>>298
correction

You wrote: “gcd(x,y,x)=1であり、x^n+y^n^=z^nを満たすのであれば、GCD(x,y)=1になるのは明らか。”

To be precise: from (gcd(x,y,z)=1) alone, it does not follow that (gcd(x,y)=1) (e.g., ((6,10,15))). Your additional condition “(x^n+y^n=z^n)” changes the setting, so the above example is not a counterexample to that *stronger* claim. The point remains that primitive reduction ((gcd(x,y,z)=1)) is weaker than pairwise coprimality, so any step that uses (gcd(x,y)=1) (or (gcd(x,z)=1), etc.) still needs explicit justification from your stated assumptions.