## Friday, February 25, 2005

### New record prime

The Mersenne numbers are the numbers of the form "2^p - 1". Several centuries ago, some people used to think that these numbers were always primes if "p" itself is a prime. While it's true for the first few values, the general claim is, of course, false. Primality of "p" is a necessary condition for primality of "2^p - 1", but not a sufficient one. (It's necessary because if "p=qr", then "2^q - 1" and "2^r - 1" are divisors of "2^p - 1".)

In fact, only 41 Mersenne primes are officially known. The highest one, "2^24,036,583 - 1", is also the greatest officially known prime integer in the world. It was found in May 2004. Usually, the exponent roughly doubles if you want to find a new Mersenne prime. However, the Mersenne prime numbers look surprisingly dense for the exponents between 20 million and 26 million.

An international network of computers GIMPS whose website is at

is looking for new Mersenne primes. You may join. One can see that the main page of this server announces that a new record prime may have been found. It would be the 42nd known Mersenne prime number. I predict that the exponent "p" will be

• "p = 25,964,951"

Note that if you find the first Mersenne prime with at least 10 million digits, you will win at least 1/2 of the \$100k award from the EFF foundation. It takes roughly one month for a 3GHz computer to test (using the Lucas-Lehmer test) a number like "2^34,362,227 - 1" which is how a candidate for a 10-million-digit prime number looks like.

Note added later: Of course that the word "predict" was partially a joke. I knew that the exponent would be what I wrote. Congratulations to Dr. Martin Nowak from Germany (by the way, it's almost a Czech name) to the discovery of the 42nd known Mersenne prime.

1. Mersenne primes are in Topological Geometrodynamics framework the most interesting primes since they correspond to most important p-adic length scales. Only Mersennes up to M_127 =2^127-1 are interesting physically since next Mersenne corresponds to a completely super astrophysical length scale. M_127 corresponds to electron whereas M_107 corresponds to hadronic length scale. M_89 corresponds to intermediate boson length scale.

There is an interesting number theoretic conjecture due to Hilbert that iterated Mersennes M_{n+1}= M_{M_n} form an infinite sequence of primes. 2,3,7,127,;M-_{127},.... etc. Quantum computers would be needed to kill the conjecture. Physically this hierarchy would be also very interesting.

Also Gaussian primes associated with complex integers are important in TGD framework. Gaussian Mersennes exist also and correspond to powers p=about 2^k, k prime.

k=113 corresponds to the p-adic length scale of muon and atomic nucleus in TGD framework.

Neutrinos could correspond to several Gaussian Mersennes populating the biologically important length scales in the range 10 nanometers 5 micrometeres. k=151,k=157,k=163, k=167 all correspond to Gaussian Mersennes. There is evidence that neutrinos can appear with masses corresponding to several mass scales.

That Gaussian Mersennes populate the biologically most interesting length scale range is probably not an accident. The hierarchical multiple coiling structure of DNA could directly correspond to these Gaussian Mersennes.

Matti Pitkänen

2. Matti that does not even look like a joke, if it was not for your teasing of Lubos with the word "TGD". Hey Lubos are you going to ask a second time what is "TGD"?

Your "prediction" of "p = 25,964,951" is just silly. There is no prediction. Any one can just download a copy of software from GIMPS and watch what kind of data is being sent over from the GIMPS network, then you will know for sure what Mersenne number they are trying to verify right now. So if your "prediction" comes out right, no big deal. But if it is not right, then too bad for you.

Quantoken

3. Matti - well, let me admit that I don't believe any of the things about "TGD" you wrote.

Quantoken - believe me that at a given moment of time, thousands or tens of thousands of Mersenne numbers are being tested, only one in hundreds of thousands turns out to be prime, and if you download the client, you won't be able to say anything about the new one.

In other words, what you wrote is a complete stupidity once again.

4. There is not much I could say, except that I know that p is not 24142693, 24251971, 24780803, 24844957, 25487719, 25799551, 26262007, 26586961, 27050717, nor 27612973... You can choose among the rest of candidates. :o)

5. This comment has been removed by a blog administrator.

6. OK, so your "prediction" was correct. Now I would only like to know, what sources did you have? :-/

7. Hi Mormegil,

maybe I listed all numbers between 24,000,000 and 26,500,000. Then I picked all prime integers by Mathematica. That's a couple of hundreds of thousands of candidates. It's then enough to browse various blogs and collect the information from people like you who say "it's certainly not this and that". You eliminate most candidates.

You're left with roughly 10,000 candidates, and you just make the Lucas-Lehmer test for them with a pen and a piece of paper. ;-)

Thanks,
Lubos

8. Who are you kidding, Lubos? I believed you did figure it out to begin with. But nobody would believe that is how you figured it out! How much manpower does it take to search all blogs and eliminate hundreds of thousands of candidates? What kind of Mersenne number can you do Lucas Lehmer test using paper and pencil only?

Here is a challenge. I used paper and pencil to figured out a Mersenne number which is 2^26335769 - 1. You can pick and choose one of any of the three challenges below:

1.Figure out what's special about this number and why I picked this one? What's the coincidence I used? or,

2.Show us how you do paper and pencil Lucas Lehmer test of just this one Mersenne candidate. or

3.It is NOT a Mersenne prime. Figure out the smallest factor of this number, either use super computer or paper and pencil. But do it within 24 hours.

9. Oh, come on, Quantoken, don't you recognize a joke? Of course Luboš did not use that, my guess is either he just copied the guess of a GIMPS' participant who wrote the same number on a GIMPS forum, or used the same method as him – went through the list of cleared exponents and filtered them by date and other parameters; I just thought there might be more magic and less work in it. :-)

10. Hi mormegil,

the sad truth is that quantoken really does not understand jokes or anything else for that matter. He is just an obnoxious, belligerent and ignorant heckler. The prototype of a crackpot.

11. Hi Marmegil,

OK, let me admit the truth which is of course less magic. ;-) I looked at the discussions over at mersenneforum.org - and probably the only extra skill I had was to be able to distinguish which of the participants were people connected to the administration of GIMPS so that their "guess" matters. I guess that you (or others) considered these users to be random users among others who just propose random numbers.

All the best
Lubos

12. Is there any practical application for such a big prime numbers?

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The Number Empire