Applying Belyi functions to the ABC-conjecture

The ABC conjecture of Masser-Oesterlé, 1988, states that for any t>1 there exists just finite number of triples (a, b, c) of relatively prime positive integers satisfying a+b=c, such that P=(log c)/(log Rad(abc)) > t, where the radical of an integer is the product of distinct primes dividing it, e.g. Rad(12)=6. We provide an overview of the ABC conjecture and its power, namely various results and conjectures, that may follow from ABC. The main example will be based on the paper “ABC allows us to count squarefrees” by Andrew Granville, IMRN, 19, 1998, 991-1009, where Belyi functions are used to obtain the implication. We will also show how to produce the triples (a, b, c) with big P from the correspondence between Belyi pairs and dessins d’enfants.