Why there are no other known Fermat primes.

Question

Fermat primes are prime numbers of the form $2^{2^n} + 1$: $$3,~5,~17,~257,~65537$$

There are no other known Fermat primes. But why?

Answer 1

One reason is that there are probably no more Fermat primes! If you picked a random odd number near $2^{2^n}$ the chance that it would be prime is roughly $1/\log(2^{2^n})$ or $k/2^n$ for some constant $k$. The sum of $k/2^n$ converges, so there should be finitely many primes. The sum over all $n$ such that it is not known whether $2^{2^n}+1$ is prime or not is tiny, so the expected number of remaining Fermat primes is 0.

Another reason is that Fermat numbers grow so quickly that it's hard to work with them. The n-th term has about $2^n$ bits, so the 23rd Fermat number takes about 1 MB to store, the 33rd takes about 1 GB, the 43rd takes about 1 TB to store, etc. This makes any reasonable primality test very hard to carry out. (On the other hand trial division is still workable, and this is how the character of many of the Fermat numbers was discovered.)