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Short effective intervals containing primes in arithmetic progressions and the seven cubes problem


Author: H. Kadiri
Journal: Math. Comp. 77 (2008), 1733-1748
MSC (2000): Primary 11M26
DOI: https://doi.org/10.1090/S0025-5718-08-02084-X
Published electronically: February 8, 2008
MathSciNet review: 2398791
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Abstract | References | Similar Articles | Additional Information

Abstract: For any $ \epsilon>0$ and any non-exceptional modulus $ q\ge3$, we prove that, for $ x$ large enough ( $ x\ge \alpha_{\epsilon}\log^2 q$), the interval $ \left[ e^x,e^{x+\epsilon }\right]$ contains a prime $ p$ in any of the arithmetic progressions modulo $ q$. We apply this result to establish that every integer $ n$ larger than $ \exp(71\,000)$ is a sum of seven cubes.


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Additional Information

H. Kadiri
Affiliation: Département de Mathématiques et Statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, QC H3C 3J7, Canada
Address at time of publication: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, Canada T1K 3M4
Email: habiba.kadiri@uleth.ca

DOI: https://doi.org/10.1090/S0025-5718-08-02084-X
Keywords: Analytic number theory, Dirichlet $L$-functions, primes, sums of cubes
Received by editor(s): August 29, 2006
Received by editor(s) in revised form: July 7, 2007
Published electronically: February 8, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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