  Partitions of the set of primes are introduced based on the Chebyshev
polynomials at rationals. The prime densities of all such partitions are
established. Euler's Criterion for $SL(2,\mathbb Q)$ is formulated, which is
the bridge between the algebra of Chebyshev polynomials and number-theoretic
properties of the partitions. It is shown how to obtain in this way some of the
classical theory of Lucas sequences. A hidden symmetry of the problem is
revealed by the new language. As an application number-theoretic properties of
simple dynamical systems (rotations and certain interval maps) are discussed.
