p-adic logarithmic forms and a problem of Erdős

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 $$\frac{P(2^n-1)}{n}\to \infty \mathrm{as}n\to \infty$$(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.