# SUMS OF TWO ${S}$-UNITS VIA FREY-HELLEGOUARCH CURVES

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 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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Type de document :
Pré-publication, Document de travail
Missing solution in Prop. 5.4 added. To appear in Mathematics of Computation. 2015

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https://hal.archives-ouvertes.fr/hal-01292805
Contributeur : Nicolas Billerey <>
Soumis le : lundi 23 mai 2016 - 09:20:33
Dernière modification le : mercredi 25 mai 2016 - 01:07:15
Document(s) archivé(s) le : mercredi 24 août 2016 - 10:19:03

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• HAL Id : hal-01292805, version 2
• ARXIV : 1603.07922

### Citation

Michael Bennett, Nicolas Billerey. SUMS OF TWO ${S}$-UNITS VIA FREY-HELLEGOUARCH CURVES. Missing solution in Prop. 5.4 added. To appear in Mathematics of Computation. 2015. 〈hal-01292805v2〉

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