1. Sums of two $S$-units via Frey-Hellegouarch curves
- Author
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Michael A. Bennett, Nicolas Billerey, Department of Mathematics [Vancouver] (UBC Mathematics), University of British Columbia (UBC), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Grant from NSERC, and ANR-14-CE25-0015,Gardio,Géométrie d'Arakelov et géométrie diophantienne(2014) more...
- Subjects
Primary 11D61, Secondary 11G05 ,Discrete mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,Logarithm ,Perfect power ,Exponential equations ,Applied Mathematics ,Carry (arithmetic) ,Modularity (biology) ,010102 general mathematics ,0102 computer and information sciences ,Galois module ,01 natural sciences ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Exponential function ,Combinatorics ,Computational Mathematics ,010201 computation theory & mathematics ,Elliptic curves over global fields ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Finite set ,Mathematics - 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}., Comment: Missing solution in Prop. 5.4 added. To appear in Mathematics of Computation more...
- Published
- 2016
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