42nd Mersenne Prime Probably Discovered

RTKfan writes "Chalk up another achievement for distributed computing! MathWorld is reporting that the 42nd, and now-largest, Mersenne Prime has probably been discovered. The number in question is currently being double-checked by George Woltman, organizer of GIMPS (the Great Internet Mersenne Prime Search). If this pans out, GIMPS will have been responsible for the eight current largest Mersenne Primes ever discovered."

Comments

Uses?

[UncleJam]

What uses are there for gignatic prime numbers like this other than showing off computing power?
Encrypting?

Re:Uses?

[selectspec]

Chics dig it.

Re:Uses?

[Husgaard]

I don't the any real use for this except finding large primes.

The theory is that there is an infinite number of these numbers, but they are unlikely to prove the theory by finding them all...

Re:Uses?

[ackthpt]

Chics dig it.

Either that or their eyes glaze over and you sneak a quick peck before they slap you silly.

"ah, l'amour"

Re:Uses?

[Jeremy Erwin]

Testing distributed primality algorithms. I should have thought this was obvious.

And, no, one does not encrypt with Mersenne primes. The rarity of such numbers makes a "brute force" crypto-analysis seem rather plausible. Best to use an ordinary prime number-- there are, after all, many more of them to choose from.

Re:Uses?

[adeyadey]

Hi darling, ooh is that a gigantic Mersenne Prime, or are you just pleased to see me?

Re:Uses?

[ArsonSmith]

no, it just seams that way. Chick's will put out just to shut you up.

Re:Uses?

[pclminion]

Yeah, except that the Mersenne primes are well known and thus useless for cryptography -- at least, if you want any security.

Re:Uses?

[jaoswald]

You are arguing about a different domain.

Encryption discussions have to take place in a "computing" domain, where a prime only exists if it has been computed to be prime by at least one computer somewhere in the world, and where the prime number can fit on a distribution medium.

Arguing that there are as many Mersenne primes as regular primes is only possible in a theoretical domain in which countably infinite sets can be said to exist.

Re:Uses?

[uberdave]

The X axis is infinite. So are the Y and Z axes. Therefore, there must be an infinite number of regular solids. Oh, wait! There's only five. Gee, I guess the mere fact that numbers are infinite doesn't imply that subsets of those numbers are infinite.

Re:Uses?

[naff]

Wrong theory. That is for primes in general. "These numbers" refers to just the Mersenne primes.

Re:Uses?

[Seculus]

That's not actually the argument for why the number of primes is infinite. Rather, assume there are only finitely many prime numbers. Multiply all of them together. Add one to this number. It is easy to show that this number is not divisible by any of the finitely many primes you started with. Hence it must be a prime number as well.

Re:Uses?

[Anonymous Coward]

Actually, it doesn't have to be. It only suffices so say, that when you multiply all primes in your list, and add one, you get a number not divisable by any of the number in the list. Hence, one of TWO things can hold true:

1) The number in question really is prime, as you suggested

2) The number in question isn't prime. Then it has prime divisors, none of them in your list (because none of the primes in the list divided our new number).

In both cases, we have derived a way to find at least one new prime from any list of primes, and hence, the collection of primes is non-finite because we can always find "another one".

Re:Uses?

[novakyu]

Actually, this theory has already been proven.

IMNSHO, but that was the worst proof of infinite number of primes. Why introduce unique factorizability when you don't need to? Why introduce something foreign that you are not going to prove when there is absolutely no need for it?

The most elegant proof I've seen so far (but I don't know any website showing it, so I can't link to it) is this: For any given N, an integer, consider N!+1, which is greater than N (where N! is defined by N! = 1 * 2 * 3 * ... * N). If this number is divisible by no other number than 1 and N!+1, then we are done (i.e. we have proven that given any arbitrary integer, there is a prime greater than tat integer). If this number is divisible by a prime, than that prime can't be less than or equal to N, since any integer (not equal to 1) less than or equal to N divides N! (see the definition of N!) but does not divide 1. Therefore, the prime that divides N! is greater than N. QED.

This proof involves no assumption (additional to assumptions (i.e. axioms) of the set of integers) other than this (which also happens to be much easier to prove than factorizability into primes): if n divides a + b and n divides a, then n divides b as well.

Of course...

[rackhamh]

... the moment they discovered the 42nd prime, the world was immediately destroyed to make way for an intergalactic superhighway.

Re:Of course...

[micromoog]

Now that was a prime rib!

Re:Would a math geek...

[haluness]

From mathworld (whose link is in the summary)

A Mersenne prime is a Mersenne number, i.e., a number of the form

2^n - 1

that is prime. In order for it to be prime, n must itself be prime.

Re:Would a math geek...

[Smallpond]

A Mersenne number is all ones when written in binary. If its prime, it is a Mersenne prime.

