# Calculating Pythagorean triples

If $$x$$ and $$y$$ are even, then of course $$z$$ is too, and $$\left(\frac{x}{2}, \frac{y}{2}, \frac{z}{2} \right)$$ is also a Pythagorean triple. For this question we assume that $$(x, y, z)$$ is a Pythagorean triple in which $$x$$ is odd, so that $$y$$ is even and $$z$$ is odd. For similar reasons, we assume that $$p$$ and $$q$$ are coprime. The theory of Pythagorean triples then tells us that there are nonzero integers $$p$$ and $$q$$ such that

$$x + iy = (p + iq)^2 \hspace{1.5cm} z = |p + iq|^2 = p^2 + q^2.$$

If $$x$$ is odd then one of $$p$$ and $$q$$ must be even and the other is odd. Otherwise all possibilities occur. Write down the Pythagorean triples for such $$p$$ and $$q$$, where $$1 \leq q < p \leq 8.$$

I'm not really sure how to calculate these triples. I first decided to start by picking say $$q = 1, p = 2$$, which then I then get $$z = 1^2 + 2^2 = 5$$. How would I then use this to calculate $$x$$ and $$y$$?

$(p+iq)^2=(2+i)^2=2^2+2\cdot 2i+i^2=4+4i-1=3+4i$, so $x=3$, $y=4$.
• OHHHHH!!!! Ok! And then I try this for all $p,q$ in that interval? Does this always work? Apr 13, 2013 at 19:41