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Sums of two $ S$-units via Frey-Hellegouarch curves


Authors: Michael A. Bennett and Nicolas Billerey
Journal: Math. Comp. 86 (2017), 1375-1401
MSC (2010): Primary 11D61; Secondary 11G05
DOI: https://doi.org/10.1090/mcom/3129
Published electronically: August 18, 2016
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Abstract: In this paper, we develop a new method for finding all perfect powers which can be expressed as the sum of two rational $ S$-units, where $ S$ is a finite set of primes. Our approach is based upon the modularity of Galois representations and, for the most part, does not require lower bounds for linear forms in logarithms. Its main virtue is that it enables us to carry out such a program explicitly, at least for certain small sets of primes $ S$; we do so for $ S = \{ 2, 3 \}$ and $ S= \{ 3, 5, 7 \}$.


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Additional Information

Michael A. Bennett
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada
Email: bennett@math.ubc.edu

Nicolas Billerey
Affiliation: Laboratoire de Mathématiques, Université Clermont Auvergne, Université Blaise Pascal, BP 10448, F-63000 Clermont-Ferrand, France — and — CNRS, UMR 6620, LM, F-63171 Aubière, France
Email: Nicolas.Billerey@math.univ-bpclermont.fr

DOI: https://doi.org/10.1090/mcom/3129
Received by editor(s): July 21, 2015
Received by editor(s) in revised form: October 19, 2015
Published electronically: August 18, 2016
Additional Notes: The first-named author was supported in part by a grant from NSERC
The second-named author acknowledges the financial support of CNRS and ANR-14-CE25-0015 Gardio. He also warmly thanks PIMS and the Mathematics Department of UBC for hospitality and excellent working conditions
Article copyright: © Copyright 2016 American Mathematical Society