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Large Gaps Between Consecutive Prime Numbers Containing Perfect Powers

  • Kevin Ford
  • D. R. Heath-Brown
  • Sergei Konyagin

Abstract

For any positive integer k, we show that infinitely often, perfect kth powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size
$$\displaystyle{c_{k}\frac{\log p\log _{2}p\log _{4}p} {(\log _{3}p)^{2}},}$$
where p is the smaller of the two primes.

Notes

Acknowledgements

Research of the third author was partially performed while he was visiting the University of Illinois at Urbana-Champaign. Research of the first author was partially performed while visiting the University of Oxford. Research of the first and third authors was also carried out in part at the University of Chicago, and they are thankful to Prof. Wilhelm Schlag for hosting these visits.

The first author was supported by NSF grant DMS-1201442. The research of the second author was supported by EPSRC grant EP/K021132X/1. The research of the third author was partially supported by Russian Foundation for Basic Research, Grant 14-01-00332, and by Program Supporting Leading Scientific Schools, Grant Nsh-3082.2014.1.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Kevin Ford
    • 1
  • D. R. Heath-Brown
    • 2
  • Sergei Konyagin
    • 3
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Mathematical InstituteRadcliffe Observatory QuarterOxfordUK
  3. 3.Steklov Mathematical InstituteMoscowRussia

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