Analytic Number Theory pp 83-92 | Cite as

# Large Gaps Between Consecutive Prime Numbers Containing Perfect Powers

## Abstract

*k*, we show that infinitely often, perfect

*k*th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size

*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.

## References

- 1.H. Cramér, On the order of magnitude of the difference between consecutive prime numbers. Acta Arith.
**2**, 396–403 (1936)Google Scholar - 2.K. Ford, B. Green, S. Konyagin, T. Tao, Large gaps between consecutive prime numbers. arXiv:1408.4505Google Scholar
- 3.A. Granville, Harald Cramér and the distribution of prime numbers. Scand. Actuar. J.
**1**, 12–28 (1995)MathSciNetCrossRefGoogle Scholar - 4.H. Halberstam, H.-E. Richert,
*Sieve Methods*(Academic, London, 1974)zbMATHGoogle Scholar - 5.D.R. Heath-Brown, A mean value estimate for real character sums. Acta Arith.
**72**, 235–275 (1995)zbMATHMathSciNetGoogle Scholar - 6.H. Heilbronn, On real zeros of Dedekind
*ζ*-functions. Can. J. Math.**25**, 870–873 (1973)zbMATHMathSciNetCrossRefGoogle Scholar - 7.J.C. Lagarias, A.M. Odlyzko, Effective versions of the Chebotarev density theorem, in
*Algebraic Number Fields*(Proceedings of the Symposia, University Durham, Durham, 1975) (Academic, London, 1977), pp. 409–464Google Scholar - 8.H. Maier, C. Pomerance, Unusually large gaps between consecutive primes. Trans. Am. Math. Soc.
**322**(1), 201–237 (1990)zbMATHMathSciNetCrossRefGoogle Scholar - 9.J. Maynard, Large gaps between primes. arXiv:1408.5110Google Scholar
- 10.J. Pintz, Very large gaps between consecutive primes. J. Number Theory
**63**(2), 286–301 (1997)zbMATHMathSciNetCrossRefGoogle Scholar - 11.R. Rankin, The difference between consecutive prime numbers. J. Lond. Math. Soc.
**13**, 242–247 (1938)MathSciNetCrossRefGoogle Scholar - 12.R.A. Rankin, The difference between consecutive prime numbers. V. Proc. Edinb. Math. Soc.
**13**(2), 331–332 (1962/1963)Google Scholar