Distribution of primes and dynamics of the w function

Abstract

Let P be the set of all primes. The following result is proved: For any nonzero integer a, the set a + P contains arbitrarily long sequences which have the same largest prime factor. We give an application to the dynamics of the w function which extends the “seven” in Theorem 2.14 of [Wushi Goldring, Dynamics of the w function and primes, J. Number Theory 119 (2006) 86–98] to any positive integer. Beyond this we also establish a relation between a result of congruent covering systems and a question on the dynamics of the w function. This implies that the answer to Conjecture 2.16 of Goldring's paper is negative. Two conjectures are posed.