[Christian Goldbach|Goldbach]], pointed him towards some of Fermat's work on the subject.{{sfn|Weil|1984|pp=2, 172}}{{sfn|Varadarajan|2006|p=9}} This has been called the "rebirth" of modern number theory,{{sfn|Weil|1984|pp=1–2}} after Fermat's relative lack of success in getting his contemporaries' attention for the subject.<ref>{{harvnb|Weil|1984|p=2}} and {{harvnb|Varadarajan|2006|p=37}}</ref> Euler's work on number theory includes the following:<ref>{{harvnb|Varadarajan|2006|p=39}} and {{harvnb|Weil|1984|pp=176–189}}</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 himself{{sfn|Weil|1984|pp=178–179}}); 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">{{harvnb|Weil|1984|p=174}}. Euler was generous in giving credit to others {{harv|Varadarajan|2006|p=14}}, not always correctly.</ref> He wrote on the link between [[simple continued fraction|continued fractions]] and Pell's equation.{{sfn|Weil|1984|p=183}}
* ''First steps towards analytic number theory.'' In his work of sums of four squares, [[Partition function (number theory)|partitions]], [[pentagonal numbers]], and the [[Distribution (number theory)|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>{{harvnb|Varadarajan|2006|pp=45–55}}; 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]].{{sfn|Varadarajan|2006|pp=44–47}}{{sfn|Weil|1984|pp=177–179}}{{sfn|Edwards|1983|pp=285–291}}
* ''Diophantine equations''. Euler worked on some Diophantine equations of genus 0 and 1.{{sfn|Varadarajan|2006|pp=55–56}}{{sfn|Weil|1984|pp=179–181}} 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.{{sfn|Weil|1984|p=181}} He did notice there was a connection between Diophantine problems and [[elliptic integral]]s,{{sfn|Weil|1984|p=181}} whose study he had himself initiated.
[[File:Andrew wiles1-3.jpg|thumb|upright=0.8|"Here was a problem, that I, a ten-year-old, could understand, and I knew from that moment that I would never let it go. I had to solve it."<ref name=pbs>{{cite web |url=https://www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html |title=Andrew Wiles on Solving Fermat |date=November 2000 |publisher=[[WGBH-TV|WGBH]] |access-date=16 March 2016 |archive-date=17 March 2016 |archive-url=https://web.archive.org/web/20160317012127/http://www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html |url-status=live}}</ref> —Sir [[Andrew Wiles]] about [[Wiles's proof of Fermat's Last Theorem|his proof]] of [[Fermat's Last Theorem]]]]

==== Lagrange, Legendre, and Gauss ====
[[File:Disqvisitiones-800.jpg|upright=0.68|thumb|Carl Friedrich Gauss's Disquisitiones Arithmeticae, first edition]]

[[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 form]]s in full generality (as opposed to <math>m X^2 + n Y^2</math>)—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>{{sfn|Weil|1984|pp=327–328}} and worked on quadratic forms along the lines later developed fully by Gauss.{{sfn|Weil|1984|pp=332–334}} 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]]).{{sfn|Weil|1984|pp=337–338}}

[[File:Carl Friedrich Gauss.jpg|thumb|left|Carl Friedrich Gauss]]

In his ''Disquisitiones Arithmeticae'' (1798), Carl Friedrich 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.{{sfn|Goldstein|Schappacher|2007|p=14}} The last section of the ''Disquisitiones'' established a link between [[roots of unity]] and number theory:
<blockquote>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 {{harvnb|Goldstein|Schappacher|2007|p=16}}</ref></blockquote>

