If you were to plot all prime Pythagorean triples (so for example, once you do $3, 4, 5$, you can't do $6, 8, 10$ )on a 3 dimensional graph (so $x$, $y$ and $z$ axis'), what sort of patterns would occur, if any, would this be useful for anything, and would there be an equation to test if a triple can be factored (that's a bit of a side note).
Thanks
Here is a different way of looking at things than the one you suggest - but it may be of some interest.
If you have a Pythagorean Triple $$x^2+y^2=z^2$$you can divide through by $z^2$ to get a rational equation $$\left(\frac xz\right)^2+\left(\frac yz\right)^2=1$$and this represents a point on the unit circle. We find that the primitive Pythagorean Triples represent rational points on the unit circle and correspond to values of $\theta$ for which $\sin \theta$ and $\cos \theta$ are both rational.
Now if we write the equation $x^2+y^2=1$ and take $t=\tan \frac {\theta}2$ the points become $$x=\frac {1-t^2}{1+t^2}, y=\frac {2t}{1+t^2}$$ and for any rational value of $t$, we find a rational point on the circle. We can clear denominators and get a primitive pythagorean triple. Likewise every rational point on the unit circle is obtained in this way (the point $(-1,0)$ is a point as $t\to \pm\infty$, and you will find that if you join $(-1,0)$ to a rational point on the circle that the angle with the $x$-axis is $\frac {\theta}2$).