Prove that $1989\mid n^{n^{n^{n}}} - n^{n^{n}}$

Question

Having difficulty in proving this: $1989\mid n^{n^{n^{n}}} - n^{n^{n}}$ for all $n \in \Bbb N$.

Prime factorization of $1989$ is $3^2 \times 13 \times 17$.

Please Help!

Answer 1

This wrong for $n=2,\;$ because $1989$ does not divide $2^{2^{2^{2}}} - 2^{2^{2}}=2^{16}-16=65520$.