Editing Number theory (section)

== History ==
Number theory is the branch of mathematics that studies [[Integer|integers]] and their [[Property (mathematics)|properties]] and relations.<ref>{{Cite web |last=Karatsuba |first=A.A. |date=2020 |title=Number theory |url=https://encyclopediaofmath.org/wiki/Number_theory |access-date=2025-05-03 |website=Encyclopedia of Mathematics |publisher=Springer}}</ref> The integers comprise a [[Set (mathematics)|set]] that extends the set of [[Natural number|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 number|prime numbers]] as well as the properties of [[Mathematical object|mathematical objects]] constructed from integers (for example, [[Rational number|rational numbers]]), or defined as generalizations of the integers (for example, [[Algebraic integer|algebraic integers]]).<ref name=":5">{{Cite book |last=Moore |first=Patrick |title=The Gale Encyclopedia of Science |publisher=Gale |year=2004 |isbn=0-7876-7559-8 |editor-last=Lerner |editor-first=K. Lee |edition=3rd |volume=4 |pages= |language=en |chapter=Number theory |editor-last2=Lerner |editor-first2=Brenda Wilmoth}}</ref><ref name=":1" />

Number theory is closely related to arithmetic and some authors use the terms as synonyms.<ref>{{multiref|{{harvnb|Lozano-Robledo|2019|p=[https://books.google.com/books?id=ESiODwAAQBAJ&pg=PR13 xiii]}}|{{harvnb|Nagel|Newman|2008|p=[https://books.google.com/books?id=WgwUCgAAQBAJ&pg=PA4 4]}}}}</ref> However, the word "arithmetic" is used today to mean the study of numerical operations and extends to the [[Real number|real numbers]].<ref>{{multiref|{{harvnb|Romanowski|2008|pp=302–303}}|{{harvnb|HC staff|2022b}}|{{harvnb|MW staff|2023}}|{{harvnb|Bukhshtab|Pechaev|2020}}}}</ref> In a more specific sense, number theory is restricted to the study of integers and focuses on their properties and relationships.<ref>{{multiref|{{harvnb|Wilson|2020|pp=[https://books.google.com/books?id=fcDgDwAAQBAJ&pg=PA1 1–2]}}|{{harvnb|Karatsuba|2020}}|{{harvnb|Campbell|2012|p=[https://books.google.com/books?id=yoEFp-Q2OXIC&pg=PT33 33]}}|{{harvnb|Robbins|2006|p=[https://books.google.com/books?id=TtLMrKDsDuIC&pg=PR12-IA1 1]}}}}</ref> Traditionally, it is known as higher arithmetic.<ref>{{multiref|{{harvnb|Duverney|2010|p=[https://books.google.com/books?id=sr5S9oN1xPAC&pg=PR5 v]}}|{{harvnb|Robbins|2006|p=[https://books.google.com/books?id=TtLMrKDsDuIC&pg=PR12-IA1 1]}}}}</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, {{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> The term number means whole numbers, which refers to either the natural numbers or the integers.<ref name=":4">{{Cite book |last1=Effinger |first1=Gove |title=Elementary Number Theory |last2=Mullen |first2=Gary L. |publisher=CRC Press |year=2022 |isbn=978-1-003-19311-1 |location=Boca Raton |language=en |chapter=}}</ref><ref name=":6">{{Cite book |last=Weisstein |first=Eric W. |title=CRC Concise Encyclopedia of Mathematics |publisher=Chapman & Hall/CRC |year=2003 |isbn=1-58488-347-2 |edition=2nd |pages= |language=en |chapter=}}</ref><ref>{{Cite book |last=Weisstein |first=Eric W. |title=CRC Concisse Encyclopedia of Mathematics |publisher=Chapman & Hall/CRC |year=2003 |isbn=1-58488-347-2 |edition=2nd |pages=3202 |language=en |chapter=Whole Number}}</ref>

[[Elementary number theory]] studies aspects of integers that can be investigated using elementary methods such as [[Elementary proof|elementary proofs]].<ref name=":3">{{multiref|{{harvnb|Page|2003|pp=[https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032 18–19, 34]}}|{{harvnb|Bukhshtab|Nechaev|2014}}}}</ref> [[Analytic number theory]], by contrast, relies on [[complex numbers]] and techniques from analysis and calculus.<ref>{{multiref|{{harvnb|Page|2003|p=[https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032 34]}}|{{harvnb|Karatsuba|2014}}}}</ref> [[Algebraic number theory]] employs [[algebraic structures]] such as [[Field (mathematics)|fields]] and [[Ring (mathematics)|rings]] to analyze the properties of and relations between numbers. [[Geometric number theory]] uses concepts from geometry to study numbers.<ref>{{multiref|{{harvnb|Page|2003|pp=[https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032 34–35]}}|{{harvnb|Vinogradov|2019}}}}</ref> Further branches of number theory are [[probabilistic number theory]],<ref>{{harvnb|Kubilyus|2018}}</ref> [[combinatorial number theory]],<ref>{{harvnb|Pomerance|Sárközy|1995|p=[https://books.google.com/books?id=5ktBP5vUl5gC&pg=PA969 969]}}</ref> [[computational number theory]],<ref>{{harvnb|Pomerance|2010}}</ref> and applied number theory, which examines the application of number theory to science and technology.<ref>{{multiref|{{harvnb|Yan|2002|pp=12, 303–305}}|{{harvnb|Yan|2013a|p=[https://books.google.com/books?id=74oBi4ys0UUC&pg=PA15 15]}}}}</ref>

