A note on the number of perfect powers in short intervals

Rafael Jakimczuk
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 20, 2014, Number 3, Pages 31—35
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Authors and affiliations

Rafael Jakimczuk
División Matemática, Universidad Nacional de Luján
Buenos Aires, Argentina

Abstract

Let N(x) be the number of perfect powers that do not exceed x. In this note we obtain asymptotic formulae for the difference N(x + xθ) − N(x), where 1/2 < θ < 2/3 + 1/7. We also prove that if θ = 1/2 the difference N(x + xθ) − N(x) is zero for infinite x arbitrarily large.

Keywords

  • Distribution of perfect powers
  • Short intervals

AMS Classification

  • 11A99
  • 11B99

References

  1. Jakimczuk, R., On the distribution of perfect powers, Journal of Integer Sequences, Vol. 14, 2011, Article 11.8.5.

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Cite this paper

Jakimczuk, R. (2014). A note on the number of perfect powers in short intervals. Notes on Number Theory and Discrete Mathematics, 20(3), 31-35.

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