What is Chris Caldwell saying here? There are certainly conjectures
that Mersenne primes are infinite in number - see the page CC hosts at
http://primes.utm.edu/mersenne/index.html#unknown which says:
Are there infinitely many Mersenne primes?
Equivalently we could ask: Are there infinitely many even perfect
numbers? The answer is probably yes (because the harmonic series
diverges).
This nebulous claim has been around for a number of years, and I'm not
sure it holds water.
Elsewhere, there is the Lenstra–Pomerance–Wagstaff conjecture,
suggesting that the number of Mersenne primes with exponent p less
than x is asymptotically approximated by
e^y x log_2(x) where y is the Euler-Mascheroni constant.
The mainstream view over the years in this mailing list seems to have
been that there is an infinite number of MPs, and those (like me) who
think otherwise have sometimes been treated like flat-earthers. So has
there been a change of mood, or is Chris saying simply that a *proof*
is a long way away? - Yes, I know that mathematics is all about
proofs.
My view is that MPs are so sparse, and getting sparser the more we
know, that it's unlikely that they are infinite in number. But if a
proof should appear, one way or the other, I'd be happy to take it on
board.
Ian
On 02/11/2007, Chris Caldwell <[EMAIL PROTECTED]> wrote:
> > It's a while since this discussion has had an airing. Good to
> > hear that you're acknowledging that there is still no proof
> > of an infinite number of Mersenne primes.
>
> Of course he is! There is some *hope* number theorists
> are now closing in on a proof that there are infinitely
> many twin primes. But a sequence as sparse as the
> Mersennes? That is not even a good pipe-dream right
> now.
>
> CC
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--
Ian W Halliday, BA Hons, SA Fin, MIMIS, ATMG, CL
+44 772 546 2965 (GMT)
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