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
where p is the smaller of the two primes.
For Helmut Maier in celebration of his 60th birthday
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Notes
- 1.
As usual in the subject, log2 x = loglogx, log3 x = logloglogx, and so on.
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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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Ford, K., Heath-Brown, D.R., Konyagin, S. (2015). Large Gaps Between Consecutive Prime Numbers Containing Perfect Powers. In: Pomerance, C., Rassias, M. (eds) Analytic Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-22240-0_5
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DOI: https://doi.org/10.1007/978-3-319-22240-0_5
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