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

=== Maturity and division into subfields ===
[[File:ErnstKummer.jpg|upright|thumb=0.8|[[Ernst Kummer]]]]
[[File:Peter Gustav Lejeune Dirichlet.jpg|upright=0.8|left|thumb|[[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 {{harvnb|Goldstein|Schappacher|2007}}. 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 {{harvnb|Weil|1984|p=25}}).</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 number theory|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 (algebra)|valuation]] theory; see below. A conventional starting point for analytic number theory is [[Dirichlet's theorem on arithmetic progressions]] (1837),{{sfn|Apostol|1976|p=7}}{{sfn|Davenport|Montgomery|2000|p=1}} whose proof introduced [[L-functions]] and involved some asymptotic analysis and a limiting process on a real variable.<ref>See the proof in {{harvnb|Davenport|Montgomery|2000|loc=section 1}}</ref> The first use of analytic ideas in number theory actually goes back to Euler (1730s),{{sfn|Iwaniec|Kowalski|2004|p=1}}{{sfn|Varadarajan|2006|loc=sections 2.5, 3.1 and 6.1}} 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 [[Riemann zeta function|zeta function]] is the canonical starting point;{{sfn|Granville|2008|pp=322–348}} [[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 form]]s).<ref>See the comment on the importance of modularity in {{harvnb|Iwaniec|Kowalski|2004|p=1}}</ref>

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 ==
=== Elementary number theory ===
The term ''[[elementary proof|elementary]]'' generally denotes a method that does not use [[complex analysis]]. For example, the [[prime number theorem]] was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by [[Paul Erdős|Erdős]] and [[Atle Selberg|Selberg]].{{sfn|Goldfeld|2003}} The term is somewhat ambiguous: for example, proofs based on complex [[Tauberian theorem]]s (for example, [[Wiener–Ikehara theorem|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.
[[File:Paul Erdos with Terence Tao.jpg|thumb|upright=1.22|Number theorists [[Paul Erdős]] and [[Terence Tao]] in 1985, when Erdős was 72 and Tao was 10]]
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.<ref>See, for example, the initial comment in {{harvnb|Iwaniec|Kowalski|2004|p=1}}.</ref>

=== Analytic number theory ===
{{Main|Analytic number theory}}

[[File:Complex zeta.jpg|thumb|[[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 (complex analysis)|argument]].]]
[[File:ModularGroup-FundamentalDomain.svg|thumb|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 analysis|real]] and [[Complex analysis|complex]] analysis;{{sfn|Apostol|1976|p=7}} or
* in terms of its concerns, as the study within number theory of estimates on size and density, as opposed to identities.<ref>{{harvnb|Granville|2008|loc=section 1}}: "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>

Some subjects generally considered to be part of analytic number theory, for example, [[sieve theory]],<ref group="note">Sieve theory figures as one of the main subareas of analytic number theory in many standard treatments; see, for instance, {{harvnb|Iwaniec|Kowalski|2004}} or {{harvnb|Montgomery|Vaughan|2007}}</ref> are better covered by the second rather than the first definition: some of sieve theory, for instance, uses little analysis,<ref group="note">This is the case for small sieves (in particular, some combinatorial sieves such as the [[Brun sieve]]) rather than for [[large sieve]]s; the study of the latter now includes ideas from [[harmonic analysis|harmonic]] and [[functional analysis]].</ref> yet it does belong to 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 conjecture]]s), the [[Waring problem]] and the [[Riemann hypothesis]]. Some of the most important tools of analytic number theory are the [[circle method]], [[sieve theory|sieve methods]] and [[L-functions]] (or, rather, the study of their properties). The theory of [[modular form]]s (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 {{harvnb|Iwaniec|Kowalski|2004|p=1}}: "However much stronger...".</ref>

One may ask analytic questions about [[algebraic number]]s, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define [[prime ideal]]s (generalizations of [[prime number]]s in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question [[Landau prime ideal theorem|can be answered]] by means of an examination of [[Dedekind zeta function]]s, which are generalizations of the [[Riemann zeta function]], a key analytic object at the roots of the subject.<ref>{{harvnb|Granville|2008|loc=section 3}}: "[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, {{harvnb|Montgomery|Vaughan|2007}}, p. 1.</ref>

=== Algebraic number theory ===
{{Main|Algebraic number theory}}

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> (say) is an algebraic number. Fields of algebraic numbers are also called ''[[algebraic number field]]s'', or shortly ''[[number field]]s''. Algebraic number theory studies algebraic number fields.{{sfn|Milne|2017|p=2}} Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study.

