On 17 Jun 99, at 13:36, Halliday, Ian wrote:
> Some considerable while back, there was a lively discussion as to the
> _total_ number of Mersenne primes. I still believe that the number is
> finite, in contrast to what appears to be the majority view: that there is
> an infinity of Mersenne primes out there waiting to be discovered.
You're entitled to your viewpoint, there is no proof one way or the
other. But the circumstantial evidence seems to point to the number
of Mersenne primes being infinite - if you take any integer n, where
the exponents up to 2n have been searched completely, the "best fit"
model is that there as many Mersenne primes between 2^n and 2^(2n)
as there are between 2^(n/2) and 2^n. The "expected number" in an
"octave" is about 1. Now there are an infinite number of "octaves",
so ...
> One correspondent at that time postulated that there were no further
> Mersenne primes to be discovered and that the 37th was also the last. In the
> light of recent events, does anybody have an update on this view?
Bearing the above in mind, even if the number of Mersenne primes _is_
finite, I'd be very surprised if we've found even half of them yet.
I vaguely recall the message, my recollection is that its author
disagreed with conventional mathematicians over the meaning of
"infinity". If he's right, a lot of text books need pulping...
> So far as
> I am aware, M38 has not been confirmed yet. No flames, please: I am _not_
> posting this as an "I told you so" but as a genuine enquiry.
So far as I know, M38 is, at the moment, unconfirmed. But, again so
far as I know, this is because the verification run has not finished,
not because it has finished but proved "unsuccessful". And, again,
the verification run could fail for some reason - in which case, at
least one more double-check/verification run would be needed to
decide the status of the "unverified prime", one way or the other.
Please bear in mind that the chance of a "false positive" being
output from a LL test program is _minute_ - exactly 1 in 2^p-1, in
fact, if the error was a random bit-flip event. It's far more
probable that the "unverified prime" is a consequence of someone
"nobbling" the server. But, seeing that you couldn't possibly lay
hands on any prize money that way, I can't see any reason why a sane
person would want to do that.
Regards
Brian Beesley
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