Date: Nov 29, 2012 12:15 PM
Author: Pubkeybreaker
Subject: Re: From Fermat little theorem to Fermat Last Theorem

On Wednesday, November 28, 2012 3:07:56 PM UTC-5, quasi wrote:
> John Jens wrote: >Corrections was made. > >It's sufficient that only a < p. But you never _proved_ the inequality a < p, so you don't get to use it. Moreover, the equation a^p + b^p = c^p with the restrictions a,b,c positive integers p prime does not imply min(a,b) < p. To see this, just use p = 2 with a,b,c = 3,4,5. You tried to argue that you can't have p=2 since the inequality min(a,b) < p would then force min(a,b) = 1, leading to an easy contradiction. But you can't use the inequality min(a,b) < p without proving it, and the example p = 2 with a,b,c = 3,4,5 makes it clear that you can't prove it. quasi

A global remark:

It was demonstrated a number of years ago that FLT can not be proved
by modular considerations such as the one presented by the O.P.

The reason is as follows: The Hasse-Minkowski theorem that allows diophantine
analysis over a local field (such as Z/pZ) to be lifted to a global field
(such as Q) is blocked for FLT by the fact that SHA (The Tate-Sharfarevic
group) is non-trivial except in the case n = 4 (which is also why descent
argument works for n = 4). It is similar to the Brauer group obstruction
that prevents Abelian Varieties over local fields from being lifted to Q.

(I realize that the level of the discussion has been raised to the research
level of algebraic geometry; which, of course, is what Wiles used to
prove the thoerem)