## New Largest Known Prime Number: 2^57,885,161-1 254

Posted
by
timothy

from the sorry-won't-fit-on-vanity-plate dept.

from the sorry-won't-fit-on-vanity-plate dept.

An anonymous reader writes with news from Mersenne.org, home of the Great Internet Mersenne Prime Search:

*"On January 25th at 23:30:26 UTC, the largest known prime number, 257,885,161-1, was discovered on GIMPS volunteer Curtis Cooper's computer. The new prime number, 2 multiplied by itself 57,885,161 times, less one, has 17,425,170 digits. With 360,000 CPUs peaking at 150 trillion calculations per second, GIMPS — now in its 17th year — is the longest continuously-running global 'grassroots supercomputing' project in Internet history."*
## Wrong (Score:5, Informative)

## Re: (Score:3, Interesting)

I'm not trying to be inflammatory here... this is a genuine question.

Why? I mean... why bother with calculating this.. What has it added to study if prime numbers? And if it's added to the study of primes... then what use is that?

## Re:Wrong (Score:5, Informative)

As is well known, there is no direct mathematical benefit from finding these primes.

It is, however, a very useful "driving problem" to developing new algorithms, software, and distributed computing infrastructure which have wide ranging real-world applications.

Check out the Mersenne Forum [mersenneforum.org] where all types of interesting mathematical, software and computer issues are discussed.

## Re:Wrong (Score:5, Funny)

I can't speak for other people, but I often use (2^57,885,161)-1 in casual conversation.

## Re: (Score:2)

And this is better than something remotely useful like Folding@home in what way?

## but most of math's "usefulness" is handwaving (Score:3)

its an aesthetic discipline not an objectively useful one.

alot of it turns out to be useful, sometimes hundreds or thousands of years after its first 'discovery'.

Quaternions would be a perfect example, imaginary numbers another.

so right now we have no idea what the usefulness of finding high primes is.

but in 200 years it might allow us to calculate the control points necessary to create an anti-matter damper for protonium neon flux casings in hyperdimensional warp transferrence beam generators.

## Re: (Score:2)

## Re:Wrong (Score:5, Informative)

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

## Re:Wrong (Score:5, Insightful)

Your first question: "What has it added to the study of prime numbers?", I'm not sure but...

Your second question: "What use is that? (the study of prime numbers)"

Well... Nearly all modern public key cryptography (SSL / TLS / SSH etc.) is based on the asymmetry in time between the multiplication of two prime numbers (very fast operation) and the factorization of the result back into these two primes (very very slow: the goal being to make so slow that it become impractical to do).

Actually, it's "worse" than that: the "proof" that most modern PKCS crypto works is: "It's hard to find the (prime) factors of the product of two primes".

In other words: the study of prime numbers is one of the most important area of mathematics when it comes to computer security.

## Re:Wrong (Score:5, Interesting)

Actually, it's "worse" than that: the "proof" that most modern PKCS crypto works is: "It's hard to find the (prime) factors of the product of two primes".

Small correction, there is actually no proof that RSA reduces to factoring. It is true that if you can factor you can break RSA, but you may also be able to break RSA without factoring. http://en.wikipedia.org/wiki/RSA_problem [wikipedia.org]

## Re:Wrong (Score:5, Insightful)

Do I believe Wikipedia? Or do I believe Donald Knuth?

Why should you believe either? One of the basic (meta-)principles of mathematics is that you don't have to take anyone's word for something. You can check out their claim yourself, using existing reference material and your own mind. This isn't facetious; mathematical and scientific results have occasionally been shown to be erroneous.

If you are unwilling to do these things, you're abandoning your (and your children's) future to those who are willing to do them (or to pay others to do them ;-). The dominant power and wealth in the human society now belongs to those who can handle technology. If you demand

beliefrather thanunderstanding, you are handing control over to those willing to gain the understanding.If you can't appreciate the aesthetic justifications for learning about math and science, you certain must be aware of the history of successes and improvements in our world by their practitioners. Few of us would like to go back to stone-age hunter-gatherer societies. It's the experimenters and reasoners that got us past that stage.

