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A first observation is that there are other cycles if negative integers are allowed in the domain of the function. There is a cycle of period 1 starting from

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

The argument of Böhm and Sontacchi is a very general one that makes use only of the fact that the necessary condition (2.24) for a cycle has a unique solution when the values are fixed. In fact, considerably more can be proved about the nonexistence of nontrivial cyclic trajectories using special features of the necessary condition (2.24). For example, several authors have independently found a much more efficient computational procedure for proving the nonexistence of nontrivial cyclic trajectories of period ; it essentially makes use of the inequality

which must hold for satisfying (2.24). This approach also allows one to check the truth of the Coefficient Stopping Time Conjecture for all(Terras).For eachkthere is a finite bound given bysuch that implies that whenever . Consequently:

- (i)
If for all , then there are no non-trivial cycles of length .- (ii)
If for all , then implies .

Theorem H can be used to show the nonexistence of nontrivial cycles of small period by obtaining upper bounds for the and checking that condition (i) holds. This approach has been taken by Crandall [28], Garner [34], Schuppar [61] and Terras [67]. In estimating , one can show that the quantities are never very large, so that the size of is essentially determined by how large

can get. The worst cases occur when is a very close approximation to , i.e., when is a very good rational approximation to . The best rational approximations to are given by the convergents of the continued fraction expansion of . Crandall [28] uses general properties of continued fraction convergents to obtain the following quantitative result.(Crandall).Let be the minimal element of a purely periodic trajectory of periodk. Thenwhere is any convergent of the continued fraction expansion of with .

As an application, use Yoneda's bound [2] that
and choose **j = 13** in (2.26),
noting that and , to conclude that
* there are no nontrivial cycles with period length less than*
275,000.

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

(Steiner).The only cycle that is a circuit is the trivial cycle.

The most remarkable thing about Theorem J is the weakness of its conclusion
compared to the strength of the methods used in its proof.
The proof of Theorem J does have the merit that it shows that the
coefficient Stopping Time Conjecture holds for the * infinite set*
of admissible vectors of the form (2.27).

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