Hello all, I'm trying to learn more about Fermat Numbers. My textbook claims that:

Fermat Numbers are defined by F[m]= (2^(2^m)) + 1. It says that it can be problem that for m!=n that (F[m],F[n])=1 , where [ ] denotes subscripts.

It says the proof involved first proving F[m+1]=F[0]*F[1]*...*F[m]+2 , and that this can be proved by trying to represent F[m+1] in terms of F[m]. How does this work?

Next question (related):

My book also says that by assuming that the Fermat Numbers F[m] are pairwise relatively prime, that we can prove that there are infinitely primes. It says this can be proved only involvingthe resultthat the Fermat numbers are pairwise relatively prime. It says the result is the only thing utilized and not the details of the proof. Can anyone explain how this is actually proven?

Much thanks!