New Mersenne Prime Discovered, Largest Known Prime Number: 2^74,207,281 - 1 (mersenne.org)

Dave Knott writes: The Great Internet Mersenne Prime Search (GIMPS) has discovered a new largest known prime number, 2^74,207,281-1, having 22,338,618 digits. The same GIMPS software recently uncovered a flaw in Intel's latest Skylake CPUs, and its global network of CPUs peaking at 450 trillion calculations per second remains the longest continuously-running "grassroots supercomputing" project in Internet history. The prime is almost 5 million digits larger than the previous record prime number, in a special class of extremely rare prime numbers known as Mersenne primes. It is only the 49th known Mersenne prime ever discovered, each increasingly difficult to find.

Comments

PrimeCoins

[darkain]

Are these people mining PrimeCoins? What's the motivation to be part of this network? Just curious is all.

Re:PrimeCoins

[Anonymous Coward]

Don't ask why people are doing math, even they don't know.

Re:PrimeCoins

[dasunt]

You were the first comment, and you didn't take the opportunity to say "prime post!" :p

Re:PrimeCoins

[Anonymous Coward]

Once you have enough for food, shelter and healthcare, only the most boring people do things for money.

Re:PrimeCoins

[suso]

There is a US$3000 award for the finder.

And a $3000 power bill for those who don't find it.

Re:PrimeCoins

[Anonymous Coward]

Why do anything? Nothing really matters if you stand back far enough.

If we're going to nitpick what others do though, I think they should concentrate their efforts on finding a formula instead of brute forcing new numbers.

Re: PrimeCoins

[Anonymous Coward]

What's the motivation? Apart from basic human curiosity and wanting to be part of something bigger than yourself, however pointless?

Not much.

But however pointless this may seem they still have a) discoverd a new prime number and b) uncovered bugs in hardware, so it's not a complete waste of time.

Re:PrimeCoins

[Livius]

Obviously prime posts start with the second one.

Re: PrimeCoins

[ememisya]

I wanna be a perfect number! 2^74,207,281 - 1, you're so fucking special. Hahaha! That ol' monk got more credit than he deserved didn't he?

Re:PrimeCoins

[DesertNomad]

this is likely true as the number 1 is not a prime number. https://primes.utm.edu/notes/f...

Re:PrimeCoins

[chmod a x mojo]

1: to help, kind of like SETI / FOLDING@Home / et al.

2: finding a large prime that hasn't been found yet pays ~$5k

3: finding million digit+ primes can pay out up to ~$50K when verified.

4: they have an old PC just hanging around and feel like helping out the math fields.

Re:PrimeCoins

[Njorthbiatr]

Who is paying for prime numbers!!!

Re:PrimeCoins

[armanox]

No coins. Just bragging rights.

Re:PrimeCoins

[WalksOnDirt]

There are only 49 of these known. I don't see how it can be used in encryption.

Re:PrimeCoins

[fahrbot-bot]

Who is paying for prime numbers!!!

Mexico.

Re:PrimeCoins

[91degrees]

If we only use Mersenne primes in encryption, we'd be trivially easy to crack. Even without a list of the primes, you only need to work through a few million of them. That doesn't take long even for a desktop PC. And you'd need a key almost a megabyte long!

Re:PrimeCoins

[SuricouRaven]

4b: Also a parent/college/workplace paying the power bill.

Re: PrimeCoins

[wonkey_monkey]

I wanna be a perfect number!

I'd rather be happy than perfect.

Re:PrimeCoins

[sectokia]

Prime numbers are very useful for puesdo random number generators, In this case the merssene twister. The is a huge benefit in making good rng from prime, because it can be made with an incredible simply algorithm that can run fast etc.

Re:PrimeCoins

[jafiwam]

4b: Also a parent/college/workplace paying the power bill.

4c: Resistance electric heating where the "extra" power goes to something you gotta do anyway.

4d: Testing a new rig.

4e: Annoys the sanctimonious douchebags in the audience.

Re: PrimeCoins

[JonahsDad]

I wanna be a perfect number! 2^74,207,281 - 1, you're so fucking special. Hahaha! That ol' monk got more credit than he deserved didn't he?

Largest known perfect number found (credit to Euclid).
(2^74,207,281 - 1) * 2^74,207,280.

Re:PrimeCoins

[Bob the Super Hamste]

4d: Testing a new rig.

Glad to see I'm not the only one who does this. Let it run full on for a few days then after that let it blast away at night when not in use for a week or 2 to make sure everything is good and doesn't have problems.

