Distributional properties of powers of matrices
- 70 Downloads
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.
Keywordslarger sieve pseudorandom number finite field special linear group of degree 2 general linear group of degree 2
MSC 201011N36 11C20 11Z05 11L05
Unable to display preview. Download preview PDF.
- J. Friedlander, H. Iwaniec: Opera de Cribro. American Mathematical Society Colloquium Publications 57, Providence, 2010.Google Scholar
- H. Iwaniec, E. Kowalski: Analytic Number Theory. American Mathematical Society Colloquium Publications 53, Providence, 2004.Google Scholar
- L. Kuipers, H. Niederreiter: Uniform Distribution of Sequences. Pure and Applied Mathematics, John Wiley & Sons, New York, 1974.Google Scholar
- W. C. W. Li: Number Theory with Applications. Series on University Mathematics 7, World Scientific, River Edge, 1996.Google Scholar
- R. A. Mollin: Advanced Number Theory with Applications. Discrete Mathematics and Its Applications, CRC Press, Boca Raton, 2010.Google Scholar