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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.
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.
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.
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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
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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.
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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.
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).