The main theme in this thesis is the ABC conjecture. We prove some partial results towards it and we find new applications of this conjecture, mainly in the context of B\"uchi's n squares problem (which has consequences in logic related to Hilbert's tenth problem) and squarefree values of polynomials. We also study related topics, such as arithmetic properties of additive subgroups of Hecke algebras, function field and meromorphic value distribution, and undecidability of the positive existential theories over languages of arithmetic interest. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2014-04-28 10:47:54.064