Slashdot Mirror
45th Known Mersenne Prime Found?
An anonymous reader writes "The Great Internet Mersenne Prime Search (GIMPS) has apparently discovered a new world-record prime number. A GIMPS client computer reported the number on August 23rd, and verification is currently under way. The verification could take up to two weeks to complete. The last Mersenne prime discovered was over 9.8 million digits long, strongly suggesting that the new value may break the 10 million digit barrier — qualifying for the EFF's $100,000 prize!"
http://en.wikipedia.org/wiki/Mersenne_prime
Fun.
How we know is more important than what we know.
From the GIMPS website:
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.
You seem to misunderstand the meaning of prime. Either that, or you're a horrible comedian. In either case, 77 isn't prime, and neither is 77777777.
However, even if a string of 7's were prime, that may not be enough. As stated previously, if n = 2^x - 1 is prime, then x must be prime. However, the converse is not true. That is, x being prime does not guarantee that n is prime. E.g., if x = 11, then n = 2^{11} - 1 = 2047 = 23 x 89.
But when it comes to primes in the 10 million digit range (I couldn't even guess how many bits you would require for a number that large)
About 33 million bits ( ln(10)/ln(2) = 3.3219 ) or 4MB.
Actually for a number of 2^n-1 to be prime, n must be prime also. There is no chance that 2^32,582,658-1 is prime.
FTFA:
Still, it's not hard to think of some more, well, "electronic frontier-ish" applications for that kind of money. The FPGA-based DES keysearch engine they built a few years back was cool. OTOH, this Mersenne prime business just sounds like they're paying for something that will become trivial soon enough anyway.
Dahlmann tightly grips the knife, which he may have no idea how to use, and steps out into the plain.
You forgot to divide by 8...bits != bytes
On slashdot, even the submitters don't RTFA apparently:
Or you didn't read it properly: the bit about being just short of 10 million is about the 44th when it was new, not the new new prime.
"previously unknown"?
Uh, no. You have a very clunky Sieve of Eratosthenes. Digital roots isn't even a thing.
Also: numerological techniques would pertain to the spirituality of numbers and their mystic powers. Numeric techniques is what real computational mathematicians use.
Testing a single candidate Mersenne prime takes a month of straight computation on a single 2.4 GHz Pentium 4 (assuming a 10 million digit prime, which would be the minimum to win the prize). This assumes the use of only one core, but you'd need at least 100 cores to make it anything resembling "quick" (~7 hours), if you could even parallelize the procedure that much.
Never mind the fact that only about one in 150,000 exponents will yield a prime, meaning that on average, 150,000 months of computation is required for a single prime to emerge, and furthermore, finding giant Mersenne primes is easier than most other kind of primes. So, I don't think your computer will find these giant primes "pretty damn quick".
Pessimism aside, I think this is a pretty impressive achievement considering that GIMPS doesn't have nearly the power of larger efforts like Folding@Home (GIMPS has around 500 GFLOPS while F@H has around 3372 TFLOPS, or 3372000 GFLOPS).
I for one am glad that the EFF isn't using my donations for this award, beautiful mathematics or not. When I donate my hard-earned money to the EFF, I expect them to use it for something worthwhile. From TFA:
Prize money comes from a special donation provided by an individual EFF supporter, earmarked specifically for this project. Prize money does NOT come from EFF membership dues, corporate or foundation grants, or other general EFF funds.
Ah, arrogance and stupidity, all in the same package. How efficient of you. -- Londo Mollari
Very Wrong
Large prime numbers are used in cryptography because it gives you massive amounts of digits that are not divisible by any other number. There's types of crypto that use multiple large prime numbers to build the keys from. If you use one of these as your seeds it will take a long time to crack anything encrypted as the number is huge and has no smaller factors. At 10 million digits you're going to take a long time to "guess" what it is.
The weighted FFT algorithm used by GIMPS is called the Irrational Base Discrete Weighted Transform (IBDWT), and it was designed with Mersenne numbers in mind.
You're almost right about GPUs getting enough memory to start tackling this sort of problem, but the issue isn't memory, it's floating point precision.
The FFTs being done by GIMPS are fairly large, and the overall error introduced by the FFT operations is roughly proportional to the log of the size of the dataset. So as the numbers become larger and larger, less and less data can be stored within each of the floating-point values. Which leads to larger transforms, which leads to less data that can be stored, etc. It's a vicious circle.
In the case of a 10 million-digit prime, the worst-case scenario ends up with 1 bit of the original value per float, and a transform size of about 2 ** 25. The values themselves will take up about 128 megs of video memory.
(Side note: the most efficient FFT algorithms involve a "scratch" buffer that stores intermediate results-- so for a 128-meg data set, we would really need to have at least 256 megs of usable space on the video card.)
In this case, if we have floating-point values with only 23 bits of mantissa (as is the case with most graphics cards), we would have to be sure that no more than 22 bits of error could creep in over the course of the calculations. But given the size of the transforms, that's not very likely. In reality, we're likely to lose the results to rounding errors.
The only real way out of this problem is to use higher-precision numbers. Some graphics cards offer 64-bit floats, but they come with a huge performance hit. There are also some techniques for "emulating" higher precision on a graphics card, but they're pretty application-specific, and I don't know that they'd work for FFTs.
So, yeah-- the long and the short of it is that graphics cards just don't have the required precision for large-integer multiplication at the size GIMPS is doing. Smaller numbers are okay (in fact, I've written code that uses GPUs to accelerate large-integer multiplication-- there IS a speedup), but 10 million digits just isn't possible yet.
Hope that clears things up!
No, that would be Max Planck.
That's not Picasso, that's Kandinsky!
His point was that while a number like 193 has a prime number of digits in decimal, if you change it to binary (11000001) it is no longer a prime number of digits long. So no, it is not an inherent property of the number. Every prime number has a prime number of digits in some base.
We hope your rules and wisdom choke you / Now we are one in everlasting peace
Can you prove that?
Sure. Every prime p has two digits in base p. QED.