Japanese Journal of Mathematics

, Volume 7, Issue 2, pp 235–294

Modular hyperbolas

Original articles

DOI: 10.1007/s11537-012-1140-8

Cite this article as:
Shparlinski, I.E. Jpn. J. Math. (2012) 7: 235. doi:10.1007/s11537-012-1140-8


We give a survey of a variety of recent results about the distribution and some geometric properties of points (x, y) on modular hyperbolas \({xy \equiv a\;(\mod m)}\). We also outline a very diverse range of applications of such results, discuss multivariate generalisations and suggest a number of open problems of different levels of difficulty.


modular hyperbola congruences exponential sums character sums 

Mathematics Subject Classification (2010)

11D79 11L07 11L40 11N69 

Copyright information

© The Mathematical Society of Japan and Springer Japan 2012

Authors and Affiliations

  1. 1.Department of ComputingMacquarie UniversitySydneyAustralia

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