Distribution of primes and dynamics of the w function

Let P be the set of all primes. The following result is proved: For any nonzero integer a ,theset a + P contains arbitrarily long sequences which have the same largest prime factor. We give an application to thedynamics of the w function which extends the “seven” in Theorem 2.14 of [Wushi Goldring, Dynamics ofthe w function and primes, J. Number Theory 119 (2006) 86–98] to any positive integer. Beyond this wealso establish a relation between a result of congruent covering systems and a question on the dynamicsof the w function. This implies that the answer to Conjecture 2.16 of Goldring’s paper is negative. Twoconjectures are posed. © 2008 Elsevier Inc. All rights reserved. MSC: 11A41; 37B99; 11N36 Keywords: Largest prime factor; Dynamics; Sequence; Parent 1. Introduction Let P be the set of all (rational) primes. For any positive integer n let P(n) denote the largestprime factor of n .Let P( 0 ) =2. Let A 3 = pqr 3 p,q,r ∈ P \ p p ∈ P. ✩ Supported by the National Natural Science Foundation of China, Grant No. 10771103.