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Re: 2^57,885,161 1
Posted:
Feb 9, 2013 1:28 PM


On 02/08/2013 07:45 AM, David Bernier wrote: > On 02/07/2013 03:56 PM, Pubkeybreaker wrote: >> On Feb 7, 2:00 pm, Transfer Principle<david.l.wal...@lausd.net> >> wrote: >>> On Feb 7, 7:17 am, Frederick Williams<freddywilli...@btinternet.com> >>> wrote: >>> >>>> Sam Wormley wrote: >>>>> Largest Prime Number Discovered [to date] >>>>>> http://www.scientificamerican.com/article.cfm?id=largestprimenumber... >>>>>> >>>>>> The number ? 2 raised to the 57,885,161 power minus 1 ? was >>>>>> discovered by University of Central Missouri mathematician Curtis >>>>>> Cooper as part of a giant network of volunteer computers devoted to >>>>>> finding primes, similar to projects like SETI@Home, which downloads >>>>>> and analyzes radio telescope data in the Search for Extraterrestrial >>>>>> Intelligence (SETI). The network, called the Great Internet Mersenne >>>>>> Prime Search (GIMPS) harnesses about 360,000 processors operating at >>>>>> 150 trillion calculations per second. This is the third prime number >>>>>> discovered by Cooper. >>>> By Cooper or by GIMPS? >>> >>> By Cooper. GIMPS itself has discovered 14 primes. >>> >>> http://en.wikipedia.org/wiki/Great_Internet_Mersenne_Prime_Search >> >> Cooper found the specific prime, BUT it was a GIMPS >> *** group effort** that sifted through many many thousands of >> candidates >> and eliminated them as possibilities. > > Yes, I agree Dr Silverman. > > This 48th known Mersenne prime has been added to Chris Caldwell's > Prime Pages: > > http://primes.utm.edu/notes/faq/NextMersenne.html > > There's the heuristic that if y_n = log_2(log_2(M_n)), where > M_n is the n'th Mersenne prime, then the y_n resemble the > arrival times in a Poisson process. > > For the known y_n, n=1 to 48, I get: > > 0.664 > 1.489 > 2.308 > 2.805 > 3.700 > 4.087 > 4.247 > 4.954 > 5.930 > 6.475 > 6.741 > 6.988 > 9.025 > 9.245 > 10.320 > 11.105 > 11.155 > 11.651 > 12.054 > 12.110 > 13.242 > 13.279 > 13.452 > 14.283 > 14.405 > 14.502 > 15.441 > 16.396 > 16.753 > 17.010 > 17.721 > 19.529 > 19.713 > 20.262 > 20.415 > 21.505 > 21.526 > 22.733 > 23.682 > 24.323 > 24.518 > 24.630 > 24.857 > 24.957 > 25.147 > 25.345 > 25.361 > 25.786 (48 values). > > Chris Caldwell states that this gives: > "a correlation coefficient R^2 = 0.9919". > > I get that this is the "Coefficient of determination" R^2: > http://en.wikipedia.org/wiki/Coefficient_of_determination > > the square of Pearson's r: > http://en.wikipedia.org/wiki/Pearson_productmoment_correlation_coefficient#For_a_sample > > > These commands show my computation of R^2: > > ? ybar = (sum(X=1,48,y[X]))/48.0000000000000 > %94 = 14.538509898083550641780687795589356676 > > ? A = sum(X=1,48, (x[X]  xbar)*(y[X]ybar)) > %95 = 5088.1949729587201638754833533863947090 > > ? B = sum(X=1,48, (x[X]xbar)^2) > %96 = 9212.0000000000000000000000000000000000 > > ? C = sum(X=1,48, (y[X]ybar)^2) > %97 = 2833.5016529801736003925746709139375204 > > ? B=sqrt(B) > %98 = 95.979164405614617749102200287187885447 > > ? C = sqrt(C) > %99 = 53.230645806529283553703786993969466052 > > ? r = A/(B*C) > %100 = 0.99592135484160784915459792443098549831 > > ? R2 = r*r > %101 = 0.99185934502954377404245361220842196837 > > ? R2 > %102 = 0.99185934502954377404245361220842196837 > > Let's compare: Caldwell obtains R^2 = 0.9919 (Ok). > > === > > Now, Poisson processes are memoryless > http://en.wikipedia.org/wiki/Poisson_process > > So, if Y_i for i=1 to 48 are the arrival times (of the event number > i) in a Poisson process, a sample, > > and X_i = i for i = 1 to 48, what is the distribution of > Z= R^2, the square of Pearson's r ? > > And in particular what are the chances that Z >= 0.9919 ? > (the R^2 for the Mersennes).
In 1,000,000 trials with 48 arrival times each,
the R^2 surpasses 0.9919 189,700 times or about 19% of the time.
David Bernier
 dracut:/# lvm vgcfgrestore File descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID 993: sh Please specify a *single* volume group to restore.



