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

## TL;DR

This paper introduces a new implementation in SageMath for solving the $S$-unit equation, enabling extensive computations that lead to applications such as an asymptotic Fermat's Last Theorem for certain cubic fields and solutions to Ramanujan-Nagell equations.

## Contribution

The paper provides a robust implementation for solving the $S$-unit equation in SageMath, with mathematical foundations and extensive computational results for various number fields.

## Key findings

- Bounded solutions for small degree fields and sets $S$
- Proof of an asymptotic Fermat's Last Theorem in specific cubic fields
- Complete solutions to certain Ramanujan-Nagell equations

## Abstract

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