(Terras). (a) The set of integers with coefficient stopping time k are exactly the set of integers in those congruence classesfor which there is an admissible vector
of length k with
.
(b) Let
for some vector
of length k. If
is admissible, then all sufficiently large integers congruent to
have stopping time k. If
is not admissible, then only finitely many integers congruent to
have stopping time k.
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 k, then
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and so all elements of
have stopping time at least k. Now define
by
![]()
where
, and note that (2.11) implies that
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for all admissible
. Now for
for an admissible
, (2.4) may be rewritten as
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Hence when
is admissible, those n in
with
![]()
have stopping time k, 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
is inflating. If
is inflating,
so that
for all n in
by (2.4), so that no elements of
have stopping time k or less. In the remaining case
has an initial segment
with i < k -1 which is admissible. Now
and all sufficiently large elements of
have stopping time i + 1 < k by the argument just given.