Questions about Pythagorean triples, positive integer solutions to $a^2 + b^2 = c^2$.

Pythagoras' Theorem states that in any right angle triangle, the length of the hypotenuse squared is equal to the sum of the squares of the lengths of the two shorter sides. If we denote the side lengths of a right angle triangle by $a$, $b$, and $c$, where $c$ is the length of the hypotenuse, then we can restate Pythagoras' Theorem as $a^2 + b^2 = c^2$.

A *Pythagorean triple* is a triple of positive integers $(a, b, c)$ such that $a^2 + b^2 = c^2$. For any such triple, there is a right angle triangle with side lengths $a$, $b$, and $c$. A Pythagorean triple is called *primitive* if $\gcd(a,b,c)=1$. There are infinitely many primitive Pythagorean triples: $(3,4,5)$, $(5,12,13)$, $(8,15,17)$, and so on.