# Number Theory

December 14, 2022

In 1909, Thue proved that when $F(x,y) \in \mathbb{Z}[x,y]$ is irreducible, homogeneous, and has degree at least 3, the inequality $|F(x,y)| \leq h$ has finitely many integer-pair solutions for any positive $h$. Because of this result, the inequality $|F(x,y)| \leq h$ is known as Thue’s Inequality and much work has been done to find sharp bounds on the number of integer-pair solutions to Thue’s Inequality. In this talk, I will describe different techniques used by Baker; Mueller and Schmidt; Saradha and Sharma; Thomas; and Akhtari and Bengoechea to make progress on this general problem. After that, I will discuss some improvements that can be made to a counting technique used in association with ``the gap principle’’ and how those improvements lead to better bounds on the number of solutions to Thue’s Inequality.