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

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

doi:10.1007/978-3-030-80914-0_1

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

Alejandra Alvarado, Christopher Rasmussen + Show 4 more

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https://doi.org/10.1007/978-3-030-80914-0_1

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Publication Date: Jan 1, 2021

Citations: 8

Affiliation: Eastern Illinois University, Wesleyan University, University of British Columbia, University of Vermont, Colorado College, University of Wisconsin–Eau Claire

#Finite Set Of Places #Infinite Places + Show 8 more

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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.

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