Diophantine Triples and Linear Forms in Logarithms
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This is the thesis for my master's degree in mathematics which I undertook with Dr. Omar Kihel. Over the last couple of years I have studied number theory with the aim being to develop a broader understanding of the theory of Diophantine equations and their (at times) elusive solutions. I begin my thesis by establishing some of the preliminary results while touching on their place within the history of number theory. This section finishes with an account of Alan Baker's work on linear forms in logarithms and some of its applications, after which the two theorems on Diophantine triples that this paper will aim to prove are stated. In the second section, I list a series of definitions and results of which the reader must be aware, but which I could not fit into the first section due to its historical slant. Following this, I prove a lemma on Pellian equations which generalizes the first lemma of . This requires that a mistake from the proof of that lemma be fixed. Since this lemma was used in , this section serves to buttress that result as well. In the next two sections, I prove the two main theorems using results on linear forms in logarithms of algebraic numbers, extending the main result in  to $D(9)$ and $D(64)$ triples. The thesis ends with a few words on potential generalization and improvement of the main results, as well as other potential avenues of inquiry, and draws attention to some potential difficulties. The main results closely follow a paper co-written with my brother, Benjamin Earp-Lynch.