Re: Primes of the form 2^(2^n)+1

Gerry Myerson <gerry@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
pauldepstein@xxxxxxx wrote:
Gerry Myerson <ge...@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

I think I used the phrase "more surprising," rather than "less likely."
I was making a comment on the people, not the problem. As Bob
Silverman pointed out, and Bill Dubuque has reminded us, there's
a heuristic reason for believing there are only finitely many Fermat
primes. I suppose it was be just as surprising to find that there are
infinitely many, as to find that it's undecidable.

And how about the degree of surprise if it turned out that only
finitely many Fermat numbers are composite? Apparently, that one's
unsolved too!

And what if it turned out that, from some point on, the pattern of
primes and composites among the Fermat numbers was exactly the pattern
of zeros and ones in the binary expansion of pi?

So far as I know, nothing is known about the primality of Fermat
numbers, beyond what the computers have been able to reach, so
anything you write down about them, however silly, is an unsolved
problem.

This implies that, before I reminded you of the well-known heuristic
argument for their finiteness, you recalled absolutely nothing about
them "beyond what the computers have been able to reach". Why, then,
did you state that "It would be a huge shock if something as
(logically) simple as the existence of infinitely many Fermat primes
turned out to be undecidable"? I am genuinely interested in your
reasoning process here. Is it only because the proposition is so
"(logically) simple" (whatever that means) or is there something else
behind your belief that this particular problem should be decidable?

--Bill Dubuque
.

Relevant Pages

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