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 [[ideal (ring theory)|ideals]] and
[[Norm (mathematics)|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 factorisation 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 [[Ernst Kummer|Kummer]]) seems to have come from the study of higher reciprocity laws,{{sfn|Edwards|2000|p=79}} that is, 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]]<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''&nbsp;+&nbsp;''iy'' of ''C'' to itself) and complex conjugation
(the map taking each element ''x''&nbsp;+&nbsp;''iy'' to ''x''&nbsp;−&nbsp;''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'')&nbsp;=&nbsp;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'')&nbsp;=&nbsp;0 has a Galois group that is [[solvable group|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 [[Leopold Kronecker|Kronecker]] and [[Gotthold Eisenstein|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 ===
{{Main|Diophantine geometry}}

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 [[algebraic curve|curve]], a [[algebraic surface|surface]], or some other such object in {{math|''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 whether there are finitely
or infinitely many rational points on a given curve or surface.

An example here may be helpful. Consider the [[Pythagorean equation]] <math>x^2+y^2 = 1,</math>
one would like to know its rational solutions; that is, its solutions
<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|thumb|Two examples of [[elliptic curve]]s, that is, curves of genus 1 having at least one rational point. Such curves have always infinitely many rational points.]]

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 (mathematics)|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 [[Glossary of arithmetic and diophantine geometry#H|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|{{pi}}]] and [[e (mathematical constant)|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]]'', however, 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 emphasize the connections to modern algebraic geometry (for example, in [[Faltings's theorem]]) rather than to techniques in Diophantine approximations.

== Other subfields ==
The areas below date from no earlier than the mid-twentieth century, even if they are based on older material. For example, as explained below, algorithms in number theory have a long history, arguably predating the formal concept of proof. However, the modern study of [[computability]] began only in the 1930s and 1940s, while [[computational complexity theory]] emerged in the 1970s.

=== Probabilistic number theory ===
{{Main|Probabilistic number theory}}

Probabilistic number theory starts with questions such as the following ones: Take an integer {{mvar|n}} 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 {{mvar|n}} have on average? What is the probability that it will 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 [[statistical independence|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 <math>0</math> 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, a non-rigorous, probabilistic approach leads to a number of [[heuristic]] algorithms and open problems, notably [[Cramér's conjecture]].

=== Arithmetic combinatorics ===
{{Main|Arithmetic combinatorics|Additive number theory}}

Arithmetic combinatorics starts with questions like the following ones: 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>, say? Should it be possible to write large integers as sums of elements of <math>A</math>?

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 <math>A</math> 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 <math>A</math> being studied need not be sets of integers, but rather subsets of non-commutative [[Group (mathematics)|groups]], for which the multiplication symbol, not the addition symbol, is traditionally used; they can also be subsets of [[ring (mathematics)|rings]], in which case the growth of <math>A+A</math> and <math>A</math>·<math>A</math> may be compared.

=== Computational number theory ===
{{Main|Computational number theory}}[[File:Computer History Museum (4145886786).jpg|thumb|A [[Lehmer sieve]], a primitive [[digital computer]] used to find [[primes]] and solve simple [[Diophantine equations]]]]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 early case is that of what is now called 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 <math>a x + b y = c</math>, 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 [[Aryabhata#Indeterminate equations|Āryabhaṭa]] (fifth to sixth centuries) as an algorithm called ''kuṭṭaka'' ("pulveriser"), without a proof of correctness.

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 [[primality test|testing primality]] are now known, 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 [[cryptography|encrypting messages]] (for example, [[RSA (algorithm)|RSA]]) depend on functions that are known to all, but whose inverses are known only to a chosen few, and 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.