Practical Applications/Uses?

[NitroWolf]

Can someone explain what the application/use these primes are for? Not a flame, I'm honestly curious as to what something like this could be used for, as are others, I'm sure.

Re:Practical Applications/Uses?

[robbyjo]

From here:

Finding new Mersenne primes is not likely to be of any immediate practical value. This search is primarily a recreational pursuit. However, the search for Mersenne primes has proved useful in development of new algorithms, testing computer hardware, and interesting young students in math.

Re:Practical Applications/Uses?

[vivin]

It's a mathematical curiosity in some cases - just to find it for the sake of finding it, or for the glory of finding it. You know, like being the first to do something cool.

Also, necessity is the mother of invention. Even if Big Primes aren't really a necessity, it has brought forth some really innovative algorithms and methods to finding prime numbers. Furthermore, it has developed newer and faster ways for multiplying integers.

In 1968, Strassen figured out how to multiply integers quickly by using Fast Fourier Transforms. Strassen, along with Schönhage improved on the method and published a refined version in 1971. GIMPS now uses an improved version of their algorithm. This improved version was developed by Richard Crandall (a longtime researcher of Mersenne Primes).

Another application of finding Prime Numbers is to test computer hardware. Since testing Primes involves a thorough excercise of basic mathematical operations, it provides a good test for hardware. Software routines from GIMPS were used by Intel to test the PII and the Pentium Pro chips before they were shipped. The search for prime numbers was also indirectly responsible for the discovery of the infamous FDIV bug on the Pentium, during the calculation of the twin prime constant (by Thomas Nicely).

Re:Practical Applications/Uses?

[Sloppyjoes7]

Uncommon and unique numbers of varying types are usually useful for mathematics in general. Usually only mathematicians know why.

Whatever the case, this must be a more useful application for CPU power than Seti@home, which is a total waste of energy. Literally.

What we need are more projects that use distributed computing for useful calculations that could further science or solve problems. Universities build giant supercomputers to help their students calculate equations and solve problems. Maybe the students should release the problems over a network, and have home users calculate the answer for them. It'd save the Universities a lot of money.

I don't think it would work for code cracking, or government projects, as these contain sensitive information.

Re:Practical Applications/Uses?

[pclminion]

Can someone explain what the application/use these primes are for?

Communicating with alien species, perhaps.

Mersenne primes have two interesting properties that might catch the attention of alien species: when written in binary, they are entirely composed of '1' bits; and, of course, they are prime.

A sure way to prove to another being that you are intelligent is to spew a bunch of numbers which all happen to be prime. The fact that they can be tranmitted using only '1' bits means the modulation is simple -- just send a series of pulses.

Re:Practical Applications/Uses?

[myowntrueself]

So... the main reason for searching for large primes is to develop better techniques for... searching for large primes?

Re:Practical Applications/Uses?

[geoffspear]

Transmitting the same binary signal over and over seems unlikely to impress anyone. You're as likely to be sending a really boring all-white image as a really big prime number.

If anything, anyone receiving the signal will wonder how you managed to build such a powerful transmitter when you haven't discovered binary numbers yet and are apparently using some sort of unary mathematics that really shouldn't work. They're bound to be disappointed when they find out you actually know about "0", but just weren't using it.

Re:Practical Applications/Uses?

[burns210]

large prime numbers are used in encryption techniques, also.

Re:Would a math geek...

[IvyMike]

A prime of the form (2^n)-1. This means that in binary, it's a big string of "1"s.

The reason that mersenne primes are usually the biggest is because for primes of this form, there is a primality test (Lucas-Lehmer) that is exceedingly efficient.

A Mersenne Prime is...

[vivin]

A mersenne Prime is a prime number that is one less than the power of two. Hence:

Mn = 2^n - 1.

Mersenne primes have a connection with Perfect Numbers (numbers that are equal to the sum of their proper divisors) where by if M is a Mersenne prime, then M(M+1)/2 is a perfect number.

Re:A Mersenne Prime is...

[tbjw]

You can say even more. If M can be written as 2^n - 1, then M is said to be a Mersenne number. If M is also prime, then it is a Mersenne prime. For 2^n - 1 to be a Mersenne prime, n must be a prime number, since we have

2^(ab) - 1 = (2^a-1)(2^(a(b-1)) + 2^(a(b-2)) + ... + 2^(a))

For instance, 2^6-1 is (in binary) 111111 = 1001 * 111, as predicted by the above factorisation.

If p is a prime number and if 2^p-1 is a Mersenn prime, then, as was pointed out above, 2^(p-1)(2^p-1), is a perfect number. Moreover, if N is an even perfect number, then N can be written (uniquely) as 2^(p-1)(2^p-1) where p is a prime number and 2^p-1 is a Mersenne prime.

