How many primes are in this set?

Question

From an old number theory book:

Consider all numbers of the form $$ x_n={2^n+1}, n=0,1,2,\dots $$ How many of them are prime? Mathematica gave $x_0=2,x_1=3,x_2=5,x_4=17,x_8=257, x_{16}=65537$ and is still running!

We know that if it finds a prime like that it will be of the form $x_{2^k}+1$. So we only need to search for index a power of two. Has this been answered yet?

Thanks a bunch!

Answer 1

Those are called Fermat primes, and it is not known how many there are. Only the first 5 are known.