The correct usage of exponent towers

Let $P_n$ be the power tower involving $n$ 2's, for example, $P_3=2^{2^2}$ ; note that we have, recursively, $P_n=2^{P_{n-1}}$ . We claim that $P_{n+1}\equiv P_n \pmod{2018}$ for $n\geq 4$ , meaning that our sequence $P_n \pmod{2018}$ becomes stationary at $n=4$ . Note that $P_m$ is even for positive $m$ and divisible by 16 for $m\geq 3$ , so that we can reduce our moduli below to the largest odd factors.

To show that $P_{n+1}\equiv P_n \pmod{2018}$ or $\pmod{1009}$ is suffices to consider the exponents and show that $P_n\equiv P_{n-1} \pmod{1008}$ , by Fermat's Little Theorem. Now $1008=16\times63$ , and it suffices to show that $P_n\equiv P_{n-1} \pmod{63}$ .

To show that $P_n\equiv P_{n-1} \pmod{63}$ , it suffices to show that $P_{n-1}\equiv P_{n-2} \pmod{6}$ or $\pmod{3}$ , by Euler's Theorem, since $\phi(7)=\phi(9)=6$ . Now $P_{n-1}$ and $P_{n-2}$ can be written as powers of 4, so that $P_{n-1}\equiv P_{n-2}\equiv 1 \pmod{3}$ .

The tower becomes stationary at $P_4= 2^{16}=2048\times32\equiv30\times32= \boxed{960} \pmod{2018}$ .