Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
Currently learning about primitive Pythagorean triples and I'm having trouble approaching the following proof.
Given that $x, y, z$, is a primitive Pythagorean triple with $y$ even, I need to show:
$$x + y \equiv x - y \equiv 1,7 \pmod 8$$
So far all I've noticed is that $x+y$ and $x-y$ must be odd but I'm not sure how to determine the equivalence to each other $\bmod 8$. Could anyone give me a small push in the right direction?
A start: Note that $x$ and $z$ are odd, so $x^2\equiv 1\pmod{8}$ and $z^2\equiv 1\pmod 8$. It follows that $z^2-x^2\equiv 0\pmod{8}$. Thus $y^2$ is divisible by $8$, and therefore $y$ is divisible by $4$.
Now we can show that $x+y\equiv x-y\pmod{8}$. For $(x+y)-(x-y)=2y$, and since $y$ is divisible by $4$, it follows that $2y$ is divisible by $8$.
