47th Mersenne Prime Confirmed

radiot88 writes to let us know that he heard a confirmation of the discovery of the 47th known Mersenne Prime on NPR's Science Friday (audio here). The new prime, 2^42,643,801 - 1, is actually smaller than the one discovered previously. It was "found by Odd Magnar Strindmo from Melhus, Norway. This prime is the second largest known prime number, a 'mere' 141,125 digits smaller than the Mersenne prime found last August. Odd is an IT professional whose computers have been working with GIMPS since 1996 testing over 1,400 candidates. This calculation took 29 days on a 3.0 GHz Intel Core2 processor. The prime was independently verified June 12th by Tony Reix of Bull SAS in Grenoble, France..."

Comments

Re:Cool processor

[Jarlsberg]

It was found through the GIMPS (The Great Internet Mersenne Prime Search). The site http://www.mersenne.org/prime.htm is currently down.

Odd's prime

[fph il quozientatore]

So, all primes greater than two are odd, but only one of them is Odd's!

Re:Odd's prime

[Melkman]

That's odd when you think about it.

Re:Odd's prime

[bcrowell]

His brother, Even Magnar Strindmo, is also an IT professional. Even, like his brother Odd, has been testing candidates since 1996. The latest candidate in Even's search was 2^42,643,801-2, which was found to be composite. The very next number, 2^42,643,801-1, was the one his brother found to be prime. "Yeah, it kind of hurts to get so close and not be the one who got it," admits Even, "but I gave it my best game. We agreed back in '96 that we'd split up the work and go even-odd. I guess it was just a matter of luck that he got the first prime. I'm going to keep on trying, though. He's ahead now, 1-0, but if we keep going, I figure at some point I'll pull ahead."

Re:Cool processor

[Kurusuki]

They're crunching 13-million-digit numbers with a desktop processor? Do they realize that they can put eight quad-core xeons in a machine and finish the calculation in a single shift instead of waiting a month?

I don't know about you, but the last 13 or so mersenne primes have been found using prime95 as a conduit for a mass distributed effort. I'm not sure where you live, but in most other places people can't just go out and put 8 quad-core xeons in a home machine.

Re:Cool processor

[Nutria]

Do they realize that they can put eight quad-core xeons in a machine and finish the calculation in a single shift instead of waiting a month?

What makes you think they aren't?

And what makes you think this man would pony up the serious coin for such a beast just to find a prime number?

Re:Cool processor

[JoshuaZ]

The system used for this is GIMPS, the Great Internet Mersenne Prime Search. The system uses a distributed computing system using unused computing power in personal computers to search for various candidate primes. Computers do one of two things: Either trying to factor candidate Mersenne numbers or running a Lucas-Lehmer test on candidates without any small prime factors (the Lucas-Lehmer test is a special primality test for Mersenne numbers that is very fast). They use modular arithmetic and a variant of the Fast Fourier Transform to handle the multiplications which might otherwise become too difficult. The procedure is naturally a problem that can be made into a parallel processing problem like this since there are so many different candidate numbers to look at.

The summary doesn't mention but it is worth noting that the Lucas-Lehmer test allows one to check the primality of Mersenne numbers (numbers of the form 2^p-1, p prime) much faster than you can test the primality of generic numbers (or almost any other specialized form). Thus, for most of the last hundred years the largest primes known have been Mersenne primes. Currently the largest known prime is a Mersenne prime and the next 4 largest are also Mersenne primes. The GIMPS website - http://mersenne.org/ has a lot more details of both the math and software and explains how you can join in to help the project.

Re:Cool processor

[Nutria]

I'm not sure where you live

He lives at home with his parents (or maybe in a dormitory room) and doesn't have a clue as to what it actually costs to run a home.

The joys of untested code

[tqft]

The admins missed the prime for about a month
http://mersenneforum.org/showthread.php?t=11996
Apparently the email that was supposed to be sent wasn't when the prime was reported

"telescope" from Intel

[moon3]

Discovering a prime number that distant from the zero is like discovering a Pluto like planet in outer space. But instead of Hubble telescope you need a powerful mathematical one..

