Great work!! And a big thank you for confirming some of my own theories . I pretty much used the same reasoning to prove several mathematical conjectures such as Fermat’s Last Theorem. I know. Wiles has already proved it but certainly not like what Fermat would have. Upon seeing Fermat’s equation I immediately said that’s just the Pythagorean theorem. Its almost exactly like you said except the “factor” is N^n-2. But since you have a,b, and c then N^n-2 is a different factor for each variable. But you can get around that by expressing all the variables in terms of c or the hypoteneuse, i.e., a=csine and b =ccos. Gives a very short sweet margin proof more in keeping with what Fermat actually came up with.
I am convinced that if mathematician would actually sit down and think about some of these problems intuitively rather than trying to impress other mathematicians with “rigor” then far more long standing mathematical problems would be solved.