David Gale wrote:
Sorry, I didn't understand the Kleber non-argument. Presumably at some point it must have used the fact that the Mersennes grow at most exponentially. Where?
Let me rephrase it. The non-argument part is passing from "the ("expected") numer of Mersenne primes" to "the expected number of primes in a sequence of integers (a0,a1,a2,...), where a_n is around 2^p(n)." Here p(n) means the nth prime. This expected value is what is infinite "because the harmonic series diverges." Hope that's clear(er).
So where does this leave us? Does the same kind of (non)-argument suggest that there are only finitely many Fermat primes?
Well, only sort of. Turn it around: obviously, if there were infinitely many Fermat primes, then primes would be more plentiful among Fermat numbers 2^(2^n)+1 than among other similarly-sized numbers. That would be interesting. The non-argument just says "If nothing interesting happens, this is what you'd expect." Luckily for us, math is full of cases where interesting things *do* happen! --Michael Kleber