The abc conjecture is about triples of positive integers, a, b and c that are relatively prime and satisfy a + b = c.
The quality q(a,b,c) of the triple (a,b,c) is defined as

q(a,b,c) = log(c)/log(rad(a*b*c)),

where the radical rad(a*b*c) is the product of the distinct prime factors of the product a*b*c.

*abc conjecture:
For every positive real number epsilon, there exist only finitely many triples (a,b,c) of coprime positive integers with a + b = c such that q(a,b,c) > 1 + epsilon.*

The amazingly important consequences of the abc conjecture are described on wikipedia.

The ABC@home non-profit network computing project with its more than 7,300 active participants from 114 countries did calculate all abc-triples of at most 18 digits.
Concerning the data that it has collected, the abc conjecture implies that the following graphs of the number of abc triples of quality q above the respectively fixed threshold (plotted over the number of digits they admit minus 1) are each one bounded by some constant:

On Prof. Bart de Smit's ABC triples page, it can be seen that the growth of the first curve (q > 1.4) stops when continuing to 19 and 20 digits (which still does not prove that there is a constant bounding the curve).
But for the abc conjecture to be true, also the curves for q > 1.01, q > 1.001, 1.0001, ... need to be bounded by a constant each.

An infinite sequence of abc triples of quality above a fixed threshold greater than 1 would disprove the abc conjecture.

**Question.** *How close can one get to counterexamples to the abc-Conjecture?*

**Goal:**
We do not aim at finding an actual counterexample.
Rather, we want to use the database collected by the ABC@home grid computing project
in order to see how close we can get to a counterexample, constructing infinite sequences using a certain strategy of the project's supervisor, which has been checked back not to be known already by Prof. Bart de Smit, who was in charge of the ABC@home network.

**Schedule:** Summer semester 2019.

**Supervision:**
Alexander D. Rahm.

**Difficulty level:**
EML 1, EML 3 or Bachelor thesis project.

**Prerequisites:**
Linear Algebra, Algebra, ability to use a programming language.

**Tools:**
A programming language of the participant's choice, and the database collected by the ABC@home grid computing project.

**Expected results:**
We do not aim at finding an actual counterexample.
We just want to find out how close we can get to one.

**Print version of this description:**
here.