Further information about the nonexistence of nontrivial cyclic trajectories
can be obtained by treating the necessary condition (2.24) as an
nonexponential Diophantine equation.
Davidson  calls a purely periodic trajectory of period k
a circuit if there is a value i for which
i.e., the parity vector
has the special form
The cycle starting with is a circuit.
Davidson observed that each solution to the exponential
gives rise to a circuit of length k = a + b with
and , and conversely.
(The equation (2.28) is the necessary condition (2.24) specialized
to the vector (2.27).
R. Steiner  showed that
is the only solution of (2.28), thus proving the following result.