Some things may not be computable at all; in fact, this can be proven in some instances. For instance, in 1970, it was proven, as a solution to [[Hilbert's tenth problem]], that there is no [[Turing machine]] which can solve all Diophantine equations.<ref>{{cite book |editor=Felix E. Browder |editor-link=Felix Browder |title=Mathematical Developments Arising from Hilbert Problems |series=[[Proceedings of Symposia in Pure Mathematics]] |volume=XXVIII.2 |year=1976 |publisher=[[American Mathematical Society]] |isbn=978-0-8218-1428-4 |pages=323–378 |first1=Martin |last1=Davis |author-link1=Martin Davis (mathematician) |first2=Yuri |last2=Matiyasevich |author-link2=Yuri Matiyasevich |first3=Julia |last3=Robinson |author-link3=Julia Robinson |chapter=Hilbert's Tenth Problem: Diophantine Equations: Positive Aspects of a Negative Solution |zbl=0346.02026}} Reprinted in ''The Collected Works of Julia Robinson'', [[Solomon Feferman]], editor, pp. 269–378, American Mathematical Society 1996.</ref> In particular, this means that, given a [[computably enumerable]] set of axioms, there are Diophantine equations for which there is no proof, starting from the axioms, of whether the set of equations has or does not have integer solutions. (i.e., Diophantine equations for which there are no integer solutions, since, given a Diophantine equation with at least one solution, the solution itself provides a proof of the fact that a solution exists. It cannot be proven that a particular Diophantine equation is of this kind, since this would imply that it has no solutions.)

== Applications ==
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, {{isbn|978-0-8218-5501-0}}</ref> 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 scientist]]s. 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, {{isbn|978-3-540-10382-0}}</ref>

Number theory has now several modern applications spanning diverse areas such as:
* [[Cryptography]]: Public-key encryption schemes such as RSA are based on the difficulty of factoring large composite numbers into their prime factors.<ref>{{Cite book |url=https://www.taylorfrancis.com/books/9781351664110 |title=An Introduction to Number Theory with Cryptography |edition=2nd |date=2018 |publisher=Chapman and Hall/CRC |isbn=978-1-351-66411-0 |doi=10.1201/9781351664110 |access-date=2023-02-22 |archive-date=2023-03-01 |archive-url=https://web.archive.org/web/20230301144259/https://www.taylorfrancis.com/books/mono/10.1201/9781351664110/introduction-number-theory-cryptography-james-kraft-lawrence-washington |url-status=live}}</ref>
* [[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>{{cite book | last=Krishna | first=Hari | title=Digital Signal Processing Algorithms | publisher=Routledge | date=2017 | location=London | isbn=978-1-351-45497-1}}</ref>
* [[Physics]]: The [[Riemann hypothesis]] has connections to the distribution of prime numbers and has been studied for its potential implications in physics.<ref>{{cite journal |title=Physics of the Riemann Hypothesis |journal=Reviews of Modern Physics |volume=83 |issue=2 |pages=307–330 |first1=Daniel |last1=Schumayer |first2=David A. W. |last2=Hutchinson |year=2011 |arxiv=1101.3116 |doi=10.1103/RevModPhys.83.307 |bibcode=2011RvMP...83..307S |s2cid=119290777}}</ref>
* [[Error correction code]]s: The theory of finite fields and algebraic geometry have been used to construct efficient error-correcting codes.<ref>{{Cite book |last=Baylis |first=John |url=https://www.taylorfrancis.com/books/9781351449847 |title=Error-Correcting Codes: A Mathematical Introduction |date=2018 |publisher=Routledge |isbn=978-0-203-75667-6 |doi=10.1201/9780203756676 |access-date=2023-02-22}}</ref>
* Communications: The design of cellular telephone networks requires knowledge of the theory of [[modular form]]s, which is a part of analytic number theory.<ref>{{Citation |last=Livné |first=R. |title=Communication Networks and Hilbert Modular Forms |date=2001 |url=http://link.springer.com/10.1007/978-94-010-1011-5_13 |work=Applications of Algebraic Geometry to Coding Theory, Physics and Computation |pages=255–270 |editor-last=Ciliberto |editor-first=Ciro |place=Dordrecht |publisher=Springer |doi=10.1007/978-94-010-1011-5_13 |isbn=978-1-4020-0005-8 |access-date=2023-02-22 |editor2-last=Hirzebruch |editor2-first=Friedrich |editor3-last=Miranda |editor3-first=Rick |editor4-last=Teicher |editor4-first=Mina}}</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>{{Cite journal |last1=Cartwright |first1=Julyan H. E. |last2=Gonzalez |first2=Diego L. |last3=Piro |first3=Oreste |last4=Stanzial |first4=Domenico |date=2002-03-01 |title=Aesthetics, Dynamics, and Musical Scales: A Golden Connection |url=http://dx.doi.org/10.1076/jnmr.31.1.51.8099 |journal=Journal of New Music Research |volume=31 |issue=1 |pages=51–58 |doi=10.1076/jnmr.31.1.51.8099 |hdl=10261/18003 |s2cid=12232457 |issn=0929-8215 |hdl-access=free}}</ref> This has been studied using number theory and in particular the properties of the 12th root of 2.

