(Terras). (a)The set of integers with coefficient stopping timekare exactly the set of integers in those congruence classes for which there is an admissible vector of lengthkwith .(b)

Let for some vector of lengthk. If is admissible, then all sufficiently large integers congruent to have stopping timek. If is not admissible, then only finitely many integers congruent to have stopping timek.

The assertions made in (a) about coefficient stopping times follow from the definition of admissibility, because that definition asserts thatTo prove (b), first note that if is admissible of length

and so all elements of have stopping time at leastk, thenk. Now define by where , and note that (2.11) implies that for alladmissible. Now for for an admissible , (2.4) may be rewritten as Hence when is admissible, thosenin with have stopping timek, and in this case.Now suppose is not admissible. There are two cases, depending on whether or not some initial segment of is admissible. No initial segment of is admissible if and only if

and when (2.15) holds say that isinflating. If is inflating, so that for allnin by (2.4), so that no elements of have stopping timekor less. In the remaining case has an initial segment withi < k -1which is admissible. Now and all sufficiently large elements of have stopping timei + 1 < kby the argument just given.