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Let $K$ be a number field, and $S$ a finite set of places in $K$ containingall infinite places. We present an implementation for solving the $S$-unitequation $x + y = 1$, $x,y \in\mathscr{O}_{K,S}^\times$ in the computer algebrapackage SageMath. This paper outlines the mathematical basis for theimplementation. We discuss and reference the results of extensive computations,including exponent bounds for solutions in many fields of small degree forsmall sets $S$. As an application, we prove an asymptotic version of Fermat'sLast Theorem for totally real cubic number fields with bounded discriminantwhere 2 is totally ramified. In addition, we use the implementation to find allsolutions to some cubic Ramanujan-Nagell equations.

A robust implementation for solving the $S$-unit equation and several applications