Variations on a theme of Mirsky

Date
2022-07-05
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Source Title
International Journal of Number Theory
Print ISSN
1793-0421
Electronic ISSN
1793-7310
Publisher
World Scientific Publishing
Volume
19
Issue
1
Pages
1 - 39
Language
English
Type
Article
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Abstract

Let k and r be non-zero integers with r≥2. An integer is called r-free if it is not divisible by the rth power of a prime. A result of Mirsky states that there are infinitely many primes p such that p+k is r-free. In this paper, we study an additive Goldbach-type problem and prove two uniform distribution results using these primes. We also study certain properties of primes p such that p+a1,…,p+aℓ are simultaneously r-free, where a1,…,aℓ are non-zero integers and ℓ≥1 .

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Keywords
Hardy–Littlewood circle method, r-free shifted primes, Goldbach-type additive problems
Citation
Published Version (Please cite this version)