Editing Ancient Greek mathematics (section)

=== Arithmetic ===
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]]. 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". Ancient Greek mathematicians conventionally separated ''numbers'' (mostly positive integers but occasionally rationals) from ''magnitudes'' or ''lengths'', with only the former being the subject of arithmetic.

The Pythagorean tradition spoke of so-called [[Polygonal number|polygonal]] or [[figurate numbers]]. The study of the sums of triangular and pentagonal numbers would prove fruitful in the [[early modern period]]. Building on the works of the earlier Pythagoreans, [[Nicomachus of Gerasa]] wrote an ''Introduction to Arithmetic'' which would go on to receive later commentary in late antiquity and the Middle Ages. The continuing influence of mathematics in Platonism is shown in [[Theon of Smyrna|Theon of Smyrna's]] ''Mathematics Useful For Understanding Plato'', written around the same time. [[Diophantus]] also wrote on [[polygonal number]]s in addition to a work in pre-modern algebra (''[[Arithmetica]]'').<ref>{{Cite journal |last=Acerbi |first=F. |date=2011 |title=Completing Diophantus, De polygonis numeris, prop. 5 |journal=Historia Mathematica |volume=38 |issue=4 |pages=548–560 |doi=10.1016/j.hm.2011.05.002 |doi-access=free}}</ref><ref>{{Cite journal |last1=Christianidis |first1=J. |last2=Oaks |first2=J. |date=2013 |title=Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria |journal=Historia Mathematica |volume=40 |issue=2 |pages=127–163 |doi=10.1016/j.hm.2012.09.001 |doi-access=free}}</ref>

An [[epigram]] published by [[Gotthold Ephraim Lessing|Lessing]] in 1773 appears to be a letter sent by [[Archimedes]] to [[Eratosthenes]]. 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.