Editing Number theory (section)

==== 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}}