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Topic: 2^57,885,161 -1
Replies: 10   Last Post: Feb 9, 2013 1:28 PM

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David Bernier

Posts: 3,276
Registered: 12/13/04
Re: 2^57,885,161 -1
Posted: Feb 8, 2013 7:45 AM
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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=largest-prime-number...
>>>>> 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_product-moment_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.



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