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 2006-09-15, 16:27 #45 ewmayer ∂2ω=0     Sep 2002 República de California 5·7·331 Posts Please Don't Flame each other over something this silly.
 2006-09-15, 19:42 #46 philmoore     "Phil" Sep 2002 Tracktown, U.S.A. 45E16 Posts Ernst, it wasn't intended as a flame, I really just didn't understand what Phil was referring to. Thanks for the explanation, Drew. I should teach probability sometime and learn the proper terminology, as my knowledge of it is more based on my experience in physics. However, there is an interesting point about Poisson processes, and that is the long tail that skews the mean to be considerably higher than the median. So if the mean ratio between successive Mersenne prime exponents is roughly 1.48 (assuming none have been missed less than the most recent), the median should be 1.48^(ln 2), or about 1.31. Someone please correct me if I have made a mistake, but it seems to me that we would expect the ratio to be less than this about half the time. Looking at the actual data, we see a median of about 1.41, but this may be somewhat skewed by the small exponents. If we consider only the Mersenne primes discovered by computer, the median is about 1.322, which is pretty close to the median of our guesses so far; I think wpolly's guess of M41991811 is the actual median (ratio of 1.29). So as a group, it looks like our guesses are fairly reasonable, although I detect a faint bias toward the "we're due for a big gap" way of thinking.
2006-09-15, 19:52   #47
ewmayer
2ω=0

Sep 2002
República de California

5×7×331 Posts

Quote:
 Originally Posted by philmoore Ernst, it wasn't intended as a flame, I really just didn't understand what Phil was referring to.
No worries - I was mainly trying to find a cute alternative acronymical mapping for "PDF."

Quote:
 as a group, it looks like our guesses are fairly reasonable, although I detect a faint bias toward the "we're due for a big gap" way of thinking.
Part of this is likely due to the "tens tails in a row .. I'm due for a heads..." style of gambler's fallacy, but perhaps there's also a simple conservatism at work here - guess a big gap, and hope to be pleasantly "disappointed."

Last fiddled with by ewmayer on 2006-09-15 at 19:53

 2006-11-20, 04:10 #48 billymac00   Nov 2006 SW Virginia USA 2 Posts 45th MPE from basic fit, "prediction" puts at 8.6E7 assuming no previous gaps
 2006-11-23, 02:20 #49 PrimeCrazzy     Dec 2005 168 Posts Predict m45 I predict it will be less than 40,000,000 and found in September 2007 Last fiddled with by PrimeCrazzy on 2006-11-23 at 02:21
 2006-12-23, 15:19 #50 davieddy     "Lucan" Dec 2006 England 2×3×13×83 Posts After consulting Chris Caldwell's page "Where is the next Mersenne Prime" It seems there is a 50% chance that it will be less than 43,000,000. However, I don't think he has taken all the testing up to 37,000,000 which has been done since the page was last updated. David Eddy
 2006-12-23, 17:21 #51 davieddy     "Lucan" Dec 2006 England 194A16 Posts Where is 45th Mersenne Prime 50% chance of it being less than 2^43,000,000 judging by Chris Caldwell's page "Where is the next Mersenne Prime" But I don't think it takes into account all the testing of exponents <37,000,000. David Eddy
 2007-01-11, 07:57 #52 T.Rex     Feb 2004 France 39216 Posts Chris Caldwell's page URL. Where is the next Mersenne Prime Within figures 1 and 2, I see "waves" around the theoretical curve. Really clear for fig 2. It really does not look as random values above or below the curve. I guess there is a way to check if the values appear to be random or not. Who can provide the theory and do the computation ? T. Last fiddled with by T.Rex on 2007-01-11 at 08:05
2007-01-11, 08:44   #53
davieddy

"Lucan"
Dec 2006
England

2·3·13·83 Posts

Quote:
 Originally Posted by T.Rex Where is the next Mersenne Prime Within figures 1 and 2, I see "waves" around the theoretical curve. Really clear for fig 2. It really does not look as random values above or below the curve. I guess there is a way to check if the values appear to be random or not. Who can provide the theory and do the computation ? T.
Much of the page is devoted to confirming that the occurence
of Mersenne primes is a "Poisson" process, i.e. random.

For example, the discovery of M44 didn't alter the "expected"
primes in each range of the status page at mersenne.org
in the slightest.

the easiest way to keep up to date on the probability
(or lack of it!) of finding a new prime.

If the expected number of primes in a range is x, then the
probability of finding none is e-x

David

 2007-01-11, 08:55 #54 davieddy     "Lucan" Dec 2006 England 2×3×13×83 Posts I do see what you mean about "waves" however. My guess is that if you chose 44 mumbers at random between one and a thousand, then plotted them in order and drew the best fit straight line, you would see a similar effect. David
2007-01-11, 09:12   #55
drew

Jun 2005

38210 Posts

Quote:
 Originally Posted by T.Rex Where is the next Mersenne Prime Within figures 1 and 2, I see "waves" around the theoretical curve. Really clear for fig 2. It really does not look as random values above or below the curve. I guess there is a way to check if the values appear to be random or not. Who can provide the theory and do the computation ? T.
It's our nature as humans to look for patterns in randomness. You may see waves, but if you tried to derive the wavelength, you'd realize there's no real way to characterize the behavior, assuming you have a large enough sample. A fourier transform of the deviation wouldn't reveal discernable pattern.

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