Prove $c,c^c,c^{c^c},c^{c^{c^c}},\ldots \pmod p$ with $p$ prime has period $1$ or $2$

Question

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

Answer 1

update
If you actually wanted to prove that $ \,^nc \pmod p$ converge to a constant sequence then the following might help for a proof:

assume $p=7$, $c=5$
Then what you ask for is whether $\{5, 5^5 , 5^{5^5}, ...\} \pmod 7$ converge to a constant sequence.
This is a question of recursive application of "order of cyclic subgroup" :

• we know $5^k \pmod 7= 5 ^ { k \pmod {\varphi(7)}}\pmod 7= 5 ^ { k \pmod 6}\pmod 7$
• Now if $k$ is itself a power of $5$ then we ask furtherly for the residues of $k=5^j \pmod 6$.
• we find that this is $5^{j \pmod 2} \pmod 6$
• And one step higher this becomes a constant sequence.

Here it is surely meaningful to try a proof. (I think, this should be easy to derive one from that example)

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old version (removed. You can see it in the "edit-history")

Answer 2

That’s false. Consider $c=3$ and $p=7$. Note that $c^2=2$, $c^4=4$ mod $7$. So depending on the value of $a_0$, you can have two constant sequences. Worse: with $a_0=3$, the sequence is a cycle $3,6,1,3,6,1,\ldots$.