- Let x^n+y^n=z^n
Then (x+y)^n-z^n = x^n+y^n-z^n +  = 
where  is the 'innards' of (x+y)^n
However every term in  contains at least an xy component.
x+y=Z is greater than z, and  doesn't change regardless of how we assign x
So let x=z, and we have a contradiction:
(x+y)^n-z^n will not be divisible by z unless y=kz, k>0.