I remember reading that one of the faults of false methods of proving FLT is that they disprove it for n=2 as well. What I've found doesn't do that but it does establish interesting limitations on x,y and z for n=2.
When I first started with this I amused myself by noting that a solution to x^2 + y^2 = z^2 could be found for any odd x by making y=(x^2 - 1)/2
like 3,4,5; or 5,12,13; or 7,24,25;
I also found the case 8,15,17 from a math book. So, I've wondered if there were any possibilities where z-x or z-y had prime factors other than one or 2.
I'm saying now that they can't for the same reasons that I give which prove FLT.
No natural number counter-examples exist for the above.