From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Counterexample to FLT! :-) (Was: Re: JSH: Afterword)
Date: 11 Dec 1998 14:58:59 GMT


On Thu, 10 Dec 1998 "James Harris" <jmail@JSHent.com> wrote:
>I've also come up with (I hope) a better way to handle the 
>problem of defining f.  Instead of the complex field I'm 
>certain that I can define it as a complex irrational....

chorn5751 <chorn5751@orcaweb.com> wrote:
>The problem with your approach is much more fundamental than 
>that.  Take the case of cubes.  The basic equation implies the
>existence of specific integers F and M such that 
>
>              F^3 = (z-y)                       (1)
>
>              M^3 = (z^2 + zy + y^2)            (2)

[valid objections to JSH deleted]

Just to underscore the difficulties here, let me observe a counterexample
to FLT for p=3: we have  x^3 + y^3 = z^3   if

        x = -333396 + 403727 sqrt(-5)
        y = -333396 - 403727 sqrt(-5)
        z = 1158822 = 2 * 3^2 * 7 * 17 * 541

Of course these are not rational integers, so they're not really a
counterexample to FLT, but there's no point in JSH's argument in which he
uses the fact that  x, y, z  were in the ring of integers; indeed he
himself begins discussing elements in rings like Z[sqrt(-5)] later on.

Just to pre-empt cries of "Foul!" from the experts: Yes, there are early
parts of JSH's argument in which he relies on unique factorization, which
is lacking in this ring. I deliberately choose this solution over the
simpler example
        2 ^3  =  (4 + sqrt(-5)) ^3  +  (4 - sqrt(-5)) ^3
so as to get a solution (x,y,z) to which the first part of JSH's argument
does apply: in this example, z-y  is indeed a cube  (of  f = 78+41 sqrt(-5))
as is  z^2+zy+y^2  (of  m=7507 - 1230 sqrt(-5) ). 
(The big solution is the triple of the small one on the corresponding
elliptic curve over  Z[sqrt(-5)]. )

This particular ring is perfect for this application because it also makes 
JSH's arguments about "3's in the denominator" fall apart: we have an identity
        3 / (2+sqrt(-5)) = (2-sqrt(-5)) / 3
in which there are more  3's  in the denominator on one side than on the other.
Is this number a multiple of 3 or isn't it?  

But you can find examples in UFDs, too, e.g. 
	(10 sqrt(-2)) ^3 = (1 + 8 sqrt(-2))^3 + (-1 + 8 sqrt(-2))^3.
Even if, by some miracle, he gets around to grappling with the inevitable 
non-principal ideals, there are further hurdles to FLT involving units.
If he ever gets that far, I can find examples in rings with unit groups of
positive rank, too, e.g. using  x=-12+43 sqrt(15)  or  -246924+168275 sqrt(11).

dave