if $L \geq 2$ has the property that whenever $n\equiv m\mod L$, we have $n^n \equiv m^m\mod p$, then $p(p - 1) \leq L.$
Let $p$ be prime. Consider the sequence: $1, 2^2, 3^3, 4^4, 5^5, 6^6 \cdots \pmod p$. Prove the sequence is periodic with smallest period $p(p-1).$