On values taken by the largest prime factor of shifted primes

Let 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 < η < (4 + 3√2)/4, there exists a constant c(η) > 1 such that for every integer a ≠ 0, the set {p ∈ P : p = P(q - a) for some prime q with p^η < q < c(η) p^η} has relative asymptotic density one in the set of all prime numbers. Moreover, in the range 2 ≤ η < (4 + 3 √2)/4, one can take c(η) = 1 + ε for any fixed ε > 0. In particular, our results imply that for every real number 0.486 ≤ v ≤ 0.531, the relation P(q -a) equivalent to q^v 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 mapping 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⁺⁰⁽¹⁾ times before it terminates.