Editing Number theory (section)

==== 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.