Re:Cool processor

[bergonom]

What are the odds this odd odd would be found by Odd?

Hmm

[Anenome]

I honestly forget why I'm supposed to care about Mersenne primes. Like, I read something about them awhile back, it was somewhat interesting... and then--yeah. So:

http://en.wikipedia.org/wiki/Mersenne_prime
In mathematics, a Mersenne number is a positive integer that is one less than a power of two.

A Mersenne prime is a Mersenne number that is prime. As of June 2009[ref], only 47 Mersenne primes are known; the largest known prime number (243,112,609 1) is a Mersenne prime, and in modern times, the largest known prime has almost always been a Mersenne prime.[1] Like several previously-discovered Mersenne primes, it was discovered by a distributed computing project on the Internet, known as the Great Internet Mersenne Prime Search (GIMPS). It was the first known prime number with more than 10 million base-10 digits.

For those who can't even remember what a prime is, it's a number that can only be divided (evenly) by 1 and itself. Here's a list of the first primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

The Mersenne primes are the largest known primes.

Prime numbers have applications in electronic security and encryption breaking. I'm not sure what other purpose there is to knowing them, other than knowing them. The Mersenne in particular seem to be merely mathematical curiosities right now.

I was much more excited by the discovery that the the Fibonnacci sequence is contained within the 1/89 calculation.
http://www.geom.uiuc.edu/~rminer/1over89/

Re:Hmm

[Bart Coppens]

Actually, you can apparently use larger Mersenne Primes to improve results in totally different but very useful fields, like privacy-related schemes. For example, this paper http://eccc.hpi-web.de/eccc-reports/2006/TR06-127/index.html uses large Mersenne primes to get interesting results on Locally Decodable Codes and Private Information Retrieval Schemes...

Why is this useful?

[Abreu]

The answer is not in the summary, nor in the first page of the FA...

Buried somewhere in the linked site is this FAQ:

http://primes.utm.edu/notes/faq/why.html

However all the answers are a bit unsatisfactory, IMHO...

So, I ask the great Slashdot hive-mind... What are the practical applications of Mersenne Primes and why are people paying money to find them?

Re:Why is this useful?

[Rockoon]

Well, for one thing if you need a prime divisor, 2^n-1 primes have some good properties...

Modulus or Division by such numbers can be accomplished with a few fast operations (bitwise Shift/And, a comparison, and maybe a subtraction) instead of a single very slow one (an actual division.)

Re:Cool processor

[rbarreira]

Do they realize that they can put eight quad-core xeons in a machine and finish the calculation in a single shift instead of waiting a month?

Do you realize that that's less efficient than using those 32 cores to calculate 32 independent numbers?

Mersenne Primes correspond to Perfect Numbers

[JoshuaZ]

The historical reasons for caring about Mersenne Prime are twofold: First, Mersenne primes correspond to perfect numbers (numbers that are the sum of their positive less than the number. So for example, 6 has as proper divisors 1,2 and 3 and 1+2+3=6). The ancient Greeks were fascinated by perfect numbers but could not do much to understand them. Euclid showed that if one had a Mersenne prime one can construct an even perfect number. In particular, if 2^n-1 is prime then (2^n-1)*2^(n-1) is perfect. Almost 2000 years later, Euler showed that every even perfect number is of Euclid's form. Thus, investigating Mersenne primes tells us more about perfect numbers. The oldest unsolved problems in math are 1) are there any odd perfect numbers? and 2) are there infinitely many even perfect numbers? Thus, investigating Mersenne primes helps us get closer to solving one of the two oldest unsolved problems in mathematics.

Re:Mersenne Primes correspond to Perfect Numbers

[Anenome]

Well, they may be unsolved problems, but again, they look like they have no relevance to anything, no application, other than being unanswered questions. But, like so many things, knowledge is valuable for its own sake, and who knows what revolution may result from what is now just a mathematical curiosity. Stealth-flight technology was originally harvested from a little known paper on radar written by an obscure Russian scientist. Kind of ironic that we were the ones to develop it. What you're really talking about is finding a proof for why there could or could not be any odd perfect numbers, and a proof for whether there are infinite perfect numbers or not. Typically, proofs like this that elude us lead into new forms, new paradigms of mathematics--which themselves result in great leaps forward in other areas once those proofs have been realized. That is certainly a story that has been repeated time and time again, most notably with calculus which is virtually the foundation of the modern world.

