Type: Article
Publication Date: 2007-02-01
Citations: 19
DOI: https://doi.org/10.1017/s1446788700017511
Abstract Let P denote the set of prime numbers, and let P ( n ) denote the largest prime factor of an integer n > 1. We show that, for every real number , there exists a constant c (η) > 1 such that for every integer a ≠ 0, the set has relative asymptotic density one in the set of all prime numbers. Moreover, in the range , one can take c (η) = 1+ε for any fixed ε > 0. In particular, our results imply that for every real number 0.486 ≤ b.thetav; ≤ 0.531, the relation P ( q − a ) ≍ q θ holds for infinitely many primes q . We use this result to derive a lower bound on the number of distinct prime divisor of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map q ↦ P ( q - a ) for a > 0, and show that for infinitely many primes q , this map can be iterated at least (log log q ) 1+o(1) times before it terminates.