Effective irrationality measures for real and p-adic roots of rational numbers close to 1, with an application to parametric families of Thue–Mahler equations
We show how the theory of linear forms in two logarithms allows one to get very good effective irrationality measures for nth roots of rational numbers a/b, when a is very close to b. We give a p-adic analogue of this result under the assumption that a is p-adically very close to b, that is, that a...
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Published in | Mathematical proceedings of the Cambridge Philosophical Society Vol. 164; no. 1; pp. 99 - 108 |
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Main Author | |
Format | Journal Article |
Language | English |
Published |
Cambridge, UK
Cambridge University Press
01.01.2018
Cambridge University Press (CUP) |
Subjects | |
Online Access | Get full text |
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Summary: | We show how the theory of linear forms in two logarithms allows one to get very good effective irrationality measures for nth roots of rational numbers a/b, when a is very close to b. We give a p-adic analogue of this result under the assumption that a is p-adically very close to b, that is, that a large power of p divides a−b. As an application, we solve completely certain families of Thue–Mahler equations. Our results illustrate, admittedly in a very special situation, the strength of the known estimates for linear forms in logarithms. |
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ISSN: | 0305-0041 1469-8064 |
DOI: | 10.1017/S0305004116000864 |