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

Abstract

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 Yann BUGEAUDAbstract. We show how the theory of linear forms in two logarithms allows one to get effective irrationality measures for n-th 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.

Irrationality measures for cubic irrationals whose conjugates lie on a curve

“…This paper deals with irrationality measure of algebraic irrational numbers. There is a vast bibliography on this subject, the interested reader can refer for instance to [8] for further references. Let us recall only some basic facts.…”