We are discussing Fermat numbers in class, and one of the claims brought up is as follows:

"For any integer $n \ge 1$, the $n$th Fermat number is $F(n)$ = $2 + \prod\_{i=0}^{n-1}F(i)$."

I have not been able to find any proofs online, but I would really like to know how a Fermat number can be represented in this form (as 2 plus the product of past Fermat numbers). Are Fermat numbers not prime, or have I confused myself? It doesn't seem intuitive to me, but I could be wrong.