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Let $K$ be a number field and $S$ a finite set of places including the archimedian ones. Let $(a_1,a_2,a_3)\in K^*$ we consider the $S$-unit equation $a_1x+a_2y=a_3$ with $x,y\in U_S$. In this paper we prove that for fixed $a_1,a_2$ the equation has at most $\frac{|S|-1}2$ solutions provided that the height of $a_3$ is...
One of the first infinite families of Thue equations, $$F_n(X)=X^3 - (n-1) X^2Y - (n+2)XY^2 - Y^3 = \pm 1$$ for $n\in \mathbb{Z}$, was solved by Thomas in 1990. This family is associated to the simplest cubic fields $\mathbb{Q}(\lambda)$ of Shanks, where $\lambda$ is a root of $F_n(X,1)$. Levesque and Waldschmidt twist...
Let $K$ be a number field of degree $D$ with maximal order $\mathcal{O}_K$. We show that under certain conditions on $K$, which in particular are always satisfied if $D$ is odd or if $D \geq 3$ and $K$ is primitive, the set of positive integers $N_K$ that can be expressed as a sum of two units in $\mathcal{O}_K^*$ is a...
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