Define what it means for a number $N$ to be a pseudoprime to the base $b$.

Show that if there is a base $b$ to which $N$ is not a pseudoprime, then $N$ is a pseudoprime to at most half of all possible bases.

Let $n$ be an integer greater than 1 such that $F_{n}=2^{2^{n}}+1$ is composite. Show that $F_{n}$ is a pseudoprime to the base 2 .