The Fermat numbers \(F_n\) are defined by \(F_n=2^{2^n}+1\), \(n\in\mathbb{Z}^+\).

Show that, for any positive integer \(k\), \(F_m\) divides \(F_{m+k}-2\), and deduce that any two distinct Fermat numbers are relatively prime.

Deduce that there are at least \(n+1\) distinct prime numbers less than or equal to \(F_n\), for any \(n\), and hence that there are infinitely many prime numbers.