Journal d'Analyse Mathématique

, Volume 124, Issue 1, pp 117–147 | Cite as

Multiplicative congruences with variables from short intervals

  • Jean Bourgain
  • Moubariz Z. Garaev
  • Sergei V. Konyagin
  • Igor E. Shparlinski


Recently, several bounds have been obtained on the number of solutions of congruences of the type
$$({x_1} + s) \cdots ({x_v} + s) \equiv ({y_1} + s) \cdots ({y_v} + s)\not \equiv 0{\text{ (mod }}p{\text{),}}$$
where p is prime and variables take values in some short interval. Here, for almost all p and all s and also for a fixed p and almost all s, we derive stronger bounds. We also use similar ideas to show that for almost all p, one can always find an element of a large order in any rather short interval.


Nontrivial Solution Asymptotic Formula Prime Divisor Algebraic Number Multiplicative Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Ayyad, T. Cochrane, and Z. Zheng, The congruence x 1 x 2x 3 x 4 (mod p), the equation x 1 x 2 = x 3 x 4, and the mean value of character sums, J. Number Theory 59 (1996), 398–413.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    U. Betke, M. Henk and J. M. Wills, Successive-minima-type inequalities, Discrete Comput. Geom. 9 (1993), 165–175.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    J. Bourgain, M. Z. Garaev, S. V. Konyagin, and I. E. Shparlinski, On the hidden shifted power problem, SIAM J. Comput. 41 (2012), 1524–1557.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    J. Bourgain, M. Z. Garaev, S. V. Konyagin, and I. E. Shparlinski, On congruences with products of variables from short intervals and applications, Proc. Steklov Math. Inst. 280 (2013), 61–90.CrossRefzbMATHGoogle Scholar
  5. [5]
    M. C. Chang, Factorization in generalized arithmetic progressions and applications to the Erdős-Szemerédi sum-product problems, Geom. Funct. Anal. 13 (2003), 720–736.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    M. C. Chang, The Erdős-Szemerédi problem on sum set and product set, Ann. of Math. (2) 157 (2003), 939–957.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    M. C. Chang, Elements of large order in prime finite fields, Bull. Aust. Math. Soc., 88 (2013), 109–176.CrossRefGoogle Scholar
  8. [8]
    J. Cilleruelo and M. Z. Garaev, Concentration of points on two and three dimensional modular hyperbolas and applications, Geom. Funct. Anal. 21 (2011), 892–904.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    P. Erdős and R. Murty, On the order of a (mod p), Number Theory, Amer. Math. Soc., Providence, RI, 1999, pp. 87–97.Google Scholar
  10. [10]
    J. H. Evertse, H. P. Schlickewei, and W. M. Schmidt, Linear equations in variables which lie in a multiplicative group, Ann. of Math. (2) 155 (2002), 807–836.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    J. von zur Gathen and J. Gerhard, Modern Computer Algebra, second edition, Cambridge University Press, Cambridge, 2003.zbMATHGoogle Scholar
  12. [12]
    H. Iwaniec, On the problem of Jacobsthal, Demonstratio Math. 11 (1978), 225–231.zbMATHMathSciNetGoogle Scholar
  13. [13]
    H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Providence, RI, 2004.zbMATHGoogle Scholar
  14. [14]
    M. Mignotte, Mathematics for Computer Algebra, Springer-Verlag, New York, 1992.CrossRefzbMATHGoogle Scholar
  15. [15]
    W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, second edition, Springer-Verlag, Berlin; PWN-Polish Scientific Publishers, Warsaw, 1990.zbMATHGoogle Scholar
  16. [16]
    C. Pomerance and I. E. Shparlinski, Smooth orders and cryptographic applications, Proc. 5-th Algorithmic Number Theory Symp., Lect. Notes in Comput. Sci., vol. 2369, Springer-Verlag, Berlin, 2002, 338–348.CrossRefGoogle Scholar
  17. [17]
    S. Shi, The equation n 1 n 2n 3 n 4 (mod p) and mean value of character sums, J. Number Theory 128 (2008), 313–321.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    T. Tao and V. Vu, Additive Combinatorics, Cambridge University Press, Cambridge, 2006.CrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2014

Authors and Affiliations

  • Jean Bourgain
    • 1
  • Moubariz Z. Garaev
    • 2
  • Sergei V. Konyagin
    • 3
  • Igor E. Shparlinski
    • 4
  1. 1.Institute for Advanced StudyPrincetonUSA
  2. 2.Centro de Ciencias MatemáticasUniversidad Nacional Autónoma de México MoreliaMichoacánMéxico
  3. 3.Steklovmathematical InstituteMoscowRussia
  4. 4.School of Mathematics and StatisticsThe University of New SouthwalesSydneyAustralia

Personalised recommendations