Wikipedia has a reasonably intelligible introduction to perfect numbers, and MathWorld contains a proof of why every even perfect number must have the form claimed above.

To see why M = 2^(p-1)(2^p-1) is a perfect number when p, 2^p-1 are primes, it suffices to note that s(n), the function that maps an integer to the sum of its divisors (e.g. s(6) = 1 + 2 + 3+6, s(8) = 1 + 2 + 4+8) is multiplicative in the number-theoretic sense, that is to say s(ab) = s(a)s(b) whenever a, b have no prime factors in common. Then evaluating s(M) is simply a case of evaluating it on the factors, which are relatively prime since one is a power of 2, 2^(p-1), and the other is an odd prime, 2^p-1. s(2^p-1) = 2^p-1 + 1 = 2^p (since we have a prime number), and 2^(p-1) = 2^p -1 is an easy formula that is true of all powers of 2. Hence s(M) = 2^p(2^p-1) = 2 ( 2^(p-1) (2^p-1) = 2s(M). That is to say, the sum of all the divisors of M add up to twice M, and if we leave the divisor M itself out of the sum, we see that M is a perfect number.

Mersenne Primes? Bah!

[Guano_Jim]

Call me when a distributed computing project finds Fruit Fucker Prime.

Probably silly reference

[serutan]

Reminds me of the first BlackAdder episode

Lord Percy: "The King is dead! L-"
Prince Harry [interrupting]: "Probably dead."
Lord Percy: "The King is probably dead!"

Spoiler alert about the number

[Anonymous Coward]

Don't read any farther if you don't like spoilers.






Seriously, don't reead any farther....






It only has two factors.

Re:Great. Now what is it???????

[Anonymous Coward]

Binary is pretty easy. The number is:

11111...1111

where "..." means some number of 1s.

One practical use of Mersenne Primes...

[William_Lee]

I'm not sure what else they're actually good for, but searching for these with Prime95 is a great way of putting the flame to your CPU.

Prime95 (which searches for these primes) really puts a load on the CPU and raises the temperature in a hurry. It's commonly used to test the stability of overclocking configurations since it stresses the chip and is able to detect if there is an error in the computation.

Generally, if you can run Prime95 for 24 hours straight, most people will consider the overclocked PC a stable configuration.

Re:Immoral use of computing power

[Stanistani]

>What is number going to for us? Is it going to feed us? No. It would be better if the computer power was used for cancer research or finding aliens.

Because of course aliens will feed us...
They even will bring a cookbook with them, "To Serve Mankind."

Re:Mersenne Primes - Definition

[jandrese]

Well, yeah, if you encode the Prime number in Binary it will not look Random at all. It will look like a giant string of 1s though... Aliens might mistake it for filler or something.

Yes!

[Anonymous Coward]

If this pans out, GIMPS will have been responsible for the eight current largest Mersenne Primes ever discovered.

In your face, Photoshop!

Two unknowns

[MaGogue]

This has not yet been confirmed, therefore there could be less than 42 known Mersenne primes.

Hovewer, according to MathWorld, there is a chance that it is not the 42nd Mersenne prime at all for another reason :

"However, note that the region between the 39th and 40th known Mersenne primes has not been completely searched, so it is not known if M20,996,011 is actually the 40th Mersenne prime.."
Looks like the big math guys don't exactly know how to count at all ;)

That brings back some memories...

[Anonymous Coward]

Back in the dark ages when I was in university, I took a class called "Mathematics and Poetry". I thought it would be a useful bird course in my senior year, but it turned out to be both interesting and challenging.

As part of the course, we studied Mersenne primes. At the time, I was dabbling in x86 assembler, and I decided to write a program to calculate the then largest known Mersenne prime number: 2^31 - 1, which worked out to 65,050 digits.

The size worked out perfectly, as in 1989 that meant it could fit into one 65KB segment on my blazing-fast 8Mhz 8088. As I recall, the runtime was about two days. The program still works--I can't remember how long it took to run on a 3Ghz P4, but I think it was just a few minutes.

I'm sure any competent programmer (read--not me) could calculate the result much faster, but at the time I was very proud of my little creation.

Re:That brings back some memories...

[djmurdoch]

As part of the course, we studied Mersenne primes. At the time, I was dabbling in x86 assembler, and I decided to write a program to calculate the then largest known Mersenne prime number: 2^31 - 1, which worked out to 65,050 digits.

I don't think it actually did bring back those memories. 2^31-1 is 2147483647. You're thinking of Mersenne prime 31, which is 2^216091 - 1.

Re:That brings back some memories...

[nizo]

Actually it did bring back memories, just not the right ones :-)

the trouble is

[pmike_bauer]

The number in question is currently being double-checked by George Woltman

Ok...lets see here...

