Czechoslovak Mathematical Journal

, Volume 64, Issue 3, pp 801–817 | Cite as

Distributional properties of powers of matrices

  • Fernando ChamizoEmail author
  • Dulcinea Raboso


We apply the larger sieve to bound the number of 2×2 matrices not having large order when reduced modulo the primes in an interval. Our motivation is the relation with linear recursive congruential generators. Basically our results establish that the probability of finding a matrix with large order modulo many primes drops drastically when a certain threshold involving the number of primes and the order is exceeded. We also study, for a given prime and a matrix, the existence of nearby non-similar matrices having large order. In this direction we find matrices of large order when the trace is restricted to take values in a short interval.


larger sieve pseudorandom number finite field special linear group of degree 2 general linear group of degree 2 

MSC 2010

11N36 11C20 11Z05 11L05 


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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2014

Authors and Affiliations

  1. 1.Department of Mathematics and ICMAT, Faculty of ScienceUniversidad Autónoma de MadridMadridSpain

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