"are one less than another number to the power of two"
Ummmm... Maybe the other way around, one less than two to the power another number?
No need to post this, just helping out. Cheers.
The Great Internet Mersenne Prime Search (GIMPS) has struck again, finding the largest-ever Mersenne prime number. The number, the 48th Mersenne prime found, is 17,425,170 digits long and therefore most comfortably represented as 257885161-1 . The previous record-holder was a mere 12,978,189 digits. If you want to read the …
Ah yes -- in my reckless youth I made a pilgrimage to Hamilton's plaque on Brougham Bridge north of Dublin, where he supposedly had the brainwave of skipping a three-dimensional number system, going up to four dimensions and dropping multiplicative-commutivity.
Quaternions got used for a while by physicists I believe, but seem to have been dropped more recently in favour of 2x2 matrices (presumably on the grounds that four real numbers are easier to believe in than three imaginary ones ;) ).
BTW, don't knock plain old "i". Quaternions aren't strictly speaking necessary, but i is compelling :)
If that's 1 GPU that's pretty impressive.
Usual caveats, highly specialised problem, highly tuned code probably non portable etc.
Today a big address space is 2^64 words and I'm not sure anyone has ever fully populated that for a single processor but 2^57 million is just a whole other ball game.
You'd literally need to store data in terms of individual molecules in a full 3d array to get this kind of capacity.
MP's are handy if you want to do size a hash table and you want to address calculations to be simple. They are a bit sparse (48 of them to get to this number) but a compact table to store, as long as you don't have to evaluate the whole number.
Thumbs up for the effort. Onward to 100 million digits!
"Usual caveats, highly specialised problem, highly tuned code probably non portable etc"
Yup, testing a prime number is an "embarrassingly parallel" activity I believe, so translates incredibly well to CUDA. When you consider the number of cores in a GPU it's rather unsurprising.
Not quite; the calculation is basically 58 million *consecutive* 3407872-element double-precision complex FFTs. The FFTs can be split among the cores of a GPU or of a multi-core CPU-based system, but it's not embarrassingly parallel in the normal sense of requiring lots of independent small calculations.
The 32-core server was running a completely different implementation of the large FFT needed to do the arithmetic on such huge numbers, which is not particularly aggressively tuned (in particular, it doesn't use AVX instructions), which is why it was rather slower than a six-core Sandy Bridge using AVX; the idea was to do the calculation using two completely different software implementations and check both got the same answer.
Getting Fourier transforms to run well on a GPU is not at all straightforward, but since doing it allows you to sell thousands of GPUs to people like Shell and Exxon because the work of converting seismic reflection data to 3D images is made of Fourier transforms, nVidia has done it.
Understanding the behaviour of prime numbers is absolutely crucial to the current, safe, implementation of any securely networked IT system - including ecommerce and military communications.
Why would a prudent society not be spending in every way on prime number research?
Lets be clear. Hunting down the biggest primes is not going to help our understanding of prime numbers. The motive for that is just thrill seeking. Everyone enjoys a prime hunt, but we have already captured sufficient primes if we want to researching their behavior, we don't need to find ever bigger ones. Some of the existing primes haven't even been studied in any detail. If these researchers would just place 7, 23 and 39 in laboratory conditions and observe them they might learn a lot about the behaviour of prime numbers. Perhaps introduce some even numbers and see what happens.
"If these researchers would just place 7, 23 and 39 in laboratory conditions and observe them they might learn a lot about the behaviour of prime numbers"
They might even spot that 39 was there masquerading as a prime number but how long would they take to see through its disguise?
Are you paying for this research? As far as I can tell, this is a volunteer program, so you're not paying for it any more than you just paid for my lunch.
Or maybe you meant, why are we, as a species, paying for something that you, as an individual have no interest in. I'd reply with, "Why the hell should the other seven billion people who don't know you care what you think they should spend their money on?"
Curtis Cooper is running prime95 on a couple of thousand computers at the University of Central Missouri.
Running prime95 on a modern computer, rather than letting it idle when not busy, costs about $70 in electricity a year.
So it is costing UCMo about $150,000 a year; not a completely trivial sum, but if that's what they want to spend their money on, so be it.
> Excluding 1, what is the smallest integer that is not the sum of two primes,
> the product of two primes or a power of a prime?
I would go with 117; it is the product of three prime numbers (3,3,13) (so not two); as a product with different factors it is not a power of a prime (and I am taking primes themselves to be excluded since they are a prime to the power 1). As an odd number, if it were to be the sum of two primes, one of them would have to be 2; but this leaves 115 which is not prime.
