Variations on a theme of Mirsky

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Akbal Y., Güloǧlu A. M.

International Journal of Number Theory, vol.19, no.1, pp.1-39, 2023 (SCI-Expanded, Scopus)

  • Publication Type: Article / Article
  • Volume: 19 Issue: 1
  • Publication Date: 2023
  • Doi Number: 10.1142/s179304212350001x
  • Journal Name: International Journal of Number Theory
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Page Numbers: pp.1-39
  • Keywords: Goldbach-type additive problems, Hardy-Littlewood circle method, r -free shifted primes
  • Open Archive Collection: AVESIS Open Access Collection
  • TED University Affiliated: Yes

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.