Update on ABC-triples

Reken mee met ABC This page serves as an update to the paper in Dutch [PDF, March 2007] that I wrote with Gillien Geuze about the ABC conjecture and the project Reken mee met ABC. This project started in September 2005 and it aims to find ABC triples through a collaborative BOINC project.

On this page we keep track of three lists of top ABC triples:

The terminology is as follows. Suppose we have a solution in coprime positive integers of the equation a+b=c with a<b.

The radical r is the product of the distinct prime divisors of abc.
The quality is q=(log c)/log r.
The size is (log c)/log 10: the number of decimal digits of c.
The merit is defined to be (q-1)2(log r) log log r.

We say that we have an ABC triple if r<c or, equivalently, q>1. One triple beats another if it has both a larger size and a larger quality.

The ABC conjecture says that the limsup of the quality when we range over all ABC triples, is 1. The unbeaten list provides the best known lower bound on how quickly this limsup tends to 1. The refined ABC conjecture of Stewart and Tenenbaum predicts the precise rate of convergence. It says that the limsup of the merit when we range over all ABC triples, is 48.