Please look at this proof and tell me what's wrong

[anastas]

I was recently bored in math class and I decided to play around with Fermat's Last Theorem. Here's what I got (with i = sqr(-1) and -x^n means the opposite of the nth power of x, NOT the nth power of -x)br]
1. Assume there are solutions to x^n + y^n = z^nbr]x^n + y^n = z^n

2. Factor as the sum of two exponentsbr](x^.5n + i*y^.5n)(x^.5n - i*y^.5n) = z^n

3. Factor out ibr]i[(i * -x^.5n + y^.5n)(i * -x^.5n - y^.5n)] = z^n

4. Simplify inner termsbr]i(-x^n - y^n) = z^n

5. Divide both sides by ibr]-x^n - y^n = (z^n)/i

6. Rationalize the denomantor of the right-hand termbr]-x^n - y^n = i*-z^n

7. Multiply both sides by -1br]x^n + y^n = i*z^n

8. Statement 7 contradicts Statement 1 as z^n does not equal i*z^n (other that z=0 of course) so the original statements must be false. Therefore, there are no solutions to x^n + y^n = z^n.

Now I know that prooving this theorem can't be that simple and I have no reason to suppose I can do it since I'm only 15 and in Algebra 2 (although I have gone through calc and stat textbooks). I can't find any mistakes, anyone care to proove my proof wrong? I just really want to know where my mistake(s) is/are.

[infamous1]

I could be wrong, but here is what I saw.

In step 3, you factor i from both parts being multiplied, so you need to square i.

3. (i^2)[(i * -x^.5n + y^.5n)(i * -x^.5n - y^.5n)] = z^n
should be the correct step.

Tell me if this makes more sense.

[anastas]

I don't think so as I am not factoring i out from both expressions in parentheses separately, I am facting i out of the entire expression at once. This can be checked by distributing i to each of the bracketed terms, which results in the equation in step 2.

Thanks for the reply though.

[.•´¯`•þasђγ•._.•´]

ummmmm, yeah...what he said....

lol

[infamous1]

2. (x^.5n + i*y^.5n)(x^.5n - i*y^.5n) = z^n

3. i(i * -x^.5n + y^.5n) i(i * -x^.5n - y^.5n) = z^n

4. (i^2)[(i * -x^.5n + y^.5n)(i * -x^.5n - y^.5n)] = z^n

If (x^.5n + i*y^.5n) = i(i * -x^.5n + y^.5n)
and (x^.5n - i*y^.5n) = i(i * -x^.5n - y^.5n)
then you must square I. One of these two assumptions may be where I went wrong.
Tell me if they are not correct.

[anastas]

Well, yes, but there's one differencebr]
If

(x^.5n + i*y^.5n) = i(i * -x^.5n + y^.5n)

and

(x^.5n - i*y^.5n) = i(i * -x^.5n - y^.5n)

then

(x^.5n + i*y^.5n)(x^.5n - i*y^.5n) = i(i * -x^.5n + y^.5n) * i(i * -x^.5n - y^.5n)

and

i(i * -x^.5n + y^.5n) * i(i * -x^.5n - y^.5n) = i[(i * -x^.5n + y^.5n) * (i * -x^.5n - y^.5n)].

This can be checked by then simplifying the last expression by reversing the previous steps, resulting in the original equation.

I could be wrong, but I'm pretty sure it's correct.

I think I figured out what's wrong with it - I assumed the equation's domain including the maginar numbers but since no domain is explicity stated, it is assumed to be the real numbers. I am working outside its domain so that's why it's incorrect.

Thank you for the feedback, however.

[infamous1]

OK, just so you found where it was wrong