Smaller Elements of Pythagorean Triple not both Odd

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Let $\left({x, y, z}\right)$ be a Pythagorean triple, i.e. integers such that $x^2 + y^2 = z^2$.

Then $x$ and $y$ cannot both be odd.


Aiming for a contradiction, suppose $x$ and $y$ are both odd such that:

$\exists z \in \Z: x^2 + y^2 = z^2$


$x^2 + y^2 \equiv 1 + 1 \equiv 2 \pmod 4$

But from Square Modulo 4:

$z^2 \equiv 0 \pmod 4$ or $z^2 \equiv 1 \pmod 4$

Thus $x^2 + y^2$ can not be square.

It follows by Proof by Contradiction that $x$ and $y$ cannot both be odd.