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

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Arithmetic Geometry, Number Theory, and Computation

Part of the book series: Simons Symposia ((SISY))

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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Notes

  1. 1.

    See also the recent translation [17] by Fuchs.

  2. 2.

    It is worth mentioning the recent results of von Känel and Matschke [42], who solve S-unit equations using modularity.

  3. 3.

    Note s d in [34] has the value t − 1 in our notation.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Eastern Illinois University, Charleston, IL, USA

    Alejandra Alvarado

  2. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

    Angelos Koutsianas

  3. Department of Mathematics and Computer Science, Colorado College, Colorado Springs, CO, USA

    Beth Malmskog

  4. Department of Mathematics and Computer Science, Wesleyan University, Middletown, CT, USA

    Christopher Rasmussen

  5. Department of Mathematics and Statistics, University of Vermont, Burlington, VT, USA

    Christelle Vincent

  6. Department of Mathematics, University of Wisconsin-Eau Claire, Eau Claire, WI, USA

    Mckenzie West

Authors
  1. Alejandra Alvarado
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  2. Angelos Koutsianas
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  3. Beth Malmskog
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  4. Christopher Rasmussen
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  5. Christelle Vincent
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  6. Mckenzie West
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Corresponding author

Correspondence to Christopher Rasmussen .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, Boston University, Boston, MA, USA

    Jennifer S. Balakrishnan

  2. Department of Mathematics, Harvard University, Cambridge, MA, USA

    Noam Elkies

  3. Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, RI, USA

    Brendan Hassett

  4. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Bjorn Poonen

  5. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Andrew V. Sutherland

  6. Department of Mathematics, Dartmouth College, Hanover, NH, USA

    John Voight

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Alvarado, A., Koutsianas, A., Malmskog, B., Rasmussen, C., Vincent, C., West, M. (2021). A Robust Implementation for Solving the S-Unit Equation and Several Applications. In: Balakrishnan, J.S., Elkies, N., Hassett, B., Poonen, B., Sutherland, A.V., Voight, J. (eds) Arithmetic Geometry, Number Theory, and Computation. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-030-80914-0_1

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