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 andNow 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.
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.