== Prizes ==
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]]''.

== See also ==
{{portal|Mathematics}}
* [[Arithmetic dynamics]]
* [[Algebraic function field]]
* [[Arithmetic topology]] 
* [[Finite field]]
* [[p-adic number]]
* [[List of number theoretic algorithms]]

== Notes ==
{{reflist|group=note|30em}}

== References ==
{{reflist}}

=== Sources ===
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* {{cite book |last=Smith |first=D. E. |year=1958 |title=History of Mathematics, Vol I |location=New York |publisher=Dover}}
* {{cite book |ref={{harvid|Tannery|Henry|1891}} |last1=Tannery |first1=Paul |author1-link=Paul Tannery |editor1=Charles Henry |editor1-link=Charles Henry (librarian) |year=1891 |last2=Fermat |first2=Pierre de |author2-link=Pierre de Fermat |language=fr, la |title=Oeuvres de Fermat |series=(4 Vols.) |location=Paris |publisher=Imprimerie Gauthier-Villars et Fils |url=https://archive.org/details/oeuvresdefermat01ferm}} [https://archive.org/details/oeuvresdefermat01ferm Volume 1] [https://archive.org/details/oeuvresdefermat02ferm Volume 2] [https://archive.org/details/oeuvresdefermat03ferm Volume 3] [https://archive.org/details/oeuvresdefermat04ferm Volume 4 (1912)]
* {{cite book |ref={{harvid|Taylor|1818}} |translator-last=Taylor |translator-first=Thomas |translator-link=Thomas Taylor (neoplatonist) |year=1818 |author1=Iamblichus |author1-link=Iamblichus |title=Life of Pythagoras or, Pythagoric Life |location=London |publisher=J. M. Watkins}} For other editions, see [[Iamblichus#List of editions and translations]]
* {{cite book |last=Truesdell |first=C. A. |author-link=Clifford Truesdell |year=1984 |translator-last=Hewlett |translator-first=John |chapter=Leonard Euler, Supreme Geometer |title=Leonard Euler, Elements of Algebra |edition=reprint of 1840 5th |location=New York |publisher=[[Springer-Verlag]] |isbn=978-0-387-96014-2 |chapter-url=https://books.google.com/books?id=mkOhy6v7kIsC}} This Google books preview of ''Elements of algebra'' lacks Truesdell's intro, which is reprinted (slightly abridged) in the following book:
* {{cite book |last=Truesdell |first=C. A. |author-link=Clifford Truesdell |year=2007 |editor1-last=Dunham |editor1-first=William |chapter=Leonard Euler, Supreme Geometer |title=The Genius of Euler: reflections on his life and work |series=Volume 2 of MAA tercentenary Euler celebration |location=New York |publisher=[[Mathematical Association of America]] |isbn=978-0-88385-558-4 |chapter-url=https://books.google.com/books?id=M4-zUnrSxNoC |access-date=2016-02-28}}
* {{cite book |last=Varadarajan |first=V. S. |year=2006 |title=Euler Through Time: A New Look at Old Themes |publisher=[[American Mathematical Society]] |isbn=978-0-8218-3580-7 |url=https://books.google.com/books?id=CYyKTREGYd0C |access-date=2016-02-28}}
* {{cite journal |last=Vardi |first=Ilan |title=Archimedes' Cattle Problem |date=April 1998 |journal=American Mathematical Monthly |volume=105 |issue=4 |pages=305–319 |doi=10.2307/2589706 |url=https://www.cs.drexel.edu/~crorres/Archimedes/Cattle/cattle_vardi.pdf |jstor=2589706 |citeseerx=10.1.1.383.545 |access-date=2012-04-08 |archive-date=2012-07-15 |archive-url=https://web.archive.org/web/20120715031904/https://www.cs.drexel.edu/~crorres/Archimedes/Cattle/cattle_vardi.pdf |url-status=live}}
* {{cite book |last1=van der Waerden |first1=Bartel L. |translator-last=Dresden |translator-first=Arnold |year=1961 |author-link=Bartel Leendert van der Waerden |title=Science Awakening |volume=1 or 2 |location=New York |publisher=[[Oxford University Press]]}}
<!-- Alternative Google books limited preview
url=https://books.google.com/books?id=S_T6Pt2qZ5YC&
Translated by Peter Huber
Publisher Springer, 1974
{{isbn|90-01-93103-0}}, {{isbn|978-90-01-93103-2}} -->
* {{cite book |last=Weil |first=André |year=1984 |author-link=André Weil |title=Number Theory: an Approach Through History – from Hammurapi to Legendre |location=Boston |publisher=Birkhäuser |isbn=978-0-8176-3141-3 |url=https://books.google.com/books?id=XSV0hDFj3loC |access-date=2016-02-28}}
{{refend}}
* {{Citizendium}}

