(Terras).For eachkthere is a finite bound given bysuch 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 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).