## Acta Mathematica

### p-adic logarithmic forms and a problem of Erdős

Kunrui Yu

#### Abstract

For any natural number m(>1) let P(m) denote the greatest prime divisor of m. By the problem of Erdős in the title of the present paper we mean the general version of his problem, that is, his conjecture from 1965 that $\frac{P(2^n-1)}{n} \to \infty \quad {\rm as}\, n \to \infty$(see Erdős [10]) and its far-reaching generalization to Lucas and Lehmer numbers. In the present paper the author delivers three refinements upon Yu [40] required by C. L. Stewart for solving completely the problem of Erdős (see Stewart [25]). The author gives also some remarks on the solution of this problem, aiming to be more streamlined with respect to the p-adic theory of logarithmic forms.

#### Dedication

Dedicated to Prof. Gisbert Wüstholz on the occasion of his 61st birthday.

#### Article information

Source
Acta Math., Volume 211, Number 2 (2013), 315-382.

Dates
Received: 3 June 2011
Revised: 8 November 2012
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892740

Digital Object Identifier
doi:10.1007/s11511-013-0106-x

Mathematical Reviews number (MathSciNet)
MR3143893

Zentralblatt MATH identifier
1362.11071

Rights
2013 © Institut Mittag-Leffler

#### Citation

Yu, Kunrui. p -adic logarithmic forms and a problem of Erdős. Acta Math. 211 (2013), no. 2, 315--382. doi:10.1007/s11511-013-0106-x. https://projecteuclid.org/euclid.acta/1485892740

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