On the order of unimodular matrices modulo integers
P. Kurlberg
Abstract
Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N(1-ε) for all N in a density one subset of the integers. If A is a hyperbolic unimodular matrix with integer coefficients, then the order of A modulo p is greater than p(1-ε) for all p in a density one subset of the primes. Moreover, the order of A modulo N is greater than N(1-ε) for all N in a density one subset of the integers.
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