On the order of unimodular matrices modulo integers

Details On the order of unimodular matrices modulo integers On the order of unimodular matrices modulo integers

2002-02-06

arxiv.org/abs/math/0202053v1

Mathematics

Number Theory

12 pages

Scientific paper

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-\epsilon) 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-\epsilon) for all p in a density one subset of the primes. Moreover, the order of A modulo N is greater than N^(1-\epsilon) for all N in a density one subset of the integers.

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