On values taken by the largest prime factor of shifted primes

William D. Banks*, Igor E. Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)
11 Downloads (Pure)

Abstract

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.

Original languageEnglish
Pages (from-to)133-147
Number of pages15
JournalJournal of the Australian Mathematical Society
Volume82
Issue number1
Publication statusPublished - Feb 2007

Bibliographical note

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.

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