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Problem

A Pythagorean triplet, $(a, b, c)$, is defined as a set of positive integers for which $a^2 + b^2 = c^2$. Furthermore, if the set is primitive it means that $a$, $b$, and $c$ share no common factor greater than 1.

Prove that the following identities will generate all primitive Pythagorean triplets and determine the conditions for $m$ and $n$.

$\begin{align}a &= m^2 - n^2\\b &= 2mn\\c &= m^2 + n^2\end{align}$