Re:PrimeCoins

[hendrips]

There is nothing dubious about it. Primes are defined this way because of the Fundamental Theorem of Arithmetic. If primes were defined to include 1, there would be no such thing as a unique factorization into primes, and if that did not exist, there would be no point to the concept of primes at all.

More generally, one of the principles of abstract algebra is that units (elements with a multiplicative inverse) can never be primes in any unique factorization domain (i.e., any ring that follows the Fundamental Theorem of Arithmetic).

Re:PrimeCoins

[slew]

Prime numbers are very useful for puesdo random number generators, In this case the merssene twister.

The is a huge benefit in making good rng from prime, because it can be made with an incredible simply algorithm that can run fast etc.

Kinda, but not really. The real requirement for making a good generalized linear congruent pseudo-random number generator with a long period is to find a large galois field matrix that has a *primitive* characteristic polynomial. As it turns out it is easier to *test* if a trinomial (polynomial with 3 non-zero terms) is primitive if it is generated from a parameter related to a Merssene prime number decomposition (2^k-1). This does not guarantee that any of the trinomials generated by this parameter is primitive, only that it is easy to test (some Merssene primes like M40 do not generate any usable trinomials).

Primitive trinomials are good for fast pseudo-random number generators because they will have the long period set by the primitive, yet be easier to compute because there will be lots of zero terms in the matrix (instead of a dense matrix).

However regardless of the difficulty of the mechanism to validate that the matrix is primitive, they computation that you do to *generate* random number is completely independent of the prime-ness of the parameter (and you don't really want a prime number if it isn't a Merssene prime because the slow way of testing can be sped up by factoring the parameter) only the density of the matrix. Matrices with primitive polynomials (or specifically trinomials) not generated by Merssene primes will have the same computation time.

You can kind of think of primitive polynomials as being kind of a generalization of a prime number to galois polynomials, but it's not really the same thing except in the case where your matrix is a simple scalar (e.g. the classic linear congruent scalar pseudo random number generator you learned in school).

Re:PrimeCoins

[agm]

Ok, you win. :-)

Re:PrimeCoins

[TeknoHog]

And a $3000 power bill for those who don't find it.

News flash: it takes energy and money to do research.

Re:PrimeCoins

[AthanasiusKircher]

There is nothing dubious about it. Primes are defined this way because of the Fundamental Theorem of Arithmetic.

While, I completely agree with you, lighten up a bit. Also, realize that in the real world the word "prime" comes from Latin primus, meaning first... hence prime steak, prime meridian, Optimus Prime, etc. And thus while in the mathy definition you're right, the real-world definition allows significant wordplay in this specific thread -- the first post is by definition a prime post, but in another sense (your sense), it's not.

Re:PrimeCoins

[cold fjord]

For purposes of encryption the fact that it is prime number is relevant, the fact that it is a Mersenne prime isn't. It can be used just like any other prime number. (Assuming the encryption software is written with the capacity to accommodate such a large prime which I wouldn't take as a given.)

Re:PrimeCoins

[cold fjord]

Haven't you ever been curious about something? The urge to explore? The desire to contribute to expanding the knowledge of humanity? The desire to contribute to developing useful knowledge for an important field like mathematics? The possibility of arriving at that important result yourself? The possibility of doing something useful?

Not everyone shares those interests, especially for something as abstract as math in general, and searching for a particular form of prime number in particular.. There are probably other reasons as well.

This is actually one of the mass participation projects that I've considered contributing to over the years. I actually have contributed to folding@home.

Re:PrimeCoins

[WalksOnDirt]

If it weren't so uselessly large it would definitely be in my list of famous prime numbers to try. That makes it an incredibly poor choice.

Mental note

[aaarrrgggh]

Thanks, now I need to change private key.

Re:Mental note

[Anonymous Coward]

I was going with the combination on my luggage.

Re:Mental note

[Etherwalk]

"2^74,207,281-1? That sounds like the combination an idiot would put on his luggage."
--Spaceballs 2: The Search for More Money

Re:Mental note

[LordHighExecutioner]

And of course you memorized it, isn't it ?!?

"each increasingly difficult to find."

[Anonymous Coward]

How the fuck do we know that? Maybe after the next one they get super easy, we have no fucking idea lol.

Re:"each increasingly difficult to find."

[Pseudonym]

Here you go: S. Wagstaff, "Divisors of Mersenne numbers," Math. Comp., 40:161 (January 1983) 385--397. MR 84j:10052

It's true that we don't know for sure, but it's not true that we have no fucking idea.

Re:"each increasingly difficult to find."

[m.alessandrini]

Math is fascinating because it can prove that a thing is possible or impossible, even if it has no idea what that thing is like. Take for example the proof that you cannot find an algebraic formula to solve equations of degree 5 or higher.

