So I was looking for an old video in numberphile (one about a particular kind of prime) and re-saw this
https://www.youtube.com/watch?v=yGsIw8LHXM8 (two square theorem)
or if you like the original video that one comes from
https://www.youtube.com/watch?v=SyJlRUBoVp0 The Prime Problem with a One Sentence Proof )
Anyway I found myself rewatching it. and decide to see if I can make a simpler version of the proof. not less space mind you, simpler as in more straight forward. I think I've done that here (it's probably not new, most things in math aren't) but regardless I submit it for your entertainment/utility.
Also it's worth saying that the you can then figure out exactly what form the X and Y (K or what have you) need to have by just unraveling all that. Here is that little bit of extra formula stuffs spelled out.