## Re: (Score:2)

Actually there is a variety of encryption that uses these numbers.

## Re:Wrong (Score:4, Funny)

That would be 3:

2 multiplied by itself is 4. No matter if you do it once or 57,885,160 times. Then minus one.

Or it could be 4 if you only do it 57,885,159 times. But I think 4 is not a prime number.

## Re: (Score:3)

## Re: (Score:3)

Yeah, it's better to get the math completely wrong.

Well, it's difficult to get the math

partiallywrong!## And what shall we call this number? (Score:2)

## Re: (Score:2)

Post responses of cool names.

Instead of the dry M(48), could one of those cool names, at the least, include something about The Gimp? This one is their 14th.

## Re: (Score:2)

The goatse number = number of seared eyeballs requiring mind bleach. 2 ** 57885161 - 1 is probably about right. Either that or its the decimal representation of the future ipv8 standard which expands the puny ipv6 addressing space from 2 ** 128 to 2 ** 1e100 aka when we replace the internet with the google-net. All hail the almighty GOOG! huzzah!

## They would have more primes to choose from ... (Score:2)

... if they didn't limit the search to just Mersenne primes [wikipedia.org].

## Re: (Score:2, Informative)

There isn't a 100% correct primality proving method for general numbers that's anywhere near as efficient as the Lucas-Lehmer test for Mersenne numbers of the same size.

## Re: (Score:2)

So Mersenne primes are just low-hanging fruit?

## Re:They would have more primes to choose from ... (Score:4, Interesting)

Yes. The LL test only works on Mersenne numbers--numbers of the form 2^p-1 where p is prime. The LL test is not statistical. It can determine if a given mersenne number is prime or not without any doubt.

To protect against errors, GIMPS (Great Internet Mersenne Prime Search) uses a variety of double checks to ensure no number if mistested. Any number that passes the LL test is double (and sometimes triple checked) to verify that there wasn't a hardware or software error that caused a false positive. I had the honor of performing the double check of a record Mersenne prime some time ago.

## Re: (Score:3)

## Re: (Score:2)

## Re:They would have more primes to choose from ... (Score:4, Informative)

Mersenne primes have a structure that makes it possible to test primality for very large numbers; there's no way to test whether unrestricted numbers of that size are prime (it's theoretically possible, but there aren't enough computing resources on the planet.)

I used to run the GIMPS search application back in the 90s; you really really don't want to run it on a laptop on batteries, especially with the battery technology of the time, and eventually I decided that my laptop didn't have enough horsepower to bother, compared to desktops that could run GPU-based calculations.

## Re: (Score:3)

Mersenne primes have a structure that makes it possible to test primality for very large numbers; there's no way to test whether unrestricted numbers of that size are prime (it's theoretically possible, but there aren't enough computing resources on the planet.)

Basically, it is much easier to find a proof that p is prime (if it is indeed prime) if you have a complete factorization of p + 1. For this number as for all Mersenne primes, the complete factorization is quite obvious.

## Re: (Score:2)

Maybe we could have more success with certainty by searching for non-primes. If you got the factors (even one factor is enough), such as 23*89 for 2^11-1, which some people think has to be prime because 11 is prime, then we could have more numbers. Why are primes so great? I can get tons of very large non-prime numbers by generating Pascal's triangle.

## Re: (Score:2)

I can get tons of very large non-prime numbers by generating Pascal's triangle.

Given that the rate of primes drop as the number length goes up, you can just pick any ole random really large number and the odds eventually become ridiculously high that its composite. You can have a lot of fun Fing with people (aka Socratic method), at least if they don't know much number theory, by trying to convince them there exists a certain largest prime number above which they're all composite, I mean, just look at the graph of rate of prime numbers vs number of digits or however you wanna phrase

## Re: (Score:2)

## Re: (Score:2)

What I'm curious of is to what certainty is this proven prime?

Absolutely 100% certain.

## Re: (Score:2)

## stats? (Score:2)

So.... is there an equivalent to rc5stats for GIMPS, so we can compare, uh, our harnessed computing prowess?