Re:"each increasingly difficult to find."

[Petronius Arbiter]

Math is also fascinating because of how it can often work around impossibility proofs.

E.g., what class of polynomials is solvable depends on what elementary functions are allowed. With Jacobi theta functions, you can exactly solve quintics.

http://mathoverflow.net/questi...

For another example, with cosine and acos, you can exactly solve cubic polynomials, w/o using cube roots. Better, if the solutions are real, then the solution does not require imaginary numbers, unlike if you solve with cube roots.

Rare Primes?

[TechyImmigrant]

>In a special class of extremely rare prime numbers known as Mersenne primes.

I understood that Mersenne primes are not rare, they are common compared to primes that don't fit the 2^n -1 form. Hence searching for Mersenne primes is a more efficient way of finding big ones.

Re:Rare Primes?

[complete loony]

No, absolutely not.

Out of 74,207,281(-ish) tested values for n, this is only the 49th prime found. If you tested the first 74,207,281 odd numbers you would have found more that 5 million primes.

Re:Rare Primes?

[Anonymous Coward]

It's not so much that Mersenne numbers are much more likely to be prime than other odd numbers of their size. It's that there is a special-purpose primality test just for Mersenne numbers that is tons more efficient than verifying other primes of similar size.

Re:Rare Primes?

[TechyImmigrant]

Thank you. That makes sense. I learned something.

Re:Rare Primes?

[Anonymous Coward]

Uninteresting fact.

If 2^n-1 is represented in binary then n will be the number of set bits.
That means that if n can be divided by m then 2^n-1 can be divided by 2^m-1. (For example 2^15-1 = 32767 and can be divided by 2^3-1 and 2^5-1.)
From the headline we can tell that 74,207,281 is a prime, otherwise 2^74,207,281-1 wouldn't be a prime.

Re:Rare Primes?

[TechyImmigrant]

Learning something by making stuff up in a forum and waiting for people to correct it has a high social cost. Please find another way to learn.

There's only a high social cost if you care what other people think. I had confused that series with . An AC did eventually explain the reason for searching for Mersenne primes being the efficiency of the primality test. I'm very familiar with primality tests for key search algorithms, which need to select uniformly amongst primes for the security of the keys to be maintained, because I'm an implementor of crypto.

Many of the other commenters just showed their smug attitude by saying "You're wrong" without bothering to put in the effort to explain the efficient primality test as the reason.

Re:Rare Primes?

[TechyImmigrant]

Argh. My html got messed up. For shame! For shame!

Re:Rare Primes?

[TechyImmigrant]

No. It's interesting.

It's not new

[Anonymous Coward]

It's not new. Chuck Norris discovered in with this pocket calculator which measuring the circumference of his biceps.

Re:It's not new

[invictusvoyd]

Which was after he smacked your head . That explains why your statement does not make sense .

Re:It's not new

[Bob the Super Hamste]

Lies we all know it was first known to Bruce Schneier who long ago found all Mersenne Primes in O(1) time.

If you are going to use a internet meme at least pick the right one.

Re:Obviously

[93 Escort Wagon]

That's a pretty large number, but they still haven't found your mom's number.

That's surprising, since so many guys have it already.

It's small.

[Kwyj1b0]

It's still smaller than the box Amazon Prime uses to send me a toothpick.

Re:i thought Optimis Prime was the ultimate prime

[cfalcon]

Well, printing it in binary would be 74,207,280 numeral "1"s, so if you go that way it would be pretty big!

Hold on, let me doublecheck in Mathematica:

[Brannon]

In1> PrimeQ[2^74207281-1]

Now we wait.

Re:Hold on, let me doublecheck in Mathematica:

[Anonymous Coward]

Heh, i would expect they make a hardcoded list for them.

22,338,618 digits

[mejustme]

How "big" is 22,338,618 digits? Text file containing the prime is 22.8 MiB in size. http://www.mersenne.org/primes...

Re:22,338,618 digits

[craigm4980]

The number is 2^74,207,281-1, thus its exactly 74,207,280 bits long and all those bits are 1. That's 9,275,910 bytes, or roughly 9MiB. When talking about mersenne primes on a tech site, using base 10 versions encoded as ascii (or utf-8, its the same for that subset) seems like an odd measure of size.

Re:22,338,618 digits

[mejustme]

I was looking for a text file showing the number since obviously this prime wont fit in a uint32_t. 74,207,280 bits means nothing to me due to the size. Seeing a 22.8 MiB text file filled with digits gave it some scale.

Re: 22,338,618 digits

[Cederic]

It wasn't a joke. I can tell because they also used MiB, which tells you immediately they don't know what they're talking about.

