Pythagorean triple

Definition

A Pythagorean triple is a triple $(a,b,c)$ of positive integers such that $a^{2}+b^{2}=c^{2}$.

Typically, the term Pythagorean triple is used to refer to the triple up to the interchange symmetry of $a$ and $b$. In other words, the triples $(a,b,c)$ and $(b,a,c)$ are considered equivalent.

We are typically interested in the study of primitive Pythagorean triples, and often, the term Pythagorean triple is used for primitive Pythagorean triple. A primitive Pythagorean triple is a Pythagorean triple where the three elements are pairwise relatively prime, or equivalent, their overall gcd is $1$.

Classification

The classification of Pythagorean triples (up to the interchange symmetry of the first two elements) follows immediately from the classification of primitive Pythagorean triples. The set of Pythagorean triples is in bijection with the set of triples $(d,u,v)$ of positive integers with $u<v$, $u,v$ coprime, and one of $u$ and $v$ odd, with the bijection given by:

$\!(d,u,v)\mapsto (d(v^{2}-u^{2}),2duv,d(u^{2}+v^{2}))$