No, I'm not going to google it. I want your opinion on whether it's any good or not.

## Why 2^n-1 (Score:2)

I know that that is called a Mersenne prime. Why do they always look for primes of that form? Are these numbers more likely to be prime? Why don't they start looking for primes of the form 2^n+1?

## Re:Why 2^n-1 (Score:5, Informative)

number of the form 2^n-1 are Mersenne numbers which are much more likely to be prime than a randomly chosen odd number. Also, we have "simple" test for these number to weed out many Mersenne numbers that are not prime. Once you have a Mersenne number that passed the "simple" primality test, there is a good chance that it will really be a prime number.

## Re:Why 2^n-1 (Score:4, Informative)

There is a very fast primality test for Mersenne numbers, the Lucas–Lehmer primality test. [wikipedia.org]

2^n+1 is prime only if it's a Fermat prime, n=2^k. None of these are known to be prime for k>4, and there probably aren't any more, whereas there are probably infinitely many Mersenne primes.

## Re: (Score:2)

## Re: (Score:2)

For one thing, it is very easy to write them down. For another, they lend themselves very easily to computation using binary math. Any number of the form 2^n - 1 is just a string of n 1's. E.g., 2^5 - 1 = 31 = 11111b.

As to your other question, regarding 2^n + 1, those would be of the form of a '1', followed by (n-1) '0's, followed by another '1'. E.g., 2^5 + 1 = 33 = 1000001b. I don't know if these are any more or less likely to be prime. I suspect

## Re: (Score:2)

It is far easier to test Mersenne numbers for primality than general integers, thanks to the Lucas-Lehmer primality test [wikipedia.org]. Weeding out composite Mersenne numbers by trial factoring is also faster, thanks to a theorem [proofwiki.org] that eliminates most of the candidate factors.

As for numbers of the form 2^n+1, it's easy to show that if 2^n+1 is prime, then n must be a power of two. Such numbers are known as Fermat numbers [wikipedia.org], and there is also a fast primality test (Pepin's test [wikipedia.org]) for numbers of this form. But because of th

## A Little More Perfection (Score:5, Informative)

## Re: (Score:2)

## Re: (Score:2)

It has been proved that such perfect number would be odd

Well, that rules half of them out.

## Number of Digits (Score:5, Interesting)

"There are 10 kinds of people in the world. Those who understand binary, and those that don't."

This new number is 2^57,885,161 - 1, so naturally it has 57,885,161 digits, all of them 1. A simpler example: 2^5 - 1 is a Mersenne prime. Written in binary it's 100000 - 1 = 11111.

Oh! You meant that it has 17,425,170

decimaldigits. Booooooooring!## Re: (Score:2)

This is why I've never been very impressed with the mersennewiki.org. Why store giant files containing the decimal expression of each Mp, when a zipped up binary format would be really freaking small?

## Re: (Score:2)

## Re: (Score:3)

Oh! You meant that it has 17,425,170

decimaldigits. Booooooooring!Well, the etymology for "digit" is "finger" [wikipedia.org], so it's first and foremost used for base 10. Binary digits have a special name, that you may know, which is a portmanteau of "binary" and "digit": it's a "bit".

## Not true. (Score:2)

## Son of a bitch (Score:4, Funny)

## total energy consumed during the 17-year project? (Score:4, Interesting)

## Last Post! (Score:2)

Hey, don't believe everything you read on the Internet! I'm going to check this puppy out!

OK... 2^57885161 - 1 is not even.

2^57885161 - 1 is not divisible by 3.

2^57885161 - 1 is not divisible by 5.

Hmm... this might take a while.

## 7 Megabyte Number (Score:2)

## Global Warming? (Score:3)

I wonder what the energy use of this project has been? 17 years? How many megawatthours? And what would be the 'carbon footprint' ?

## Re: (Score:3)

## Re:Uhhh... (Score:5, Informative)

2^4-1 = 16-1 = 15.

5 * 3 = 15.

Go read it again.

## Re: (Score:3)

I see a problem though: According to wikipedia, "Some definitions of Mersenne numbers require that the exponent n be prime."