Even the concept of 'perfect numbers' is not familiar to me. Off to the Wiki!:

http://en.wikipedia.org/wiki/Perfect_number
"In mathematics, a perfect number is defined as a positive integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself), or (n) = 2n.

The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6.

The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. This is followed by the perfect numbers 496 and 8128 (sequence A000396 in OEIS).

These first four perfect numbers were the only ones known to early Greek mathematics."

Re:Mersenne Primes correspond to Perfect Numbers

[JoshuaZ]

Well, this does actually fit some of the patterns we are beginning to see. For example, there are reasons to expect that for most Mersenne primes 2^p-1 one will have p-1 having many small prime factors (this has to do with Fermat's Little Theorem). We in fact see that again in this case we see that since p-1 factors as 2^3 * 3^3 * 5^2 * 53 * 149. Also, the discoveries are are enough to be noteworthy in the same way that discovery of new elements is noteworthy. We likely won't find out much by itself from the discovery of element 117 when it occurs but it will be a big enough deal to get on Slashdot. Discoveries like these are sort of like summiting mountains. And for all we know, each one may be in fact be the last, since we can't prove there are infinitely many primes. Furthermore, primes in general are a fundamental building block of the integers. So finding very large ones is intrinsically cool.

Re:Can we have the value?

[spandex_panda]

This gives you the number, keep hitting 'More digits'. Unfortunately it gives it as an image so I can't copy paste here.

Re:Can we have the value?

[Eudial]

In base 2, it's 1111[42,643,792 more 1:s]1111.
In base 16 it's 0xffff[2,665,229 more f:s]ffff.

Re:Cool processor - No, they can't

[davmoo]

Do they realize that they can put eight quad-core xeons in a machine and finish the calculation in a single shift instead of waiting a month?

No, they can't. Each iteration of the software requires the results of the previous iteration. It cannot easily be made to run like you want on multiple cores. The best they could do on the processor you describe is run 8 separate copies of the application, each taking one month to run...they could test 8 numbers at once, but they cannot test one number 8 times as fast.

Re:Cool processor - No, they can't

[rbarreira]

Actually that's not exactly correct, each iteration is also parallelizable. Most of the work in an iteration is a FFT, which is parallelizable.

http://www.fftw.org/parallel/parallel-fftw.html

It's less efficient to do this than using each core for one independent number, so it's only used if quick checking of a number is desired (for example, when double-checking a previously found prime number).

Re:Cool processor

[Daswolfen]

As one of the IT guys who maintain the lab that found the 43rd and 44th primes at University of Central Missouri (formerly CMSU), I can tell you its one number per core. Also, these are production machines in computer labs as well as classroom, faculty and staff systems that run the GIMPS software.

We are a public university, its not like we have extra $5k machines just sitting around crunching a number. BTW, the systems that found the 43rd and 44th prime numbers were base model Dell GX280s.

Re:Cool processor

[ceoyoyo]

You don't get the $100k by searching for one prime though. You've got to be the lucky one that does the month long calculation on the number that actually happens to be prime.

It's like the lottery. You can't make a profit at it unless you're lucky. Otherwise, some big company would come in, invest a few million in number crunching, and take home all the bounties.

Re:Cool processor - No, they can't

[smallfries]

If you were going to test for primality by sieving then you could take a process that is millions of times slower than the primality test used, and speed it up by a factor of 8.

Instead the test being discussed performs a series of squares and modulo reductions. Each operand is dependent on the previous result - the entire computation is one long dependency chain and so cannot be split onto multiple cores in the way that you describe.

Although having said that, it all flips around again if you look inside the primitive operations that the primality tests uses. So they're using FFT based multiplication steps to do the squaring which obviously can be parallelised quite well..