Theorem H

(Terras). For each k there is a finite bound given by


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 .

Proof H

The existence of the bound follows 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).