Properties of large prime divisors of numbers of the form p − 1

Abstract

The main result of this paper is the fact that the fraction of primes p ≤ x satisfying the condition that p − 1 has a prime divisor q > exp(ln x/ln ln x) and the number of prime divisors of q − 1 essentially differ from ln ln(x/n), where n = (p − 1)/q, tends to zero as x increases.