Suppose $p$ is a prime number and $c$ is some constant value which is coprime to $p$. I found that $c,c^c,c^{c^c},c^{c^{c^c}},\ldots \bmod p$ have period $1$ or $2$. 

In other words, it seems sequence,$(\ ^nc )\_{\geq1}$, have period $1$ or $2$.

Can we prove this?

The code which I used is here: [link](http://wandbox.org/permlink/VLWZrrutUiQaL7NV)