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
Igor E. Shparlinski
  Department of Computing
  Macquarie University
  Sydney, NSW 2109

Let $\mathcal{P}$ 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 $32/17<\eta<(4+3\sqrt{2})/4$, there exists a constant $c(\eta)>1$ such that for every integer $a\ne 0$, the set
\[ \bigl\{p\in\mathcal{P}:p=P(q-a)\text{ for some prime }q \text{ with }p^\eta<q<c(\eta)\,p^\eta\bigr\} \]
has relative asymptotic density one in the set of all prime numbers. Moreover, in the range $2\le\eta<(4+3\sqrt{2})/4$, one can take $c(\eta)=1+\varepsilon$ for any fixed $\varepsilon>0$. In particular, our results imply that for every real number $0.486\le\vartheta\le 0.531$, the relation $P(q-a)\asymp q^{\vartheta}$ holds for infinitely many primes $q$. 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 $q\mapsto P(q-a)$ 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.
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