Abc conjecture and its implications

In this thesis, our aim is to state the importance of abc conjecture and prove the strong results we obtain with the help of abc conjecture. First, we give necessary notions and tools which are used throughout the thesis. Then we introduce Hall conjecture, Fermat's last theorem and Mordell conjecture, and their relations with abc conjecture. In particular, we give the effective proof ofMordell conjecture using abc conjecture, given in the article of Noam Elkies, and also get another height bound by combining with a different theorem. Finally, we give three examples where we use both of the height bounds.