Editing Number theory (section)

==== 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}}