What is the relation between the coefficient stopping time and the stopping time?

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Theorem C shows that generally they are equal: For any fixed k at most a finite number of those n having coefficient stopping time

have

. Terras [67] and later Garner [34] conjecture that this never occurs.

COEFFICIENT STOPPING TIME CONJECTURE.

For all , the stopping time equals the coefficient stopping time .

The Coefficient Stopping Time Conjecture has the aesthetic appeal that if it is true, then the set of positive integers with stopping time k is exactly a collection of congruence classes , as described by part (i) of Theorem C. Furthermore, the truth of the Coefficient Stopping Time Conjecture implies that there are no nontrivial cycles. To see this, suppose that there were a nontrivial cycle of period k and let be its smallest element, and note that . Then for and

Now

since

isn't a power of 2, so that (2.20) implies that

. Hence

, so that

. The following result shows that the Coefficient Stopping Time Conjecture is ``nearly true.'' I will use it later to bound the number of elements not having a finite stopping time.

Theorem E

There is an effectively computable constant such that if is admissible of length , then all elements of have stopping time k except possibly the smallest element of S.

[Proof]

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