On the unit equation $\varepsilon+δ=n$ in cubic fields

Abstract

Let $n$ be an integer not equal to $-2$, $0$ or $2$. We consider the unit equation $\varepsilon + δ= n$ in units $\varepsilon, δ$ of cubic fields. We show that this equation has no solutions for 100% of cubic fields, when ordered by discriminant. This is consistent with a recent conjecture of the authors.