Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
David Bernier

Posts: 3,210
Registered: 12/13/04
Re: 2^57,885,161 -1
Posted: Feb 9, 2013 1:28 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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=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).


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.



Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.