Type: Article

On the largest prime factor of the Mersenne numbers

Kevin Ford; Florian Luca; Igor E. Shparlinski;

Abstract:

Let P(k) be the largest prime factor of the positive integer k. In this paper, we prove that the series Sigma(n >= 1)(log n)(alpha)/P(2(n) - 1) is convergent for each constant alpha < 1/2, which gives a more precise form of a result of C. L. Stewart ['On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers', Proc. London Math. Soc. 35(3) (1977), 425-447].