J. Aust. Math. Soc.
82 (2007), 133147

On values taken by the largest prime factor of shifted primes



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.

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