Sums of two ${s}$-units via frey-hellegouarch curves
Number Theory
2016-05-25 v2
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}.
Keywords
Cite
@article{arxiv.1603.07922,
title = {Sums of two ${s}$-units via frey-hellegouarch curves},
author = {Michael Bennett and Nicolas Billerey},
journal= {arXiv preprint arXiv:1603.07922},
year = {2016}
}
Comments
Missing solution in Prop. 5.4 added. To appear in Mathematics of Computation