Usage of Fermat primes in RSA Why are Fermat primes ($2^{2^n}+1$) useful as $e$ (the public key) in RSA encryption?

I understand why $2^n+1$ primes are useful, because they would simply be 100...001 in binary, which for computers is faster in calculations. But why is it important that the exponent is also a power of 2?
All primes of the form $2^n+1$ have the form $2^{2^n}+1$.

As the wikipedia article on Fermat numbers says:


  If $2^n + 1$ is prime, and $n &gt; 0$, it can be shown that $n$ must be a power of two. (If n = ab where $1 ≤ a$, $b ≤ n$ and $b$ is odd, then $2^n + 1 = (2^a)^b + 1 ≡ (−1)^b + 1 = 0 \pmod{2^a + 1}$. See Sec. 5 for complete proof.) In other words, every prime of the form $2^n + 1$ is a Fermat number, and such primes are called Fermat primes.


The only known numbers with this format are 3, 5, 17, 257 and 65537.