Editing Number theory (section)

=== 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>
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