Fermat's Last Theorem and Euler

Date: 11/29/2004 at 21:45:39
From: Darren
Subject: x^3+y^3=z^3

I was reading through different sites about Fermat's Last Theorem, and 
I remember reading that Euler showed the case x^3 + y^3 = z^3 had no 
integer solutions easily--the key word being easily.  I haven't been 
able to find any site that explains how to show this.

Date: 11/30/2004 at 02:33:12
From: Doctor Nitrogen
Subject: Re: x^3+y^3=z^3

Hi, Darren.

You will find a proof of an important related proposition offline in 
chapter 17, Section 8, pages 284, 285, of:

  [1] "A Classical Introduction to Modern Number Theory," Second 
  Edition, Kenneth Ireland, Nathan Rosen, Springer-Verlag, 
  ISBN# 0-387-97329-X.

The proposition proved in this book is:

  Proposition: The equation x^3 + y^3 = u*z^3, where u is a fixed unit 
  in Z[omega], has no integral solution (x, y, z), xyz =/= 0, where x, 
  y, z are elements of Z[omega].

The immediate implication is that, and I am quoting from page 284 
here:

  "...a nonzero cube in Z [i.e., Z is the ring of integers] is not the 
  sum of two nonzero cubes in Z."

In Chapter 1, Section 4, pages 12-13, Z[omega] is defined as being 
this commutative ring below:

  Z[omega] := {a + b*omega | a, b are in Z},

and where
                      
  omega = -1 + (sqrt(-3))/2,

is one of the 3rd roots of unity. 

In Chapter 1, Section 4 of [1], Z[omega] is shown to be an integral 
domain as well as a Euclidean domain.

The proposition is important in proving FLT when the exponent n = 3.

Incidentally, that

  x^p + y^p + z^p = 0, where p is a prime > 2,

has no integer solutions with xyz =/= 0, is related to work by 
Taniyama, Shimura, Wiles, Ribet and Frey on elliptic curves, which are
also sometimes called algebraic varieties, under suitable conditions.

Hope this helped answer the questions you had concerning your 
mathematics problem.  Feel free to write back if you have further
questions.

- Doctor Nitrogen, The Math Forum
  http://mathforum.org/dr.math/