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: Bachelor thesis project.
Prerequisites: Linear Algebra 1, Structures mathématiques, Analysis 1.
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.