# A Lower Bound for Periods of Matrices

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## Abstract

For a nonsingular integer matrix *A*, we study the growth of the order of *A* modulo *N*. We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field. For exceptional matrices, it is easily seen that there are arbitrarily large values of *N* for which the order of *A* modulo *N* is logarithmically small. In contrast, we show that if the matrix is not exceptional, then the order of *A* modulo *N* goes to infinity faster than any constant multiple of log *N*.

## Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing
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© Springer-Verlag Berlin Heidelberg 2004