At 10:56 AM 5/16/2001 -0000, "Brian J. Beesley" <[EMAIL PROTECTED]> wrote:
>Another point - we're coming up to the second anniversary of the
>discovery of M38(?) - I think we're overdue to find another one!
It would be nice to find another soon. But I don't think we're overdue.
Long ago in Internet time I wrote:
Date: Thu, 29 Jan 1998 22:57:31 -0600
"In mersenne primality testing, exponents double in magnitude are much
more than double the work. For prime95 or the related versions,
each iteration takes about 2.1 times as long as an iteration for
half the FFT length, and there are twice as many iterations to perform,
per exponent. The odds of a number being prime diminishes as the
magnitude increases. Twice the FFT length is usable up to not quite twice
the exponent.
An interval of n to 2n contains nearly twice as many exponents as
the interval 1/2 n to n. In either of these intervals we expect
based on experience, to find about the same number of mersenne primes
on the average. (The actual rate of occurrence seems to be a little
bit in our favor for finding more primes in higher intervals, but it's
slight.) That makes about 3 factors of 2.
I think to maintain a constant rate of discovery of new primes, we would
need to maintain about an 8 or 9-fold increase of computing resources per
period of exponent doubling. Since exponent doubling has occurred in
about the past year, most of this must come in rapid growth in the cpu
pool, both by upgrades and by new membership. Otherwise we can expect
to droop back to a lower discovery rate. On average there are less than
2 mersenne primes per exponent doubling:
36 / [ln(2976221)/ln(2)] = 1.67 mersenne primes per doubling of exponent,
or about 37 / [ln(~3000000)/ln(2)] = 1.72 Mp's per doubling of p
(though we may have yet to find one in p < 2976221, or slightly above!)
In the long run the present discovery rate is unsustainable.
Even if we do drop back to a discovery rate of one per two years,
from the recent ~2 per year, we will have moved this area ahead by
years from its old curve. (The discovery rate was about 1 per 2 years
over the past 20 year interval and for the past 40 year interval.)"
Our experience described on http://www.mersenne.org/history.htm
bears this out. And there is no particular reason to expect the time
interval,
or the spacing between mersenne primes on a log or linear scale to be uniform.
It should vary about the expected curve.
Ken Kriesel
_________________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
