From: "N.R.Bruin" <nbruin@wi.leidenuniv.nl>
Newsgroups: sci.math
Subject: Q: ABC-like conjecture by Stewart
Date: 16 Feb 1995 15:24:41 GMT

First a short intro to ring some bells:

Let A,B,C be integers such that A+B=C.
h:=log(max(|A|,|B|,|C|))
r:=log(prod(primes p such that p|A,B or C))

L:= {h/r for A,B,C: A+B=C; gcd(A,B,C)=1}

Conjecture (ABC conjecture):
  L is a bounded set

More precise:
 limsup L <= 1

This is a very imortant conjecture.

Somebody told me that Stewart has conjectured something similar in which
the demand that gcd(A,B,C)=1 can be dropped. Can anyone give me
any reference to such a conjecture ?

Any other info to closely related topics are also welcome.

Since I am not a regular reader of this newsgroup, please send any reply
to me (as well as to the group, perhaps).

Thanks in advance,

Nils Bruin
nbruin@wi.leidenuniv.nl
==============================================================================

From: Chas F Brown <cbrown@batnet.com>
Subject: Re: m!=k!n!
Date: Fri, 30 Jul 1999 19:58:56 -0700
Newsgroups: sci.math
To: John R Ramsden <jr@redmink.demon.co.uk>

John R Ramsden wrote:

> Thinking about this during the week, I had reached the conclusion
> that A, B, C must be relatively prime. All the same, I still wish
> some helpful poster would "kick in" and post a _statement_ of the
> blasted conjecture!


Not that I know what it is, but from a sci.math FAQ (regarding FLT): 

"
For an excellent survey article on these subjects see the article by
Serge Lang in the Bulletin of the AMS, July 1990 entitled ``Old and new
conjectured diophantine inequalities".

Masser and Osterle formulated the following known as the ABC conjecture:

Given epsilon > 0 , there exists a number C(epsilon) such that for any
set of non-zero, relatively prime integers a,b,c such that a + b = c we
have max (|a|, |b|, |c|) <= C(epsilon) N(abc)^(1 + epsilon) where N(x)
is the product of the distinct primes dividing x .
"

If you read french (even some) and can deal with PDF documents, you can
also try:

http://www.mat.ulaval.ca/~jnfourni/abc.pdf

for a nice overview of the conjecture and some of its implications.

I assume "relatively prime" here means pairwise co-prime?

> Surely you wouldn't want to think of me covered in cobwebs, and
> stuck waist-deep in the bathroom ceiling, because that's what
> will most likely happen if I have to brave the loft ...

That's why I keep that crap in the basement!

Cheers - Chas

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