Assume that all prime periods of periodic orbits of a continuous map $f:[0:1]\to [0:1]$ are uniformly bounded (i.e. there exists N such that the prime period of every periodic orbit of f is smaller than N). What can you say about periods of periodic orbits of f? For example, can f have a periodic orbit of period 2007? Of period 2048?
This is just a problem I found online at http://www.its.caltech.edu/%7Easgor/Ma4/ in relation to what I am currently studying, so hopefully I am not missing anything to do it. I'm not sure how to approach this though. The only thing I can think of is that since $f([0,1])\subset [0,1]$ and is continuous, due to the intermediate value theorem, $f$ has a fixed point. But I am not sure on how to work this in, or if it is relevant. I figure the interval $[0,1]$ is somehow relevant, but am not positive.