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PRODUCTS OF SHIFTED PRIMES SIMULTANEOUSLY TAKING PERFECT POWER VALUES

Part of: Multiplicative number theory

Published online by Cambridge University Press:  19 September 2012

TRISTAN FREIBERG
TRISTAN FREIBERG*
Affiliation:
Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden (email: tristanf@kth.se)
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Abstract

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Let r be an integer greater than 1, and let A be a finite, nonempty set of nonzero integers. We obtain a lower bound for the number of positive squarefree integers n, up to x, for which the products ∏ p∣n(p+a) (over primes p) are perfect rth powers for all the integers a in A. Also, in the cases where A={−1} and A={+1}, we will obtain a lower bound for the number of such n with exactly r distinct prime factors.

Keywords

shifted primesEuler’s totient functionperfect powers

MSC classification

Secondary: 11N25: Distribution of integers with specified multiplicative constraints
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 2 , April 2012 , pp. 145 - 154
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

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