"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 whole …

"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.

Nice try, no cigar. 2^13 - 1 = 8191, 2^8191 - 1 is not prime.

I wondered about that as a kid, finding that 2^3 - 1 = 7, 2^7 - 1 is 127, 2^127 - 1 is a 35-digit prime.

2^(2^127-1)-1 has, as far as I know, not been tested.

**would** also be prime?

Maybe. * Could* not

(Few (2^p)-1 are prime but those that are are the Mersenne primes.)

yes, but the good professor have it written down... can you do the same ;-)

Simple imaginary numbers are so passé. Go for quaternions and you can have `i`

, `j`

, `k`

*and* a real, if you want it.

Mine's the one with the name tag for *Lee Al Jibra* - I made it myself from a pair of damaged coats, alternating pieces from each.

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 :)

Indeed, they are very useful to represent rotations in 3D space.

I can write it down in binary for you if you like. Or octal, or hex.

Y'all might enjoy this fascinating article:

http://www.scottaaronson.com/writings/bignumbers.html

"GIMPS has struck again, finding the largest-ever Mersenne prime number."

Are you sure this story isn't confusing the version of GIMP that will finally support CYMK, high bit depth and non-destructive editing out of the box?

**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.

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.

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.

Its conclusive proof of the existence of aliens right?

AC/DC 6EQUJ5

Doesn't seem to be a use for it, so why is it being pursued?

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?

^ This is a bit like saying that the safety of nuclear reactor is crucial, and it was therefore prudent to spend a billion on building the LHC to discover the Higgs boson.

The correct answer is: Because this is fun!

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.

To discover more huge prime numbers to feed to the prime number eating monster that lives under Belgium.

"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?

that's funny, 39 certainly used to be prime. I wonder when it changed. Something we'd probably know if these so-called "researchers" were actually watching the low primes instead of chasing tail up at the high end.

**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?"

"39 certainly used to be prime. "

But then the number 3 came along, and divided the sceptics. And the number 39..

"that's funny, 39 certainly used to be prime. I wonder when it changed."

It changed just after they discovered the number 13, I believe.

Colin

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.

@NomNomNom: "Everyone enjoys a prime hunt"

Are you sure about that? Have you *asked* them?

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? First person to get it wins a shiny upvote.

0, is my brain working? Please respond with the appropriate vote

11

now where is my upvote?

P.S. a prime number is a natural number that is greater than 1 and have itself and 1 as a factor. Therefore '0' doesn't count. Sorry mate.

opppsss, the question said integer not prime number. My bad :-)

I'm not sure if you mean whole positive number or a negative number, if so, this debate could go on for quite a long time....

@stranger I believe you'll find 2 is the only even prime, and its the second number in your equation there. So I'd say 3 is out.

I think I might be taking this a but too seriously... 3 posts already

> 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.

No, not eleven, since that itself is prime, so fails the test for being a power of a prime (11^1)

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)?

Damn, forgot that 43+2=45. 117 = 3 x 3 x 13 it is, then.

*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 ...

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