One can easily show that for any given length k there are only a finite
number of integers n that are periodic under iteration by T with period k,
in fact at most such integers, as observed by
Böhm and Sontacchi [13].
To see this, substitute the equation (2.4) into
to obtain the equation
There are only choices for the 0-1 vector ,
and for each choice of the equation (2.24)
determines a unique rational solution .
Consequently there are at most solutions to (2.23).
Böhm and Sontacchi also noted that this gives an (inefficient)
finite procedure for deciding if there are any cycles of a given length k,
as follows:
Determine the rational number for each of the
vectors , and for each which is an integer test if
(2.23) holds.