On the largest prime factor of the Mersenne numbers

Abstract

Let P(k) be the largest prime factor of the positive integer k. In this paper, we prove that the series $\sum_{n\ge 1}\frac{(\log n)^a}{P(2^n-1)}$ is convergent for each constant a<1/2, which gives a more precise form of a result of C. L. Stewart from 1977.