In the excellent article on Mersenne primes

http://www.utm.edu/research/primes/mersenne/

I found the following, to me, cryptic passage

"Are there infinitely many Mersenne primes? The answer is  probably[ital] yes (because the harmonic series diverges). Could someone please explain the "because"? Does the same argument work for Fermat primes? for primes of the form 10^n + 3? for (a^ n)+b, gcd[a,b]=1?