Given an in integer $n$, and let $p$ be its smallest prime divisor (you can assume that $p$ is very large ). Let $ord_n(i)$ denotes the order of $i$ as an element of $\Bbb Z_n^*$ the multiplicative group of $\Bbb Z_n$.

**My question**:

What is the best known lower bound for: $$\max_{2\leq i\leq p-1}(ord_n(i)) $$

**A lower bound**

For example it's clear that if $k=ord_n(2)$ then $$2^k>n$$ because $$2^k\equiv 1 \mod n $$

so as a lower bound we have $\log_2(n) \leq \max_{2\leq i\leq p-1}(ord_n(i))$ I'm looking for some references which discuss this problem.

Crossposted at MSE: https://math.stackexchange.com/questions/1222761