TITLE: Converse of Fermat's Little Theorem.
QUESTION [5 upvotes]: If $a^n\equiv a \pmod n$ for all integers $a$, does this imply that $n$ is prime?
I believe this is the converse of Fermat's little theorem.

REPLY [2 votes]: Here is one more example: 
We see that $341 = 11 \cdot31$ and $2^{340} = 1\mod341$ 
To show this we see that by routine calcultions the following relations hold $$2^{11} = 2\mod31  $$
$$ 2^{31} = 2 \mod 11$$
Now by using Fermat's little theorem $$ ({2^{11}})^{31} = 2^{11} \mod 31 $$ but $2^{11} = 2 \mod 31$ so I leave you to fill the details of showing $2^{341} = 2 \mod 341$.