TY - JOUR

T1 - On values taken by the largest prime factor of shifted primes

AU - Banks, William D.

AU - Shparlinski, Igor E.

N1 - Copyright 2007 Cambridge University Press. Article originally published in Journal of the Australian Mathematical Society, Volume 82, Issue 1, pp. 133-147. The original article can be found at http://dx.doi.org/10.1017/S1446788700017511.

PY - 2007/2

Y1 - 2007/2

N2 - 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 pη < q < c(η) pη} 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 qv 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.

AB - 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 pη < q < c(η) pη} 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 qv 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.

UR - http://www.scopus.com/inward/record.url?scp=34247551628&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:34247551628

SN - 1446-7887

VL - 82

SP - 133

EP - 147

JO - Journal of the Australian Mathematical Society

JF - Journal of the Australian Mathematical Society

IS - 1

ER -