Editing Pythagorean triple (section)

{{Short description|Integer side lengths of a right triangle}}
[[File:Pythagorean theorem - Ani.gif|thumb|Animation demonstrating the smallest Pythagorean triple, {{math|1=3{{sup|2}} + 4{{sup|2}} = 5{{sup|2}}}}.]]
A '''Pythagorean triple''' consists of three [[positive integer]]s {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}, such that {{math|1=''a''{{sup|2}} + ''b''{{sup|2}} = ''c''{{sup|2}}}}. Such a triple is commonly written {{math|(''a'', ''b'', ''c'')}}, a well-known example is {{math|(3, 4, 5)}}. If {{math|(''a'', ''b'', ''c'')}} is a Pythagorean triple, then so is {{math|(''ka'', ''kb'', ''kc'')}} for any positive integer {{math|''k''}}. A triangle whose side lengths are a Pythagorean triple is a [[right triangle]] and called a '''Pythagorean triangle'''.

A '''primitive Pythagorean triple''' is one in which {{math|''a''}}, {{math|''b''}} and {{math|''c''}} are [[coprime]] (that is, they have no common divisor larger than 1).<ref>{{harvtxt|Long|1972|p=48}}</ref> For example, {{math|(3, 4, 5)}} is a primitive Pythagorean triple whereas {{math|(6, 8, 10)}} is not. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple by dividing {{math|(''a'', ''b'', ''c'')}} by their [[greatest common divisor]]. Conversely, every Pythagorean triple can be obtained by multiplying the elements of a primitive Pythagorean triple by a positive integer (the same for the three elements).

The name is derived from the [[Pythagorean theorem]], stating that every right triangle has side lengths satisfying the formula <math>a^2+b^2=c^2</math>; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the [[triangle]] with sides <math>a=b=1</math> and <math>c=\sqrt2</math> is a right triangle, but <math>(1,1,\sqrt2)</math> is not a Pythagorean triple because the [[square root of 2]] is not an integer. Moreover, <math>1</math> and <math>\sqrt2</math> do not have an integer common multiple because <math>\sqrt2</math> is [[Irrational number#History|irrational]].

Pythagorean triples have been known since ancient times. The oldest known record comes from [[Plimpton 322]], a Babylonian clay tablet from about 1800 BC, written in a [[sexagesimal]] number system.<ref>{{citation|first=Eleanor|last=Robson|author-link=Eleanor Robson|title=Words and Pictures: New Light on Plimpton 322|journal=[[The American Mathematical Monthly]]|volume=109|year=2002|issue=2|pages=105–120|doi=10.1080/00029890.2002.11919845|s2cid=33907668|url=https://www.maa.org/sites/default/files/pdf/news/monthly105-120.pdf}}</ref>

When searching for integer solutions, the [[equation]] {{math|1=''a''{{sup|2}} + ''b''{{sup|2}} = ''c''{{sup|2}}}} is a [[Diophantine equation]]. Thus Pythagorean triples are among the oldest known solutions of a [[linear equation|nonlinear]] Diophantine equation.