I would also claim that it would be wrong to claim that as its factors (3,3,13) consist of only two distinct primes that disqualifies it; because, by that logic you can create any integer above 5 by repeatedly adding 2 and 3.
I say 45 = 3 x 3 x 5.
It cannot be a power of a prime, and I take this to mean it cannot be a prime itself. Power by one is a power.
It cannot be the product of two primes, so it must be the product of three primes at least.
It cannot be the sum of two primes, so it is not even, because every even number small enough to be tested is the sum of two primes. See Goldbach Conjecture.
So I choose the smallest possible three primes, none of them 2, not all of them equal, which are 3, 3 and 5.
I am excluding zero and negative integers on grounds of not being interesting.
I don't get it. Why after all the replies here has nobody mentioned that this is Goldbach's conjecture (and unsolved)?
Golbach's Conjecture is only for even numbers.
Oops. I misread the OP, then. I guess that's what I get for reading these articles first thing after waking up...
So... first odd number that's not a semi-prime or a power of a prime? I think I may need some coffee ... and a calculator ...
Whenever 2^n - 1 is prime, the larger number (4^n - 2^n) / 2 is perfect, which means that its factors add up to the number itself. n=2 gives 6 = 1+2+3, n=3 gives 28 = 1+2+4+7+14, etc.
This new Mersenne prime gives a new perfect, which will have approx twice as many digits as the prime itself.
When i was a young'un in the 80s my pa was working for Cray at the time they found the 32nd Mersenne prime.
That was a Cray 2 - the liquid cooled the size of a large fridge.
And now they're finding these exponentially larger numbers on core 2 duos and i7's like we have in our desktop machines.... which we use for reading email and playing farmville????
> Quite. It's trivial to find enormous primes.
True (for levels of confidence < 1.0) ....
> googolplex! -1, googolplex! + 1
> are both rather larger than this.
True; but is this a non-sequitir or are you suggesting that these are prime?
NB 4!+1 = 25, 5!-1=119 (7*17), 5!+1 = 121 (11*11)
It's just you. You have the algorithm wrong.
The real algorithm, which is the proof of the assertion "there is no largest prime", goes like this:
Assume there is some largest prime P.
Multiply all the primes <=P to create Q. (that's one of the steps you got wrong - it's not all the numbers, it's all the primes, so you cannot just use factoral).
By construction, Q is divisible by any of the primes up to and including P.
Subtract (or add) 1 to Q to yield R. R is NOT divisible by any prime less than or equal to P, by construction.
Therefor, R is either prime itself, or has one or more prime factor >P. (that's the second bit you got wrong. There's no guarantee R is prime.)
Ergo, P cannot be the largest prime.
Each test at that size takes about ten days on a $1500 computer which uses about $300 of electricity a year; so in a five-year lifetime it does about 200 tests and costs $3000. The chance of success is about one in a million per test, so: yes, it would cost a lot more than $50,000.
[ Blank stare ]
So basically there is no practical use for this discovery, other than nerd bragging rights. The boffins could be creating cold fusion, flying cars, or holographic 3D TV, but nooooooo. They use their so-called intelligence to find a big prime number while humanity suffers.
Intelligence is being able to build your own nuclear bomb in your garage. Wisdom is not using the bomb you just built to remove a stump in your backyard. I'd rather be wise than intelligent any day.
FYI, cold fusion has largely been dismissed as a hoax, flying cars would be impractical, principally because you wouldnt want them falling on top of you when the fleshware makes a mistake, and the concept of 'holographic 3D TV' demonstrates that you don't know what a hologram is.
Tha ability to follow instructions, and the clearance to obtain large quantities of radionuclides is being able to build a nuclear bomb in your garage. Common sense is not blowing up your back yard with a nuke. Wisdom comes from experience and the ability to learn from your mistakes, and those of others. Intelligence is the ability to work things out without the benefit of hindsight. I'd rather take the intelligence and not have to rely on canned quotes to make a point.
So basically there is no practical use for this discovery
depends, the number of calculations for this kind of thing are huge. so i suspect, the scientific drive to find these number does, in part, fund the research into making faster and faster computer systems. and eventually, that research filters down into the graphics cards that allow us to play such great looking games on our PCs and watching stunning CG fx in movies.
Quote: A more likely reason for the ongoing pursuit of primes is the $US50,000 prize on offer to the finder of the first 100-million-digit prime number.
So, if the GIMPS project finds the first 100m digit prime, would anyone whose computer was distributed a portion of that numbers workload be awarded a percentage of that?