Exercise fermat numbers

Question

Someone can help me with this problem? $F_p=2^{2^p}+1$

1. Prove that for $2^n+1$ be prime, n have to be a power of 2.
2. Prove that for $k\ge1$ $F_p \mid F_{p+k}-2$
3. Deduce that $F_p$ and $F_{p+k}$ are primes between them.
4. Deduce that there are an infinity of prime numbers.

I don't know how to do the 1 nor the 4.

Answer 1

Hint:

For 2), use induction on $k$, remembering that $2^{2^{p+k}}=\bigl(2^{2^p}\bigr)^{2^k}$.

For 4): use that each Fermat number has a prime factor.