>>297

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。”

This is false. From (gcd(x,y,z)=1) it does not follow that (gcd(x,y)=1). Counterexample: ((x,y,z)=(6,10,15)) gives (gcd(6,10,15)=1) but (gcd(6,10)=2). So “stronger vs weaker” gcd conditions are not “all the same.”

You also wrote: “z^n=abであり、aとbが互いに素であるからbがn乗数だと書いているだけ。”

From (z^n=ab) and (gcd(a,b)=1), the correct conclusion is that **both** (a) and (b) must be (n)-th powers, but that requires invoking a standard lemma (via prime factorization/unique factorization). Please state explicitly which lemma you are using, and cite where you prove (gcd(a,b)=1) in your text.