Before Christmas I was teaching a class about surds. They were able to simplify, add, multiply etc.
To give them one application of this, I wanted to give them some triangles that they would have to identify as right-angled or not.
I wanted to avoid obvious multiples of usual integer Pythagorean triples, so I used a brute force search of surds of the form $p+q\sqrt2$ to find a few like this:
$T_1: a=1+\sqrt2$, $b=1+\sqrt2$, $c=2+\sqrt2$
$T_2: a=2+\sqrt2$, $b=2+\sqrt2$, $c=2+2\sqrt2$
$T_3: a=1+2\sqrt2$, $b=1+2\sqrt2$, $c=4+\sqrt2$
$T_4: a=2+2\sqrt2$, $b=2+2\sqrt2$, $c=4+2\sqrt2$
$T_5: a=4+\sqrt2$, $b=4+\sqrt2$, $c=2+4\sqrt2$
$T_6: a=1+2\sqrt2$, $b=4+2\sqrt2$, $c=5+2\sqrt2$
$T_7: a=4+\sqrt2$, $b=4+4\sqrt2$, $c=4+5\sqrt2$
$T_8: a=2+\sqrt2$, $b=4+4\sqrt2$, $c=6+3\sqrt2$
It turns out that $T_1, T_2, T_3, T_4, T_5$ are all multiples of each other - some more obvious than others.
$T_6$ and $T_7$ are also multiples of each other.
This raises a number of questions for me:
1) Has work on this kind of thing already been done? Is there a good reference work on this topic?
2) What do we call these kinds of numbers? They aren't integers, but they aren't "Gaussian integers" because those take the form $x+yi$, whereas I want the form to be $p+q\sqrt r$
3) Is there a standard way to identify the primitive triples?
4) Instead of a brute force approach I know I can use the usual Euclid's formula to generate them, too, but then what are a "good" set of values to start with - and will I achieve all primitives using that method?
5) I have also read of a set of transformations that will generate all primitive triples starting from $(3,4,5)$ - can I use that set of transformations starting from ...?