# Boffins find 17,425,170-digit prime number

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 …

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1. "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.

1. Ummmm... Or perhaps even, one less than two raised to a prime.

1. So, forgive my lack of knowledge in this field, 2^(this mahoosive number they found)-1 would also be prime?

1. 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.

2. So, forgive my lack of knowledge in this field, 2^(this mahoosive number they found)-1 would also be prime?

Maybe. Could not would. If it is prime then it's a Mersenne prime. Why not test it and get back to us? ;o)

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

1. Thanks, thought that would make it a bit too easy.

1. #### Re: My imaginary number is....

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

1. #### Re: My imaginary number is....

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

i?

1. #### Re: My imaginary number is....@Benchops

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.

1. #### Re: My imaginary number is....@Benchops

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 :)

1. #### Re: My imaginary number is....@Benchops

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

3. #### Re: My imaginary number is....

Y'all might enjoy this fascinating article:

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

3. #### GIMP

"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?

4. #### Note the time for the GPU vs the 32 core server.

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!

1. #### Re: Note the time for the GPU vs the 32 core server.

"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.

1. #### Re: Note the time for the GPU vs the 32 core server.

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.

2. #### Re: Note the time for the GPU vs the 32 core server.

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.

5. #### So if we receive a larger MPn number from optical SETI..

Its conclusive proof of the existence of aliens right?

AC/DC 6EQUJ5

6. #### Why are we paying for this research?

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

2. #### Re: Why are we paying for this research?

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?

1. #### Re: Why are we paying for this 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!

2. #### Re: Why are we paying for this 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.

1. #### Re: Why are we paying for this research?

"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?

1. #### Re: Why are we paying for this research?

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.

1. #### Re: Why are we paying for this research?

"39 certainly used to be prime. "

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

2. #### Re: Why are we paying for this research?

"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

2. #### Are you trying to get them to multiply

or will you supply condoms?

3. #### Re: Why are we paying for this research?

@NomNomNom: "Everyone enjoys a prime hunt"

3. #### Re: Why are we paying for this research?

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

4. #### Re: Why are we paying for this research?

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?"

1. #### Re: Why are we paying for this research?

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.

7. #### Here's a fun question for you

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.

1. #### Re: Here's a fun question for you

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

2. #### Re: Here's a fun question for you

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.

1. #### Re: Here's a fun question for you

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

3. #### Re: Here's a fun question for you

Let's have a guess at 11.

1. #### Re: Here's a fun question for you

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

1. #### Re: Here's a fun question for you

Is that an African Integer, or European?

4. #### Re: Here's a fun question for you

3

1+2 = 3 but 1 is not a prime number.

1. #### Re: Here's a fun question for you

@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

5. #### Re: Here's a fun question for you

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....

6. #### Re: Here's a fun question for you

> 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.

1. #### "I would go with 117"

One shiny upvote for AC 6/2/13 13:14, as promised. It's higher than you'd think, isn't it?

7. #### Re: Here's a fun question for you

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.

1. #### Re: Here's a fun question for you

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

2. This post has been deleted by its author

8. #### Re: Here's a fun question for you

I don't get it. Why after all the replies here has nobody mentioned that this is Goldbach's conjecture (and unsolved)?

1. #### Re: Here's a fun question for you

Golbach's Conjecture is only for even numbers.

1. #### Re: Here's a fun question for you

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 ...

9. #### Re: Here's a fun question for you

2, Shirley?

8. #### damn

That's the password on my luggage.

9. #### That's just perfect

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.

10. #### an i7 ?

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????

Windows

2. #### Re: an i7 ?

It's also because in the meantime someone realized that

PRIMES is in P

11. #### is it just me?

But finding 48 primes out of a set of 56 million doesn't seem that much better than random?

1. #### Re: is it just me?

Mersenne prime number; not just a prime number

http://en.wikipedia.org/wiki/Mersenne_prime

1. #### Re: is it just me?

Quite. It's trivial to find enormous primes.

googolplex! -1

and

googolplex! + 1

are both rather larger than this.

1. #### Re: is it just me?

> 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)

2. #### Re: is it just me?

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.

12. #### \$50K for the 100M digit prime

How much would the electricity and the use of a good enough computer cost?

1. #### Re: \$50K for the 100M digit 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.

13. [ 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.

1. Yeah but, though but, when they find the "correct" prime number and analyse digits in just the "right" way, there will be the plans for this humungous machine for making the perfect pasta.

Now that has to be worth some time and money!

2. 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.

3. >> I'd rather be wise than intelligent any day.

I feel with you.

1. I'd rather be both - as indeed I am.

4. 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.

14. #### Sexy

yes really they have sexy prime numbers....

http://en.wikipedia.org/wiki/Sexy_prime

I mean they had to right? No one who works out a prime number is getting it any other way.

The internet is full of filth. Sexy prime quadruplets.....

Put your 1 next to my 0.

1. #### Re: Sexy

Quote: yes really they have sexy prime numbers....

That's just an extension of rule 34.

15. Must have been calculated on a PR1ME computer.

Shurely it would be easier[1] to simply download a JavaScript program that generated the required number, rather than transferring the 17 million digits verbatim?

[1] for small values of easier.

17. #### Prize money distribution

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?

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