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On sums of S-units and linear recurrences. (English) Zbl 0547.10008
This paper deals with some remarkable consequences of Schlickewei’s p- adic version of the method of Thue-Siegel-Roth-Schmidt. The most striking results deal with recurrent sequences \(u_ 0,u_ 1,u_ 2,..\). given by a linear recurrence relation \(u_{n+1}=a_ ku_ n+...+a_ 0u_{n- k}\), \(a_ i\) fixed. Here we assume \(a_ i,u_ n\in {\mathbb{Q}}\). Let P(a/b) be the greatest prime factor occurring in the numerator and denominator of a/\(b\in {\mathbb{Q}}\). One of the results is, that \(P(u_ r/u_ s)\to\infty \) if \(r>s\) and \(r\to\infty \). Some equally interesting results, including the ones that were obtained independently by A. J. van der Poorten, are also given.
Reviewer: F.Beukers

MSC:
11B37 Recurrences
11D61 Exponential Diophantine equations
11J81 Transcendence (general theory)
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