Here are excerpts from B19 of UPINT3: Browkin \& Brzezi\'nski generalize the $abc$-conjecture (which is their case $n=3$) to an ``$n$-conjecture'' on $a_1+\cdots+a_n=0$ in coprime integers with non-vanishing subsums. With $R$ and $P$ defined analogously, they conjecture that $\lim\sup P=2n-5$. They prove that $\lim\sup P\geq2n-5$. They give a lot of examples for the $abc$-conjecture with $P>1.4$. Their method is to look for rational numbers approximating roots of integers (note that the best example above is connected to the good approximation 23/9 for $109^{1/5}$). Abderrahmane Nitaj used a similar method. Some of these were found independently by Robert Styer ({\bf D10}). The Catalan relation $1+2^3=3^2$ gives a comparatively poor $P\approx1.22629$. Jerzy Browkin \& Juliusz Brzezi\'nski, Some remarks on the $abc$-conjecture, {\it Math.\ Comput.}, {\bf62}(1994) 931--939; {\it MR} {\bf94g}:11021. Paul Vojta, A more general $abc$-conjecture, {\it Internat.\ Math.\ Res.\ Notices}, {\bf1998} 1103--1116; {\it MR} {\bf99k}:11096. On Fri, 4 Jun 2004, Dylan Thurston wrote:

On Tue, Jun 01, 2004 at 08:22:41PM +0100, Jon Perry wrote:

...

I've not seen this anywhere else, can we generalize the abc conjecture into something whereby;

f=a+b+c+d+e

or any number of variables, and a form of the original abc conjecture still exists?

A google for 'abc conjecture' gives the page http://www.math.unicaen.fr/~nitaj/abc.html which has a generalized conjecture as you ask for:

The n-term abc conjecture for integers. In 1994, Browkin and Brzezinski [Br-Brz] proposed the following conjecture.

Given any integer n > 2 and any eps > 0, there exists a constant C(n,eps), such that for all integers a1, ..., an with a1+...+ an=0, gcd( a1,..., an)=1 and no proper zero subsum, we have max(|a1|,...,|an|) <= C(n,eps)*(rad(a1 � ... � an))^(2n-5+eps).

Peace, Dylan