Some? Huh? Mathematics does not deal in disputed definitions! I've never heard before that Mersenne numbers need a prime exponent.

## Re: (Score:3, Informative)

It's really just a matter of semantics. If n is composite, then 2^n - 1 cannot be prime.

## Re:Uhhh... (Score:5, Informative)

lotof "disputed" definitions. Mathematics deal even with a lot of "disputed" logic and "disputed" interpretations. Read about the axiom of choice, set Theory in general, Constructivism (mathematics) and Finitism and you will understand that things get quite more complicated than you thought.## Re:Uhhh... (Score:5, Funny)

## Re:Uhhh... (Score:5, Interesting)

If the exponent is not prime, then the number will not be prime.

I don't do HTML, I'll use the symbol ^ for exponent (I don't do C either). Let's suppose c = a*b, then 2^c - 1 is divisible by both 2^a - 1 and 2^b - 1. (That's true with x instead of 2, the difference being 2^1 - 1 is 1 which is not prime.) Whether the definition requires primality or not, mathematics dictates that the exponent must be prime.

## Re: (Score:3, Informative)

## Re:Uhhh... (Score:5, Insightful)

111 1111 1111 == 2047 == 23 * 89

Funny how many assertions here that number disproves

## Re: (Score:2)

That has 2 1s, which is prime (2 is prime), but 6 definitely is NOT prime...

Likewise,

That has 2 1s, which is prime, but 9 definitely is NOT prime...

## Re: (Score:2)

n=4: 2^4-1 = 15, which isn't prime.

You get nothing! You lose, GOOD DAY SIR!

## Re: (Score:2)

## YOU'RE A CROOK! (Score:3)

In the movie, that was

NOTWW's final judgement. It was a final test of Charlie's character. Charlie contested WW's brusque send-off and ended up owning a chocolate factory.What kind of lesson is this teaching our kids?!

## Re: (Score:2)

Don't be dissing the WonkMan, at least not the Gene Wilder version.

The lesson for children is to be honest even when it seems like there is no longer a need to do so.

Or that everlasting gobstoppers lose their flavor much more quickly than advertised.

## Read the article (Score:2)

## Re: (Score:2)

## Re: (Score:2)

Not all Mersenne numbers are prime. Consider 2^4-1.

## Re: (Score:2)

## Re: (Score:3)

Sigh, you need to go back and read the link you posted:

"Though it was believed by early mathematicians that Mp is prime for all primes p, Mp is very rarely prime."

## Re: (Score:2)

There must be a lot of prime numbers in your world

2^n-1 is always prime?

n=4

(2**4)-1 = 15

## Re: (Score:2)

Considering any number 2^n-1 is prime

Why don't you try that with n=4...

## Re: (Score:2)

Um, no. read that wiki article again.

## Re: (Score:2)

Considering any number 2^n-1 is prime [wikipedia.org], this isn't too impressive, except for maybe that they bothered to expand it into a full number.

Nope: 2^6 - 1 = 63. Correct me if I'm wrong, but 63 isn't prime.

## Re: (Score:2)

If you could find new primes that easily then internet banking wouldn't be secure (well...as secure as it currently is, which is "enough for the insurance companies").

## Re:Uhhh... (Score:4, Informative)

If you could find new primes that easily then internet banking wouldn't be secure (well...as secure as it currently is, which is "enough for the insurance companies").

No, it relies on factoring being much more difficult than multiplication. That is, if I have two large primes p and q I can trivially calculate p*q = n, but you can not easily find p and q from n. Being able to generate primes quickly doesn't give you anything.

## Re: (Score:2)

Banks are plenty secure when they can get federal bailouts.

## Re: (Score:2)

Uh, did you actually read the article you linked to?

Not all 2^n-1 numbers (Mersenne numbers) are prime. For instance, if n is not prime, then 2^n-1 is not prime (so no, 2^57,885,162-1 is not prime). But even if n is prime, 2^n-1 is not prime - take n=23, where 2^n-1 (8388607) is divisible by 47 and 178481. In fact, only 48 Mersenne primes are currently known, as listed in the article you linked.

