>>293
I will respond to the points you raised in this message only.

1) On your assumption about common prime factors

You wrote: “私は論文で以下のように書いているのだから、gcd(x,y,z)=1のみを仮定していることは明らかだろう。” and you quoted: “We suppose that these variables do not have a same prime factor.”

If your meaning is that no prime divides all three of (x,y,z), then yes, that corresponds to the standard primitive reduction (gcd(x,y,z)=1).

My question is not about whether (gcd(x,y,z)=1) is allowed. It is whether any later step relies (explicitly or implicitly) on a stronger coprimality claim (for example, a statement equivalent to (gcd(x,z)=1) or (gcd(y,z)=1)). If you believe it does not, the practical way forward is to identify the first place where “relatively prime” (or “no common prime factor”) is used later and check what is actually required at that step.

2) On writing something as an (n)-th power

You wrote: “where do you prove that (b) is an (n)-th power of an integer? これに関しても以下のように書いているだろう。これのどこに問題があるのか明確に述べよ。” and you quoted: “Since a and b are relatively prime and b is the nth power of some number,”

Stating “(b) is the (n)-th power of some number” is not, by itself, a proof that (b) is an (n)-th power. The key question is: is this being introduced as a new assumption, or is it derived from earlier statements?

- If it is an assumption, then the proof needs to explain why it is legitimate to restrict to that case.
- If it is derived, then please point to the exact earlier line(s) (equation/paragraph) that establish that (b) is an (n)-th power, and indicate which lemma is being used and why its hypotheses hold here.

You also wrote: “全く問題のないものにケチを付ける意図は何だ。”

The intent is simply to pin down the exact logical status of that step so it can be checked precisely, not to make personal claims.