From: "N.R.Bruin" <nbruin@wi.leidenuniv.nl>
Newsgroups: sci.math
Subject: Re: Generalization of Fermat's Last Theorem.
Date: Thu, 22 Feb 1996 10:36:29 +0100

Bruce Kaskel wrote:
> 
> In article <4gg13i$k9n@bell.maths.tcd.ie>,
> Ruadhan O'Flanagan <rof@maths.tcd.ie> wrote:
> >  a(1)^n + a(2)^n + a(3)^n + ... + a(m)^n = b^n

[snip]

> This generalization is known to be false. The best known example is
> 
> 27^5 + 84^5 + 110^5 + 133^5 = 144^5.
> 
> In fact, this example has been included in a number of relatively
> recent posts (last month).

There is a standing conjecture on this, however. It is related to the
ABC-conjecture, which bounds the largest square-free divisor of ABC from
below in terms of C for A+B=C, gcd(A,B,C)=1.

The n-conjecture is something like the following (the actual statement is
somewhat stronger, but this formulation is far from being proven as well):

if x1 ... xn are relatively prime positive integers and x1+...+xn=y.

Let R be the product of distinct prime divisors of x1,..,xn,y. Then

there are C,e such that y<=C*R^e

For n=2 we have the ABC conjecture. Actually, e=1+epsilon is conjectured.

It can be shown that the statement is false for e<=2*n-5.

The n-conjecture can be proved for polynomials. (then we must look at
exp(deg) of the polynomials involved). This is only for e quadratic in n.
There is a strong belief that e should actually be linear in n. (actually,
e=2*n-5 might very well be attainable. My masters thesis was about this
subject. It is available (full of minor mistakes, unfortunately) through:

ftp://ftp.wi.leidenuniv.nl/pub/GM/Publications/N.Bruin/scriptie.ps.gz

Greetings,

Nils Bruin