Harristotelian Logic

This is a challenge only for people who just like to do algebraic math and like an interesting problem. I previously posted a request for help and gave some information. I gave all of the information to retrace my steps. One email has shown that that might not be as easy as I had assumed. So, I'm not saying this leads to anything, how can it since we all know how hard FLT is ( it takes 130 pages to prove ) but this is for fun.

Given: (x+delta x)^n + (y+delta y)^n = (z+delta z)^n show that if delta z = delta y then delta x = (x+y-z)(x-y)/(z-x) INDEPENDENT OF N.

Here's the clue:

For n=2, 3^2 + 4^2 = 5^2
8^2 + 6^2 = 10^2

Now here's the graduate level problem.

Show that with delta z = delta x - delta y that the following MUST be divisible by z+y-x (or z+x-y, why?):

(x+y-z)^n + (x+y+z)^n - 2^n z^n

Of course, if you can show that it can't be, you've proven FLT. What I've written above has been verified by an editor for PAMS. Can you retrace my steps?

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Ok. I noted elsewhere that I didn't include enough information in this
challenge.

1. x^n + y^n = z^n


2. x + delta x = my, y + delta y = mx, z + delta z = mz


3. They're not necessarily integers unless you want to talk about FLT.


4. There's an (x-y)^n multiplying the term I give. I don't mention it because it's trivial to prove that (x-y) isn't divisible by z+y-x OR z+x-y (now we're talking Integers).


5. It's actually relatively easy to prove that the term given isn't divisible by z+y-x unless n=2, but of course, I won't just air the method to the world.

If you wish, you can do the multiplications for say n=3 and then do the division. You end up with a mess. I assume there's some high-powered math that can give the answer. But, I don't know any of the high-powered stuff…you see…I'm just an amateur.

Last thing: I haven't gotten a reply from the math historians out there. I DID get one from an editor at PAMS. He says he hasn't seen anything like this before. I guess there MUST be something I've missed because obviously, PAMS wouldn't pass up on the paper otherwise, right?

Since this is a hobby for me, I'd appreciate anyone who can show me what I'm missing. However, please nothing more from weemba.