=== Origins ===
==== Ancient Mesopotamia ====
[[File:Plimpton 322.jpg|thumb|Plimpton 322 tablet]]
The earliest historical find of an arithmetical nature is a fragment of a table: [[Plimpton 322]] ([[Larsa]], Mesopotamia, c.&nbsp;1800&nbsp;BC), a broken clay tablet, 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 numerous and too large to have been obtained by [[brute force method|brute force]]. The heading over the first column reads: "The {{tlit|akk|takiltum}} of the diagonal which has been subtracted such that the width..."<ref>{{harvnb|Neugebauer|Sachs|1945|p=40}}. The term {{tlit|akk|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 that it was constructed by means of what amounts, in modern language, to the [[Identity (mathematics)|identity]]<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>

<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 language|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 {{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>

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.{{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> Plimpton 322 tablet is the only surviving evidence of what today would be called number theory within Babylonian mathematics, though a kind of [[Babylonian mathematics#Algebra|Babylonian algebra]] was much more developed.{{sfn|van der Waerden|1961|p=63-75}}

==== Ancient Greece ====
{{Further|Ancient Greek mathematics}}

Although other civilizations probably influenced Greek mathematics at the beginning,<ref>{{harvnb|van der Waerden|1961|p=87-90}}</ref> all evidence of such borrowings appear relatively late,<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}}</ref><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> and it is likely that Greek {{tlit|grc|arithmētikḗ}} (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 Greece|Archaic]] and [[Classical Greece|Classical]] periods) comes through either the reports of contemporary non-mathematicians or references from mathematical works in the early [[Hellenistic period]].{{sfn|Boyer|Merzbach|1991|p=82}} In the case of number theory, this means largely [[Plato]], [[Aristotle]], and [[Euclid]].

Plato had a keen interest in mathematics, and distinguished clearly between {{tlit|grc|arithmētikḗ}} and calculation ({{tlit|grc|logistikē}}). Plato reports in his dialogue ''[[Theaetetus (dialogue)|Theaetetus]]'' that [[Theodorus of Cyrene|Theodorus]] had proven that <math>\sqrt{3}, \sqrt{5}, \dots, \sqrt{17}</math> are irrational. [[Theaetetus of Athens|Theaetetus]], a disciple of Theodorus's, worked on distinguishing different kinds of [[Commensurability (mathematics)|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 [[Pythagoreanism|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>

Euclid devoted part of his ''[[Euclid's Elements|Elements]]'' (Books VII–IX) to topics that belong to elementary number theory, including [[Prime number|prime numbers]] and [[Divisibility rule|divisibility]].<ref>{{Cite book |last=Corry |first=Leo |title=A Brief History of Numbers |publisher=Oxford University Press |year=2015 |isbn=978-0-19-870259-7 |language=en |chapter=Construction Problems and Numerical Problems in the Greek Mathematical Tradition}}</ref> He gave an algorithm, the [[Euclidean algorithm]], for computing the [[greatest common divisor]] of two numbers (Prop. VII.2) and a [[Euclid's theorem|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">{{harvnb|Becker|1936|p=533}}, cited in: {{harvnb|van der Waerden|1961|p=108}}.</ref> This is all that is needed to prove that [[square root of 2|<math>\sqrt{2}</math>]] is [[Irrational number|irrational]].{{sfn|Becker|1936}} Pythagoreans apparently 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, 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, {{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>{{sfn|von Fritz|2004}} This forced a distinction between ''[[number]]s'' (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 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]] (17th to early 19th centuries).

An [[epigram]] published by [[Gotthold Ephraim Lessing|Lessing]] in 1773 appears 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 Indian mathematicians. It is not known whether Archimedes himself had a method of solution.

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

Aside from the elementary work of Neopythagoreans such as [[Nicomachus]] and [[Theon of Smyrna]], the foremost authority in {{tlit|grc|arithmētikḗ}} in Late Antiquity was [[Diophantus of Alexandria]], who probably lived in the 3rd century&nbsp;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 ''{{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.

==== Asia ====
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&nbsp;AD (Yan Dunjie) or 280–473&nbsp;AD (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.

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 autochthonous 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 generalization 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:Selenographia 1647 (122459248) (cropped).jpg|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 but communicated through correspondence instead. Accordingly, his work on number theory is contained almost entirely in letters to mathematicians and in private marginal notes.{{sfn|Weil|1984|pp=45–46}} 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>{{harvnb|Weil|1984|p=118}}. This was more so in number theory than in other areas ({{harvnb|Mahoney|1994|p=|pp=283-289}}). 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 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.&nbsp;100&nbsp;AD), who wrote a very elementary but influential book entitled ''[[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 ''{{lang|la|[[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.

==== Lagrange, Legendre, and Gauss ====
[[File:Carl Friedrich Gauss 1840 by Jensen.jpg|upright=0.8|thumb|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 form]]s 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>{{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}}

[[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.{{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 forays towards [[Évariste Galois]]'s work and the area [[algebraic number theory]].

=== Maturity and division into subfields ===
[[File:Peter Gustav Lejeune Dirichlet.jpg|upright=0.8|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 [[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]]''.