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On values taken by the largest prime factor of shifted primes
Published online by Cambridge University Press: 09 April 2009
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 , there exists a constant c(η) > 1 such that for every integer a ≠ 0, the set
has relative asymptotic density one in the set of all prime numbers. Moreover, in the range
, one can take c(η) = 1+ε for any fixed ε > 0. In particular, our results imply that for every real number 0.486 ≤ b.thetav; ≤ 0.531, the relation P(q − a) ≍ qθ holds for infinitely many primes q. We use this result to derive a lower bound on the number of distinct prime divisor of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map q ↦ P(q - a) for a > 0, and show that for infinitely many primes q, this map can be iterated at least (log logq)1+o(1) times before it terminates.
- Type
- Research Article
- Information
- Journal of the Australian Mathematical Society , Volume 82 , Issue 1 , February 2007 , pp. 133 - 147
- Copyright
- Copyright © Australian Mathematical Society 2007
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