For questions about Pythagorean triples which are triples of positive integers $(a, b, c)$ satisfying $a^2 + b^2 = c^2$.

Pythagoras' Theorem states that if a right angle triangle in the plane has side lengths $a$, $b$, and $c$, with $c$ being the length of the hypotenuse, then $a^2 + b^2 = c^2$. A Pythagorean triple is a triple of positive integers $(a, b, c)$ which satisfy this equation. For example, $(3, 4, 5)$ is a Pythagorean triple since $3^2 + 4^2 = 9 + 16 = 25 = 5^2$. Note that any triple of positive real numbers $(a, b, c)$ satisfying $a^2 + b^2 = c^2$ arises as the side lengths of a right angle triangle in the plane, so Pythagorean triples are precisely the collection of side lengths of such triangles where all the lengths are integers.

Note that if $(a, b, c)$ is a Pythagorean triple, so is $(b, a, c)$. One usually chooses to write a Pythagorean triple such that $a \leq b \leq c$. Note, these inequalities must actually be strict: if $a = b$, then $2a^2 = c^2$ which would imply that $\sqrt{2}$ is rational, while if $b = c$, then $a$ must be zero (which is not positive).

If $(a, b, c)$ is a Pythagorean triple, then so is $(ka, kb, kc)$ for every positive integer $k$. A Pythagorean triple for which $a$, $b$, and $c$ are coprime (i.e. can't be obtained as $(ka', kb', kc')$ for some other Pythagorean triple $(a', b', c')$) is called a primitive Pythagorean triple. For example, the Pythagorean triple $(3, 4, 5)$ mentioned above is primitive.

Euclid showed that every primitive Pythagorean triple $(a, b, c)$ is either of the form $(m^2 - n^2, 2mn, m^2 + n^2)$ or $(2mn, m^2 - n^2, m^2 + n^2)$ where $m$ and $n$ are coprime positive integers, not both odd, with $m > n$. Note that under these conditions, $m^2 - n^2$ is odd and $2mn$ is even, so the two possible forms correspond to whether $a$ is odd or even. It follows that there are infinitely many primitive Pythagorean triples.

Looking for integer solutions of $a^2 + b^2 = c^2$ is an example of a non-linear Diophantine equation. Famously, the more general equation $a^n + b^n = c^n$ has no positive integer solutions when $n > 2$ - this is Fermat's Last Theorem.