From: "N.R.Bruin" <nbruin@wi.leidenuniv.nl>
Newsgroups: sci.math
Subject: Re: ABC conjecture
Date: Mon, 06 Jan 1997 10:23:59 +0100

Antoine Mathys wrote:
> 
> Hi everybody
> 
> Could everyone explain me what is the ABC conjecture ?
        ^^^^^^^^
I guess you don't want that to happen. It'll get your system
administrator a heart attack.

Here it goes:

For rational functions, it's a theorem by Mason.

A special case is polynomials over C:

if A,B and C are polynomials (in C[X]) (not all constant) such that A,B
and C have no zeros in common and if A+B=C, then
max deg(A), deg(B), deg(C) < # zeros of ABC.
It means that if A+B=C, then not all of A,B,C can have zeros of high
multiplicity. Especially, A,B and C cannot be 3rd powers or more.

The direct analogue for integers is not true. The next-best shot is the
ABC-conjecture for integers:

Let R(A,B,C) be the product of prime divisors of ABC

For every e>0 there is a constant K_e such that for all coprime integers
A,B,C>0 with A+B=C :

C < K_e * R(A,B,C) ^ (1+e)

It relates the multiplicative structure ( R(A,B,C) ) to the additive
structure (A+B=C).

For more information, read the first chapter of

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

Greetings,

Nils Bruin