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

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


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 \(N^\varepsilon \).


Combinatorial number theory Quadratic non-residues Combinatorial sieve 

Mathematics Subject Classification

11A15 11A41 11N35 11B75 


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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.


  1. 1.
    Balister, P., Bollobás, B., Lee, J.D., Morris, R., Riordan, O.: A note on Linnik’s theorem on quadratic non-residues, arXiv:1712.07179
  2. 2.
    Bombieri, E.: On the large sieve. Mathematika 12, 201–225 (1965)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cojocaru, A.C., Murty, M.R.: An Introduction to Sieve Methods and Their Applications, London Mathematical Society Student Texts, vol. 66. Cambridge University Press, Cambridge (2006)Google Scholar
  4. 4.
    Dickman, K.: On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Mat. Astron. Fys. 22, 1–14 (1930)zbMATHGoogle Scholar
  5. 5.
    Friedlander, J.B., Iwaniec, H.: Opera de cribro, American Mathematical Society Colloquium Publications, vol. 57. American Mathematical Society, Providence, RI (2010)zbMATHGoogle Scholar
  6. 6.
    Goldfeld, D.: The elementary proof of the prime number theorem: an historical perspective. In: Chudnovsky, D., Chudnovsky, G., Nathanson, M. (eds.) Number Theory, pp. 179–192. Springer, New York (2004)CrossRefGoogle Scholar
  7. 7.
    Linnik, U.V.: The large sieve. C. R. (Doklady) Acad. Sci. URSS (N.S.) 30, 292–294 (1941)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Linnik, U.V.: A remark on the least quadratic non-residue. C. R. (Doklady) Acad. Sci. URSS (N.S.) 36, 119–120 (1942)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Mertens, F.: Ein Beitrag zur analytischen Zahlentheorie. J. Reine Angew. Math. 78, 46–62 (1874)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Rényi, A.: On the representation of an even number as the sum of a single prime and a single almost-prime number. Dokl. Akad. Nauk SSSR (N.S.) 56, 455–458 (1947)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Rényi, A.: Un nouveau théorème concernant les fonctions indépendantes et ses applications à la théorie des nombres. J. Math. Pures Appl. 28, 137–149 (1949)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Rényi, A.: On the large sieve of Ju.V. Linnik. Compos. Math. 8, 68–75 (1951)MathSciNetGoogle Scholar
  13. 13.
    Rényi, A.: New version of the probabilistic generalization of the large sieve. Acta Math. Acad. Sci. Hung. 10, 217–226 (1959)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Roth, K.F.: On the large sieves of Linnik and Rényi. Mathematika 12, 1–9 (1965)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Tenenbaum, G.: Introduction to Analytic and Probabilistic Number Theory (3rd edn), Graduate Studies in Mathematics, vol. 163. American Mathematical Society, Providence (2015)CrossRefGoogle Scholar

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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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