Geodesic
Distributional properties of powers of matrices
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 801-817.

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We apply the larger sieve to bound the number of $2\times 2$ 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.
DOI : 10.1007/s10587-014-0133-6
Classification : 11C20, 11L05, 11N36, 11Z05
Keywords: larger sieve; pseudorandom number; finite field; special linear group of degree 2; general linear group of degree 2 
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Chamizo, Fernando; Raboso, Dulcinea. Distributional properties of powers of matrices. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 801-817. doi : 10.1007/s10587-014-0133-6. http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0133-6/

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