5465875133124687545551258898456556......98034802

BUMMER!

Here's one good reason...

[thesatch]

http://www.eff.org/awards/coop.html

Thought it takes my 1.7Ghz 3 months to test a 10mil digit prime.

Re:Mersenne Primes - Definition

[SwansonMarpalum]

Might want to check your math:

2^2 = 4
4 - 1 = 3

2^3 = 8
8 - 1 = 7

2^4 = 16
16-1 = 15

On a related note

[OverlordQ]

Seventeen or Bust (a distributed attack on the Sirpinski Problem), found the fourth largest (well fifth if this one pans out) prime at the beginning of this year, which contains 2,357,207 digits.

28433 * 2^7830457 + 1

List of all of the largest primes can be found here

Overheard in the Math Dept...

[Shadow Wrought]

"OK, I've narrowed the range down to between zero and infinity. The rest is up to you..."

Re:GOOD QUESTION!

[Surt]

It's one step closer to proving there's an infinite sequence of these numbers. Just infinity - 42 more to go and the proof is complete!

In all seriousness, they are interesting mainly because they are so simple mathematically that very very early mathematicians got interested in them. But even after hundreds of years of interest among mathematicians, there's no formula for predicting them, and very little successfully proven about them.

Since they are so rare, each find is a significant advancement for those who might be interested in trying to find a pattern.

Full text of article

[utexaspunk]

72,881,090,798,676,481,947,445,843,876,689,972,113 ,188,382,077,838,576,766,415,271,554,183,554,023,6 81,926,442,357,773,229,141,527,132,801,050,545,169 ,980,023,429,475,382,981,277,026,411,446,450,732,1 20,206,920,761,648,530,323,773,463,358,502,551,340 ,699,145,522,328,264,108,074,466,176,204,798,818,5 91,643,822,008,785,083,299,073,103,153,980,722,122 ,415,403,264,180,661,744,484,810,522,551,289,556,1 61,305,278,379,785,516,809,393,766,311,656,230,448 ,542,351,852,881,090,798,676,481,947,445,843,876,6 89,972,113,188,382,077,838,576,766,415,271,554,183 ,554,023,681,926,442,357,773,229,141,527,132,801,0 50,545,169,980,023,429,475,382,981,277,026,411,446 ,450,732,120,206,920,761,648,530,323,773,463,358,5 02,551,340,699,145,522,328,264,108,074,466,176,204 ,798,818,591,643,822,008,785,083,299,073,103,153,9 80,722,122,415,403,264,180,661,744,484,810,522,551 ,289,556,161,305,278,379,785,516,809,393,766,311,6 56,230,448,542,351,852,881,090,798,676,481,947,445 ,843,876,689,972,113,188,382,077,838,576,766,415,2 71,554,183,554,023,681,926,442,357,773,229,141,527 ,132,801,050,545,169,980,023,429,475,382,981,277,0 26,411,446,450,732,120,206,920,761,648,530,323,773 ,463,358,502,551,340,699,145,522,328,264,108,074,4 66,176,204,798,818,591,643,822,008,785,083,299,073 ,103,153,980,722,122,415,403,264,180,661,744,484,8 10,522,551,289,556,161,305,278,379,785,516,809,393 ,766,311,656,230,448,542,351,852,881,090,798,676,4 81,947,445,843,876,689,972,113,188,382,077,838,576 ,766,415,271,554,183,554,023,681,926,442,357,773,2 29,141,527,132,801,050,545,169,980,023,429,475,382 ,981,277,026,411,446,450,732,120,206,920,761,648,5 30,323,773,463,358,502,551,340,699,145,522,328,264 ,108,074,466,176,204,798,818,591,643,822,008,785,0 83,299,073,103,153,980,722,122,415,403,264,180,661 ,744,484,810,522,551,289,556,161,305,278,379,785,5 16,809,393,766,311,656,230,448,542,351,852,881,090 ,798,676,481,947,445,843,876,689,972,113,188,382,0 77,838,576,766,415,271,554,183,554,023,681,926,442 ,357,773,229,141,527,132,801,050,545,169,980,023,4 29,475,382,981,277,026,411,446,450,732,120,206,920 ,761,648,530,323,773,463,358,502,551,340,699,145,5 22,328,264,108,074,466,176,204,798,818,591,643,822 ,008,785,083,299,073,103,153,980,722,122,415,403,2 64,180,661,744,484,810,522,551,289,556,161,305,278 ,379,785,516,809,393,766,311,656,230,448,542,351,8 52

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Small steps

[Duncan3]

One small step for mathematics, one giant leap for global warming :)

Please come join Folding@home, we're actually doing something worth all that waste heat. :)

meaning of life, bah

[zenst]

All that distibuted processing power to work out how long to hold the `1` key down :)