Article Dans Une Revue Mathematics of Computation Année : 2016

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

Résumé

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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Dates et versions

hal-01292805 , version 1 (24-03-2016)
hal-01292805 , version 2 (23-05-2016)

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Michael A. Bennett, Nicolas Billerey. Sums of two ${S}$-units via Frey-Hellegouarch curves. Mathematics of Computation, 2016, 86 (305), pp.1375-1401. ⟨10.1090/mcom/3129⟩. ⟨hal-01292805v2⟩
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