The Number of Solutions to the Trinomial Thue Equation

Abstract: In this paper, we study the number of integer pair solutions to the equation $|F(x,y)| = 1$ where $F(x,y) \in \mathbb{Z}[x,y]$ is an irreducible (over $\mathbb{Z}$) binary form with degree $n \geqslant 3$ and exactly three nonzero summands. In particular, we improve Emery Thomas' explicit upper bounds on the number of solutions to this equation. For instance, when $n \geqslant 219$, we show that there are no more than 32 integer pair solutions to this equation when $n$ is odd and no more than 40 integer pair solutions to this equation when $n$ is even, an improvement on Thomas' work, where he shows that there are no more than 38 such solutions when $n$ is odd and no more than 48 such solutions when $n$ is even.