J. Aust. Math. Soc.  82 (2007), 133-147
On values taken by the largest prime factor of shifted primes

 William D. Banks   Department of Mathematics   University of Missouri   Columbia, MO 65211   USA  bbanks@math.missouri.edu
 and
 Igor E. Shparlinski   Department of Computing   Macquarie University   Sydney, NSW 2109   Australia  igor@ics.mq.edu.au

Abstract
Let denote the set of prime numbers, and let denote the largest prime factor of an integer . We show that, for every real number , there exists a constant such that for every integer , the set

has relative asymptotic density one in the set of all prime numbers. Moreover, in the range , one can take for any fixed . In particular, our results imply that for every real number , the relation holds for infinitely many primes . We use this result to derive a lower bound on the number of distinct prime divisors of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map for , and show that for infinitely many primes , this map can be iterated at least times before it terminates.