p-adic logarithmic forms and a problem of Erdo{double acute}s

For any m∈ℤ let P(m) denote the greatest prime divisor of m with the convention that P(m)=1 when m∈{1,0,-1}. By the problem of Erdo{double acute}s in the title of the present paper we mean the general version of his problem, that is, his conjecture from 1965 that (Formula Presented) (see Erdo{double acute}s [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 Erdo{double acute}s (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.