pythagorean Definition and 1 Threads

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer 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 a, b and c are coprime (that is, they have no common divisor larger than 1). For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple by dividing (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




a

2


+

b

2


=

c

2




{\displaystyle a^{2}+b^{2}=c^{2}}

; 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



a
=
b
=
1


{\displaystyle a=b=1}

and



c
=


2




{\displaystyle c={\sqrt {2}}}

is a right triangle, but



(
1
,
1
,


2


)


{\displaystyle (1,1,{\sqrt {2}})}

is not a Pythagorean triple because the square root of 2 is not an integer or ratio of integers. Moreover,



1


{\displaystyle 1}

and





2




{\displaystyle {\sqrt {2}}}

do not have an integer common multiple because





2




{\displaystyle {\sqrt {2}}}

is 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.
When searching for integer solutions, the equation a2 + b2 = c2 is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation.

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