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A note on Linnik’s theorem on quadratic non-residues

  • Paul Balister
  • Béla Bollobás
  • Jonathan D. Lee
  • Robert MorrisEmail author
  • Oliver Riordan
Article
  • 15 Downloads

Abstract

We present a short and purely combinatorial proof of Linnik’s theorem: for any \(\varepsilon >0\) there exists a constant \(C_\varepsilon \) such that for any N, there are at most \(C_\varepsilon \) primes \(p\le N\) such that the least positive quadratic non-residue modulo p exceeds \(N^\varepsilon \).

Keywords

Combinatorial number theory Quadratic non-residues Combinatorial sieve 

Mathematics Subject Classification

11A15 11A41 11N35 11B75 

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Notes

Acknowledgements

The work of the first two authors was partially supported by NSF Grant DMS 1600742, and work of the second author was also partially supported by MULTIPLEX Grant 317532. The work of the fourth author was partially supported by CNPq (Proc. 303275/2013-8) and FAPERJ (Proc. 201.598/2014). The research in this paper was carried out while the third, fourth and fifth authors were visiting the University of Memphis.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA
  2. 2.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK
  3. 3.London Institute for Mathematical SciencesLondonUK
  4. 4.Microsoft ResearchRedmondUSA
  5. 5.IMPARio de JaneiroBrazil
  6. 6.Mathematical InstituteUniversity of OxfordOxfordUK

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