For every prime $p$

Fermat's Little Theorem tells us that for each $a$ such that $(a,p)=1$, $a^p \equiv 1 \mod{p}$

Nevertheless, we can observe that some number to a power less than $p-1$ can be congruent to $1$ modulo $p$

For each integer $a$ with $(a,p)=1$, we define the order of a to be the smallest number $d$ such that$a^d \equiv 1 \mod {p}$

For example, for p=11, the order of $1$ is $1$, the order of $2$ is 12, and so on…

I found that there are many special properties of orders

1) the order always divide $p-1$

2) the number of $a$ with $1\leq a \leq p-1$ such that the order of $a$ is $d$ is exactly $\phi (d)$

Hence this leads to a fact $\sum_{d \mid p-1} \phi(d)=p-1$

There are still many amazing properties of order, hope people can discover theorems of themself!