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New Largest Known Prime Number: 2^57,885,161-1
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."
Actually it would be 2 multiplied by itself 57,885,160 times, minus 1.
2^4-1 = 16-1 = 15.
5 * 3 = 15.
Go read it again.
You might want to read that link again. Not any number 2^n-1 is prime. Just that a prime in the form of 2^n-1 is called a Mersenne prime.
It's really just a matter of semantics. If n is composite, then 2^n - 1 cannot be prime.
Mathematics does deal with a lot of "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.
They're all right here: http://mersennewiki.org/index.php/List_of_known_Mersenne_primes
Since the only known perfect numbers 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.
Have you read my blog lately?
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.
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.
Bill Stewart
New Fast-Compression-only CPR http://preview.tinyurl.com/dy575ks
Hey, has anyone told you that your post is wrong yet?
"Our two-party system is like a bowl of shit looking at itself in a mirror." - Lewis Black
Mathematics may not deal in disputed definitions, but Wikipedia certainly does.
111 1111 1111 == 2047 == 23 * 89
Funny how many assertions here that number disproves
There is a very fast primality test for Mersenne numbers, the Lucas–Lehmer primality test.
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.
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.
Live today, because you never know what tomorrow brings
"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!
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.
Using CPUs is generally a simpler approach when building a distributed system like this. Yes, per individual computer GPUs can be more efficient, but you also need to look at the number of assisting nodes you're going to get, and this number may (I'm guessing) be higher if you let any old CPU join the party rather than fewer high-end GPUs.
Please consider this account deleted, I just can't be bothered with the spam anymore.
Now I have to find a new combination to my luggage. Again.
Well, there's spam egg sausage and spam, that's not got much spam in it.
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 ). Can anyone provide a better answer or instruct those running the GIMPS homepage to do so?
Yes. And both are used for GIMPS.
See the Mersenne Forum's GPU Computing sub-forum 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).
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.