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## New Largest Known Prime Number: 2^57,885,161-1254

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

• #### Wrong (Score:5, Informative)

by Anonymous Coward on Tuesday February 05, 2013 @01:11PM (#42798277)
Actually it would be 2 multiplied by itself 57,885,160 times, minus 1.
• #### Re: (Score:3, Interesting)

by Anonymous Coward

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)

on Tuesday February 05, 2013 @01:55PM (#42799015) Homepage

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)

by Anonymous Coward on Tuesday February 05, 2013 @02:35PM (#42799617)

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)

+1. I was wondering the exact same thing; is this a "because it's there" type of thing or is there an actual use to calculating this number?
• #### Re:Wrong (Score:5, Informative)

on Tuesday February 05, 2013 @02:11PM (#42799273)
• #### Re:Wrong (Score:5, Insightful)

by Anonymous Coward on Tuesday February 05, 2013 @01:59PM (#42799093)

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)

on Tuesday February 05, 2013 @04:35PM (#42801181)

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: (Score:2)

Insert Euclid coin joke.

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

• #### Re:Wrong (Score:4, Funny)

on Tuesday February 05, 2013 @01:58PM (#42799075)

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.

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

Post responses of cool names.
• #### 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.

• #### 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)

by Anonymous Coward

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)

on Tuesday February 05, 2013 @01:51PM (#42798953)

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)

To be a little more clear about this, primality proving algorithms are generally constructed to produce a certificate that makes it easy to verify that the algorithm was indeed executed completely and indeed proved primality, and these certificates themselves are as difficult to forge as the proof itself.
• #### Re: (Score:2)

Also they have the property of having a corresponding perfect number. (sum of all its divisors except itself)
• #### Re:They would have more primes to choose from ... (Score:4, Informative)

on Tuesday February 05, 2013 @01:30PM (#42798665) Journal

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.

• #### 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)

on Tuesday February 05, 2013 @01:28PM (#42798613)

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)

by Anonymous Coward on Tuesday February 05, 2013 @01:41PM (#42798833)

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)

Actually they are a lot less likely to be prime than any random odd number, only 48 are now known and all have been tested up to above 13 million digits. But yes they are very easy to prove prime compared to "random" numbers of no special form, I *think* it was proven mathematically that there is no faster proof possible, but don't hold me to that. The highest "random" number proven to be prime is only 26,642 digits vs 17 million for this new mersenne prime. There are numbers of other special forms that a
• #### Re: (Score:2)

Why do they always look for primes of that form

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)

<FourteenerCleaner@yahoo.com> on Tuesday February 05, 2013 @01:28PM (#42798611) Homepage Journal
Since the only known perfect numbers [wikipedia.org] are derived from Mersenne Primes, this means there are also now 48 known perfect numbers. Interestingly, this property of Mersenne Primes was discovered by Euclid about 2000 years before Mersenne was born (time machine, anyone?). Finding a non-Mersenne perfect number would be a huge accomplishment.
• #### Re: (Score:2)

We also know that all even perfect numbers must have the form 2^(p-1) * (2^p - 1), where 2^p - 1 is a Mersenne prime. It's not known whether an odd perfect number exists, but we at least know that there are none less than 10^1500.
• #### Number of Digits (Score:5, Interesting)

on Tuesday February 05, 2013 @01:47PM (#42798897) Journal

The new prime number, 2 multiplied by itself 57,885,161 times, less one, has 17,425,170 digits

"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 decimal digits. 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)

Both binary and decimal-in-the-ascii-format should actually have about the same entropy. I realize that zip wont do as good job on the later format, but plenty of higher order entropy compressors would see them as nearly equivalent.
• #### Re: (Score:3)

Oh! You meant that it has 17,425,170 decimal digits. 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)

Optimus Prime is bigger.
• #### Son of a bitch (Score:4, Funny)

on Tuesday February 05, 2013 @02:08PM (#42799235)
Now I have to find a new combination to my luggage. Again.
• #### total energy consumed during the 17-year project? (Score:4, Interesting)

on Tuesday February 05, 2013 @02:09PM (#42799243)
I'm just trying to weigh the energy consumption versus potential benefit. The GIMPS homepage does a terrible job of explaining why (I'm not suggesting that there is no reason to do this) and a linked FAQ is hardly better ( http://primes.utm.edu/notes/faq/why.html [utm.edu] ). Can anyone provide a better answer or instruct those running the GIMPS homepage to do so?
• #### 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)

57,885,161 bits requires nearly 7 megabytes to represent this number in binary form. Wow!
• #### Global Warming? (Score:3)

on Tuesday February 05, 2013 @04:12PM (#42800895) Homepage

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)

I heat my apartment calculating prime numbers. I have electric heat so I'd rather have the power go to something interesting than heating up a wire.

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