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Re: 2^57,885,161 1
Posted:
Feb 8, 2013 7:45 AM


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).
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.



