(Terras). For each k there is a finite boundgiven by
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such that
implies that
whenever
. Consequently:
- (i) If
for all
, then there are no non-trivial cycles of length
.
- (ii) If
for all
, then
implies
.
The existence of the boundfollows immediately from (2.14), and (ii) follows immediately from this fact.
To prove (i), suppose a nontrivial cycle of length
exists. We observed earlier that if
is the smallest element in a purely periodic nontrivial cycle of length
, then
and
. The first part of the theorem then implies that
. This contradicts the hypothesis of (i).