From: "N.R.Bruin" 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