== Further reading ==
Two of the most popular introductions to the subject are:
* {{Cite book |first1=G. H. |last1=Hardy |author1-link=G. H. Hardy |first2=E. M. |last2=Wright |title=An introduction to the theory of numbers |year=2008 |orig-year=1938 |url=https://books.google.com/books?id=rey9wfSaJ9EC |publisher=[[Oxford University Press]] |edition=rev. by D. R. Heath-Brown and J. H. Silverman, 6th |isbn=978-0-19-921986-5 |ref=none}}
* {{cite book |last=Vinogradov |first=I. M. |author1-link=Ivan Matveyevich Vinogradov |title=Elements of Number Theory |location=Mineola, NY |publisher=Dover Publications |year=2003 |orig-year=1954 |edition=reprint of the 1954}}

Hardy and Wright's book is a comprehensive classic, though its clarity sometimes suffers due to the authors' insistence on elementary methods ([[#CITEREFApostol1981|Apostol 1981]]).
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. Other popular first introductions are:
* {{Cite book |author1=Ivan M. Niven |author1-link=Ivan M. Niven |author2=Herbert S. Zuckerman |author3=Hugh L. Montgomery |author3-link=Hugh L. Montgomery |title=An introduction to the theory of numbers |year=2008 |orig-year=1960 |url=https://books.google.com/books?id=V52HIcKguJ4C |publisher=[[John Wiley & Sons]] |edition=reprint of the 5th 1991 |isbn=978-81-265-1811-1 |access-date=2016-02-28}}
* {{Cite book |first=Kenneth H. |last=Rosen |title=Elementary Number Theory |year=2010 |url=https://books.google.com/books?id=JqycRAAACAAJ |publisher=[[Pearson Education]] |edition=6th |isbn=978-0-321-71775-7 |access-date=2016-02-28}}

Popular choices for a second textbook include:
* {{cite book |last1=Borevich |first1=A. I. |last2=Shafarevich |first2=Igor R. |author-link1=Borevich |author-link2=Igor Shafarevich |title=Number theory |volume=20 |year=1966 |url=https://books.google.com/books?id=njgVUjjO-EAC |publisher=[[Academic Press]] |location=Boston, MA |series=Pure and Applied Mathematics |isbn=978-0-12-117850-5 |mr=0195803}}
* {{cite book |last=Serre |first=Jean-Pierre |year=1996 |orig-year=1973 |author-link=Jean-Pierre Serre |title=A course in arithmetic |series=[[Graduate Texts in Mathematics]] |volume=7 |publisher=Springer |isbn=978-0-387-90040-7 |url=https://archive.org/details/courseinarithmet00serr |ref=none}}

== External links ==
{{Commons}}
{{Wikiquote}}
* [https://encyclopediaofmath.org/wiki/Number_theory Number Theory] entry in the [[Encyclopedia of Mathematics]]
* [http://www.numbertheory.org/ Number Theory Web]

{{Number theory}}
{{Number theory tables}}
{{Areas of mathematics}}
{{Computer science}}
{{Authority control}}

[[Category:Number theory| ]]

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