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2021 | OriginalPaper | Chapter

A Robust Implementation for Solving the S-Unit Equation and Several Applications

Authors: Alejandra Alvarado, Angelos Koutsianas, Beth Malmskog, Christopher Rasmussen, Christelle Vincent, Mckenzie West

Published in: Arithmetic Geometry, Number Theory, and Computation

Publisher: Springer International Publishing

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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, \(x,y \in \mathcal {O}_{K,S}^\times \) 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.
Footnotes
1
See also the recent translation [17] by Fuchs.
 
2
It is worth mentioning the recent results of von Känel and Matschke [42], who solve S-unit equations using modularity.
 
3
Note s d in [34] has the value t − 1 in our notation.
 
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Metadata
Title
A Robust Implementation for Solving the S-Unit Equation and Several Applications
Authors
Alejandra Alvarado
Angelos Koutsianas
Beth Malmskog
Christopher Rasmussen
Christelle Vincent
Mckenzie West
Copyright Year
2021
DOI
https://doi.org/10.1007/978-3-030-80914-0_1

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