Gordon Spence writes:
>>>What bothers me most I guess is that the ordinary humble joe in the street
>>>being realistic has no chance whatever of finding the next mersenne prime.
This is a slight exaggeration. If you're running LL tests or double-checks at all,
you've more chance than if you're not.
Also, the "Big Bands" seem to be largely concentrating on first tests of largish
exponents using lots of powerful systems, leaving double-checking to "ordinary
humble joes".
Taking the most recent figures / estimates available, there are approximately
20,000 Mersenne numbers less than "Spence's Number" M(2976221) which have been
tested only once. If the error rate of 1 in 200 is accepted, approximately 100 of
these residuals will be incorrect. The average of the exponents is approx. 2.4
million (by visual inspection of the table), and George's "a priori" estimate that
an untested exponent around 2.4 million is 1 in 10,000. Therefore there is about 1
chance in 100 that another Mersenne prime less than Spence's Number remains to be
found.
(All above worked to very few significant figures, since so much is (hopefully)
educated guess anyway)
A couple of associated questions:
1. Does anyone know where the estimate of an exponent yielding a Mersenne prime
that Prime95 outputs in its status pop-up comes from? (George, I can see &
understand the source code, what I would like to know is the source of the
algorithm)
2. Does our continued effort in eliminating possible Mersenne primes change that
estimate in any way? Since I joined the project 10 months ago, we have found no
new Mersenne primes (any day now?) but eliminated a great many exponents,
nevertheless the formula doesn't seem to have changed, which it probably should
have done if it's (even partly) based on empirical data.
Regards
Brian Beesley
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