Fermat's Last Theorem - short proof - please point out my mistake!

[richieroooo]

I and a few friends can't spot a mistake in the short version of FLT I've tried here. Please explain where I've gone wrong as this looks really simple, probably too simple?! You really only need to read the attached pages 1&5, but the other pages give clarifying examples.

Thanks for your time and help,

Rich

 

[JeffM]

How does \(\displaystyle x^3 + 3(a - b)x^2 + 3(a^2 - b^2)x + (a^3 - b^3) = 0,\ where\ 0 \le a < b,\ and\ a,\ b \in \mathbb Z,\ and\ x > 0 \implies x \notin \mathbb Z?\)

I do not see what you are getting at at all.

I agree that \(\displaystyle given\ y = x + a\ and\ z = x + b,\ then\ x^3 + (x + a)^3 = (x + b)^3 \iff x^3 + y^3 = z^3.\)

I agree that \(\displaystyle x^3 + (x + a)^3 = (x + b)^3 \implies x^3 + 3(a - b)x^2 + 3(a^2 - b^2)x + (a^3 - b^3) = 0.\)

I agree that all the coefficients of the lower order terms are negative. So what?

Note that x is guaranteed to have at least one real root. What shows that there is no pair of non-negative integers a and b that result in that real root belonging to the class of positive integers?