Let f be a continuous map of the circle into itself and let
P
(
f
)
P(f)
denote the set of positive integers n such that f has a periodic point of period n. It is shown that if
1
∈
P
(
f
)
1\, \in \,P(f)
and
n
∈
P
(
f
)
n\, \in \,P(f)
for some odd positive integer n then for every integer
m
>
n
m\, > \,n
,
m
∈
P
(
f
)
m\, \in \,P(f)
. Furthermore, if
P
(
f
)
P(f)
is finite then there are integers m and n (with
m
⩾
1
m\, \geqslant \,1
and
n
⩾
0
n\, \geqslant \,0
) such that
P
(
f
)
=
{
m
,
2
m
,
4
m
,
8
m
,
…
,
2
n
m
}
P(f)\, = \,\{ m,\,2\,m,\,4\,m,\,8\,m,\,\ldots ,\,{2^n}\,m\}
.