## Re: (Score:2)

## Re:Uhhh... (Score:5, Funny)

Hey, has anyone told you that your post is wrong yet?

## Re: (Score:2)

Epic fail 57885162 = 2×3×9647527 and a Mersenne prime is only a Mersenne prime if p is prime, which it clearly isn't.

2 ** 57885167 - 1 might or might not be a prime, but at least you can't rule it out automagically because 57885167 is in fact a prime.

## Re: (Score:2)

Looks to me like someone just said 'I think things tend to naturally go up' and you had to explain general relativity in full detail to prove him wrong when dropping a stone was enough

PS: this is supposed to be humorous, not meant to offend you in any way

## Re: (Score:2)

You are confusing Fermat's conjecture, which is different (and wrong), with M-primes.

## Re: (Score:2)

... except when it isn't ... like 2^11-1.

## Re: (Score:2)

Now if that were true wouldn't we be able to say the largest known prime is:

2^((2^57,885,161)-1) ? That can't be true or they would have found it already.

## Re: (Score:3)

Though it was believed by early mathematicians that Mp is prime for all primes p, Mp is very rarely prime. In fact, of the 1,622,441 prime numbers p up to 25,964,951,[5] Mp is prime for only 42 of them. The smallest counterexample is the Mersenne number

Mv11 = 2^11 1 = 2047 = 23 × 89.

## Re: (Score:2)

In fact, of the 1,622,441 prime numbers p up to 25,964,951,[5] Mp is prime for only

42of them.I can't believe you wrote that without giving credit to Ford and Arthur!

## Re: (Score:2)

error: Violation of Order of Operations Rules

actually its 15 not 8 due to it being (2^4)-1

http://en.wikipedia.org/wiki/Order_of_operations [wikipedia.org]

## Re: (Score:2)

I think, rather, that Hanlon's razor applies here:

Never attribute to malice that which is adequately explained by stupidity.## Re: (Score:2)

## Re: (Score:2)

I thought the Slashdot summary goofed on this, but the scientists did as well.

It's the largest known

Mersenneprime. For instance, 1000000! - 1 is also known to be prime, and it is much larger than this number.how do you know 1000000! - 1 is prime? I cannot find any evidence about that. Care to provide some links? I launched mathematica to check this, using command PrimeQ[1000000! - 1] but it didn't calculate yet.

## Re: (Score:2)

I thought the Slashdot summary goofed on this, but the scientists did as well.

It's the largest known

Mersenneprime. For instance, 1000000! - 1 is also known to be prime, and it is much larger than this number.And also, you are wrong: 1000000! - 1 has only 5.56e6 decimal digits, while 2^57885161 - 1 has 1.74e7 decimal digits:

In[8]:= Log[10,1000000!-1] //N //N

Out[8]= 5.56571*10^6

In[7]:= Log[10,2^57885161-1]

Out[7]= 1.74252*10^7

## Re: (Score:3, Informative)

They're all right here: http://mersennewiki.org/index.php/List_of_known_Mersenne_primes [mersennewiki.org]

## mathematians are mean (Score:2)

kill## Re: (Score:2)

Can you print it out and send me a hardcover copy? that'd be soooo cool ...well, at least until the next prime number is found.

Better idea: Tattoo

## Re: (Score:3)

I wonder if 2^(2^57,885,161-1)-1 is also prime.

Stop wondering and start dividing. Call us when done.

## Re: (Score:2)

## Re:CPUs? why not GPUs? (Score:4, Informative)

## Re:CPUs? why not GPUs? (Score:5, Informative)

Yes. And both are used for GIMPS.

See the Mersenne Forum's GPU Computing sub-forum [mersenneforum.org] for details.

There are, however, many more CPUs than GPUs out there, so most of the work is still done by CPUs. Two different GPUs using different software (CUDALucas) were used to confirm that 2^57,885,161-1 was prime, in addition to two other CPUs (one using different software than the GIMPS standard Prime95/mprime).

## Re: (Score:2)