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 [29] calls a purely periodic trajectory of period **k**
a * circuit* if there is a value **i** for which
and
i.e., the parity vector
has the special form
where .
The cycle starting with is a circuit.
Davidson observed that each solution to the exponential
Diophantine equation
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 [64] showed that
is the only solution of (2.28), thus proving the following result.