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## 2) Introducing auxiliary integers (e.g. writing something as an (n)-th power)

You object to wording like “perfect (n)-th power” or “gcd condition” as “not mathematical language.” That objection is not relevant. The logical issue is simple and standard:

- If at some step you write an integer (b) in the form (b=c^n), then you must have already established that **(b) is an (n)-th power of an integer**.
- Otherwise, you are not making a deduction; you are adding a new assumption that restricts the case.

So the concrete request is: **where do you prove that the quantity you rename as (c^n) is indeed an (n)-th power?** If the proof is “because it must be so,” that is circular. If it is proved from prior equalities plus coprimality plus a lemma (e.g., a standard unique factorization argument), then please state that lemma and show the hypotheses are satisfied.