Re:22,338,618 digits

[shawn2772]

python -c "with open('prime.txt','w') as f: f.write('%d' % (2**74207281-1))"

That's gonna take a while. This is much faster:

python -c "with open('prime.txt','w') as f: f.write('0x' + 'F' * 9275910))"

Re:22,338,618 digits

[shawn2772]

python -c "with open('prime.txt','w') as f: f.write('%d' % (2**74207281-1))"

That's gonna take a while. This is much faster:

python -c "with open('prime.txt','w') as f: f.write('0x' + 'F' * 9275910))"

Oop, but wrong. It leaves out one bit.

python -c "with open('prime.txt','w') as f: f.write('0x1' + 'F' * 9275910))"

That one is right.

Re:22,338,618 digits

[shawn2772]

(Note to self: test before posting)

python -c "with open('prime.txt','w') as f: f.write('0x1' + 'F' * 9275910)"

Re:22,338,618 digits

[ChrisMaple]

Actually, the most significant hexadecimal digit is 1.

Re:22,338,618 digits

[skastrik]

But only 11 mb in packed decimal, which will be more or less as efficient when you write the number to the terminal.

Re:i thought Optimis Prime was the ultimate prime

[Jeremi]

Well, printing it in binary would be 74,207,280 numeral "1"s, so if you go that way it would be pretty big!

binary is for pikers. Real mathematicians print it out in unary.

OMG, who cares!

[footNipple]

Isn't knowing the definition of a prime number enough?

Re:OMG, who cares!

[jandersen]

Isn't knowing the definition of a prime number enough?

Enough - for what? The definition of prime numbers is deceptively simple, but we still don't know a general way to construct all prime numbers - we don't even know if there is one. The same can be said for many other classes of numbers, I suppose, but prime numbers have turned out to be useful for our understanding of numbers and other things.

Compare with vector spces: a vector space is, to put it simply, a space with 'dimension': every point in a vector space can be represented as a tuple of numbers: a = (a1, a2, a3, ...., aN). The first thing you want to find in a vector space is the basis: a set of N vectors that point out the independent coordinate axes of the space think of R^2 or R^3, the 2- and 3-dimensional spaces we are familiar with. Every natural number has a slightly similar property: it can be written as a product of prime numbers - the prime factors. This can useful when you calculate things - if you know that 30030 = 2*3*5*7*11*13 and 136367 = 7*7*11*11*23, then it is easy to see that 136367/30030 = 2*3*5*13/7*11*23; sometimes it is easy to find prime factors, at least if you know what the prime numbers are.

Also, in the theory for finite groups, if p is a prime number, then any group with p elements is cyclic and any group with p^2 is Abelian (wikipedia is your friend, if you want to know more); cryptic, I know, but it has profound consequences.

Re:OMG, who cares!

[PvtVoid]

Isn't knowing the definition of a prime number enough?

Totally. It's not like prime numbers can be applied to anything useful like cryptography.

Re:OMG, who cares!

[ChrisMaple]

The Sieve of Eratosthenes is a general method to find all prime numbers. To the best of my knowledge, there is no algebraic function that given an integer input always produces a prime.

I don't get it

[JustNiz]

So what is the big deal? I mean how will this actually change anything?

Re:I don't get it

[Anonymous Coward]

They want to determine the bucket count for the hash map of the cosmic simulator.

Re:I don't get it

[david_thornley]

Or if it wasn't hastily hacked in Perl mostly on the Seventh Day (the one that God put a sign on the door saying "Resting - Do Not Disturb").

Re:Better link

[Anonymous Coward]

I'm pretty sure the press release is more relevant to the news of a new Mersenne Prime discovery then a link to generic information of primes of that type.

These are insanely large

[140Mandak262Jamuna]

The number of subatomic particles in the known universe is estimated to be less than 1.0e99. So if tag each subatomic particle with an integer you would run out of subatomic particles before you reach even one googol.

Re: i thought Optimis Prime was the ultimate prime

[TheRealMindChild]

Omega prime

Not to be confused with...

[Anonymous Coward]

...the very short paper: W. Sagstaff, "Divisors of Mersenne Prime numbers," Math. Comp., 41:161 (February 1983) 261--261. MR 84j:10053

luggage combination

[esperto]

Great, now i have to change my luggage lock combination.

For Science!!

[billstewart]

We've been running things like this for a couple of decades, just as SETI@Home searches for little green men, etc. I ran the GIMPS Mersenne prime search software for a couple of years on my work laptop, but it really chewed through battery life, and eventually the desktop CPUs (and GPUs, once those were supported) became enough faster that it wasn't worth contributing.